François Boulier

François Lemaire

Marc Moreno Maza

Well known theorems on triangular systems and the D 5 principle

The theorems that we present in this paper are very important to prove the correctness of triangular decomposition algorithms. The most important of them are not new but their proofs are. We illustrate how they articulate with the D 5 principle.

Introduction

This paper presents the proofs of theorems which constitute the basis of the triangular systems theory: the equidimensionality (or unmixedness) theorem for which we give two formulations (Theorems 1.1 and 1.6) and Lazard's lemma (Theorem 2.1). The first section of this paper is devoted to the proof of the equidimensionality theorem. Our proof is original since it covers in the same time the ideals generated by triangular systems saturated by the set of the initials of the system (i.e. of the form (A) : I ∞ A ) and those saturated by the set of the separants of the system (i.e. of the form (A) : S ∞ A ). The former type of ideal naturally arises in polynomial problems while the latter one naturally arises in the differential context. Our proof shows also the key role of Macaulay's unmixedness theorem [24, chapter VII, paragraph 8, Theorem 26]. Its importance in the context of triangular systems was first demonstrated by Morrison in [START_REF] Morrison | Yet another proof of Lazard's lemma[END_REF] and published in [START_REF] Morrison | The Differential Ideal [P ] : M ∞[END_REF]. In her papers, Morrison aimed at completing the proof of Lazard's lemma provided in [3, Lemma 2]. Thus Morrison only considered the case of the ideals of the form (A) : S ∞ A , which are the ideals w.r.t. which Lazard's lemma applies. The case of the ideals of the form (A) : I ∞ A was addressed in [START_REF] Aubry | On the theories of triangular sets[END_REF]. The proof of [START_REF] Aubry | On the theories of triangular sets[END_REF]Theorem 5.1] involves the same gap as that given in [3, Lemma 2]. It was fixed in [START_REF] Aubry | Ensembles triangulaires de polynômes et résolution de systèmes algébriques[END_REF]. The proof provided in [START_REF] Aubry | Ensembles triangulaires de polynômes et résolution de systèmes algébriques[END_REF] does not explicitly use Macaulay's theorem but relies on the properties of regular sequences in Cohen-Macaulay rings, which are the rings in which Macaulay's theorem applies.

What is this gap in the proofs mentioned above ? Among all the indeterminates the elements of a triangular system A depend on, denote t 1 , . . . , t m the ones which are not main indeterminates. The proofs given in [START_REF] Aubry | On the theories of triangular sets[END_REF]Theorem 5.1] and in [3, Lemma 2] rely implicitly on the assumption that the non zero polynomials which only depend on t 1 , . . . , t m are not zero divisors modulo the ideal defined by A. This assumption is indeed true but certainly deserves a specific proof.

In the case of the ideals of the form (A) : I ∞ A , let's mention the equidimensionality result of [START_REF] Shang | On the dimension of an arbitrary ascending chain[END_REF] which is not sufficient since it does not solve the problem of the embedded associated prime ideals of (A):I ∞ A . In the case of the ideals of the form (A):S ∞ A , there is a simple proof [START_REF] Ollivier | A proof of Lazard's lemma[END_REF][START_REF] Boulier | Computing representations for radicals of finitely generated differential ideals[END_REF] which unfortunately does not seem to generalize to the ideals of the form (A) : I ∞ A . The second section of this paper is devoted to the proof of Lazard's lemma. This lemma was communicated by Lazard to the first author a few days before his PhD defense in 1994, with a sketch of proof. The proof given here is very close to the original one. As stated above, Lazard's lemma was first published in [START_REF] Boulier | Representation for the radical of a finitely generated differential ideal[END_REF] but its first complete proof is due to Morrison [START_REF] Morrison | Yet another proof of Lazard's lemma[END_REF][START_REF] Morrison | The Differential Ideal [P ] : M ∞[END_REF]. Among the few other proofs published afterwards, let's mention the ones given in [START_REF] Schicho | A construction of radical ideals in polynomial algebra[END_REF][START_REF] Boulier | Computing representations for radicals of finitely generated differential ideals[END_REF][START_REF] Hubert | Factorization free decomposition algorithms in differential algebra[END_REF][START_REF] Sadik | Une note sur les algorithmes de décomposition en algèbre différentielle[END_REF].

In the remaining sections, we show how the equidimensionality theorem and Lazard's lemma apply to the so called "regular chains" [START_REF] Lazard | A new method for solving algebraic systems of positive dimension[END_REF][START_REF] Kalkbrener | A Generalized Euclidean Algorithm for Computing Triangular Representations of Algebraic Varieties[END_REF][START_REF] Yang | Searching dependency between algebraic equations: an algorithm applied to automated reasoning[END_REF][START_REF] Aubry | On the theories of triangular sets[END_REF]. We last recall a few basic algorithms which carry out a generalization of the "D 5 " principle [START_REF] Della | About a new method for computing in algebraic number fields[END_REF] for regular chains and which implictly rely on the equidimensionality theorem. Historically, the "D 5 " principle suggests to compute modulo zero dimensional ideals presented by triangular systems as if these ideals were prime (whenever a zero divisor is exhibited, the ideal is split). It is its generalization to non zero dimensional ideals which requires the equidimensionality theorem.

Throughout this paper, K denotes a commutative field of characteristic zero.

The equidimensionality theorem

In the polynomial ring R = K[x 1 , . . . , x n , t 1 , . . . , t m ], we consider a polynomial system A = {p 1 , . . . , p n }. We assume that deg(p i , x i ) > 0 and deg(p i , x k ) = 0 for all 1 ≤ i ≤ n and i < k ≤ n i.e. that A is a triangular system w.r.t. at least one ordering such that x 1 < • • • < x n and that the x indeterminates are precisely the main indeterminates of the elements of A. The initial of a polynomial p i is the leading coefficient of p i , viewed as a univariate polynomial in x i . The separant of p i is the polynomial ∂p i /∂x i .

In the following, h denotes either the product of the initials of all the elements of A or the product of the separants of all the elements of A.

We are concerned by the properties of the ideal A = (A) : h ∞ which is the set of all the polynomials f ∈ R such that, for some nonnegative integer r and some λ 1 , . . . , λ n ∈ R we have h r f = λ 1 p 1 + • • • + λ n p n . When h is the product of the initials of the elements of A, the ideal A is often denoted (A) : I ∞ A in the literature. When h is the product of the separants, the ideal A is often denoted (A) : S ∞ A . In general, the ideal A may be the trivial ideal R (take A = {x 1 , x 1 x 2 }). We assume this is not the case.

Denote R 0 = K(t 1 , . . . , t m )[x 1 , . . . , x n ] the polynomial ring obtained by "moving the t indeterminates in the base field" of R and A 0 the ideal (A) :

h ∞ in the ring R 0 . Denote M the multiplicative family K[t 1 , . . . , t m ] \ {0} so that R 0 = M -1 R. Denote M/A the image of M by the canonical ring homomorphism R → R/A.
The elements of R 0 /A 0 , which is isomorphic to (M/A) -1 (R/A), have the form a/b where a ∈ R/A and b ∈ M/A. In this section, we prove the following theorem.

Theorem 1.1. An element a ∈ R/A is zero (respectively regular 1 ) if and only if every element a/b ∈ R 0 /A 0 is zero (respectively regular).

Proposition 1.2. To prove Theorem 1.1, it is sufficient to prove that every element of M/A is regular.

Proof. This is a very classical proposition. If every element of M/A is regular then R 0 /A 0 , is a subring of the total ring of fractions of R/A [24, chapter IV, paragraph 9]. The proposition then follows [24, In a noetherian ring, every ideal is a finite intersection of primary ideals. Every representation of an ideal A as an intersection of primary ideals can be minimized by removing on the one hand the redundant primary ideals and by grouping on the other hand the primary ideals whose intersection is itself primary. The so obtained minimal primary decomposition of A is not uniquely defined. However, the number of its components and the radicals of its components (the so called "associated prime ideals" of A) are uniquely defined.

All the rings considered in this section are noetherian.

Proposition 1.4. To prove Theorem 1.1, it is sufficient to prove that no associated prime ideal of A meets M .

Proof. According to [24, chapter IV, paragraph 6, Corollary 3], if M does not meet any associated prime ideal of A then every element of M/A is regular. Theorem 1.1 then follows from Proposition 1.2.

Recall the definition of the dimension of an ideal.

Definition 1.5. The dimension dim p of a prime ideal p of a polynomial ring R with coefficients in a field K is the transcendence degree of the fraction field of R/p over K. The dimension dim B of an ideal B of R is the maximum of the dimensions of the associated prime ideals of B.

The rest of this section is completely dedicated to the proof of the following theorem which admits Theorem 1.1 as a corollary. This reformulation of Theorem 1.1 is often convenient for writing proofs.

Theorem 1.6. The associated prime ideals of A have dimension m and do not meet M .

In order to apply Macaulay's unmixedness theorem, one needs to get rid of the saturation by h. For this, one may use the Rabinowitsch trick [20, section 16.5]. One introduces some new indeterminate x n+1 and a new polynomial p n+1 = h x n+1 -1. One denotes A ′ the triangular system of R ′ = R[x n+1 ] obtained by adjoining p n+1 to A. One denotes A ′ the ideal (A ′ ) of R ′ . Consider the two following canonical ring homomorphisms:

R φ ---→ h -1 R ≃ R ′ /(p n+1 ) π ←---R ′ . The isomorphism h -1 R ≃ R ′ /(p n+1 ) is classical [7, Exercise 2.2, page 79]: every element of R corresponds to itself, x n+1 corresponds to h -1 . If B is an ideal of R, one denotes h -1 B or (φB) the ideal of h -1 R generated by φB. If B ′ is an ideal of R ′ then πB ′ is an ideal of πR ′ = R ′ /(p n+1 ). Lemma 1.7. The ideal A ′ is proper. If q ′ 1 ∩ • • • ∩ q ′ r is a minimal primary decomposition of A ′ then φ -1 (πq ′ 1 ) ∩ • • • ∩ φ -1 (πq ′ r
) is a minimal primary decomposition of A.

Proof. We use the notations of extensions and contractions defined in [24, chapter IV, paragraph 8], w.r.t. the ring homomorphism φ so that (φA) = A e . The ideal πA ′ is equal to the ideal A e since both ideals admit a same generating family: A. By [24, chapter IV, Theorem 15 (a)] we have A = A ec since A = A : h ∞ . Therefore, since A is assumed to be proper, so are A e and A ′ . Consider now a minimal primary decomposition

q ′ 1 ∩ • • • ∩ q ′ r of A ′ . According to [24, chapter IV, paragraph 5, Remark concerning passage to a residue class ring], πq ′ 1 ∩ • • • ∩ πq ′ r
is a minimal primary decomposition of πA ′ = A e . Since A = A ec , by [24, chapter IV, Theorem 15 (b) and a comment just above this theorem], the associated prime ideals of A do not meet M . By [24, chapter IV, Theorem 17] the intersection

φ -1 (πq ′ 1 ) ∩ • • • ∩ φ -1 (πq ′ r ) is a minimal primary decomposition of A.
Proposition 1.8. To prove Theorem 1.6, it is sufficient to prove that the associated prime ideals of A ′ have dimension m and do not meet M .

Proof. Let p ′ be an associated prime ideal of A ′ and p = φ -1 (πp ′ ) the corresponding associated prime ideal of A according to Lemma 1.7. Let a be an element of the subring R of R ′ . Then a ∈ p ′ if and only if a/1 ∈ πp ′ and a/1 ∈ πp ′ if and only if a ∈ p. Therefore, if p ′ does not meet M then p does not either and dim p ≥ m. If moreover dim p ′ = m then x 1 , . . . , x n must depend algebraically on t 1 , . . . , t m modulo p ′ hence they depend algebraically on t 1 , . . . , t m modulo p and dim p ≤ m. Combining both inequalities, one concludes that dim p = m.

One distinguishes two sorts of prime ideals associated to an ideal A: the isolated or minimal ones and the embedded or imbedded ones. An embedded associated prime ideal of A is an associated prime of A which contains another associated prime ideal of A. In the context of polynomial rings, its algebraic variety is included (embedded) in that of the associated prime ideal that it contains. One thus sees that, at least in the context of polynomial rings, it is much easier to get informations on the minimal associated prime ideals (they correspond to the irreducible components of the algebraic variety of the ideal [24, chapter VII, paragraph 3, Corollary 3 to Hilbert's Nullstellensatz]) than on the embedded associated prime ideals, which have no such simple geometric meaning (see however [7, section 3.8] for a geometric interpretation of embedded primes). In our case, the problem of the minimal associated prime ideals is easily solved by Lemma 1.10. The problem of the embedded associated prime ideals is solved by a difficult theorem: Macaulay's unmixedness theorem. Recall Krull's principal ideal theorem [24,

(1 ≤ k ≤ n) then dim A ≥ n -k.
Let us come back to our study of the ideal A ′ of R ′ .

Lemma 1.10. The dimension of A ′ is m. Moreover, none of the m-dimensional associated prime ideal of A ′ meets M .

Proof. Consider an associated prime ideal p ′ of A ′ .

First consider the case of h being the product of the initials of the elements of A. Then none of these initials belongs to p ′ (otherwise p ′ , which contains h x n+1 -1 would also contain 1). Thus x 1 , . . . , x n+1 are algebraically dependent on t 1 , . . . , t m over K in R ′ /p ′ (the polynomials of A ′ cannot degenerate at all).

Consider now the case of h being the product of the separants of the elements of A. Let

p ℓ = a d x d ℓ + • • • + a 1 x ℓ + a 0 be any element of A ′ . Since its separant s ℓ = d a d x d-1 ℓ + • • • + a 1
does not belong to p ′ (otherwise p ′ , which contains h x n+1 -1 would also contain 1), at least one of the coefficients a d , . . . , a 1 does not belong to it 2 . Thus x 1 , . . . , x n+1 are algebraically dependent on t 1 , . . . , t m over K in R ′ /p ′ (the polynomials of A ′ cannot completely degenerate).

In both cases, x 1 , . . . , x n+1 are algebraically dependent on t 1 , . . . , The following proposition, combined to Proposition 1.8, concludes the proof of Theorem 1.6 hence that of Theorem 1.1.

t m over K in R ′ /p ′ . One then concludes, first that dim p ′ ≤ m hence dim A ′ ≤ m, second that if dim p ′ = m then p ′ ∩ M = ∅. The ideal A ′
Proposition 1.12. The associated prime ideals of A ′ have dimension m and do not meet M .

Proof. The ideal A ′ admits a basis made of n + 1 elements in a polynomial ring in n + m + 1 indeterminates. According to Lemma 1.10, its dimension is m. According to Macaulay's unmixedness theorem, all its associated prime ideals have dimension m. According to Lemma 1.10 again, none of these prime ideals meets M .

Let us state a few easy corollaries to Theorem 1.1.

Corollary 1.13. The minimal primary decomposition of A is uniquely defined. 2 The characteristic zero hypothesis is used here.

Proof. By Theorem 1.6, the ideal A has no embedded associated prime ideal. The corollary then follows [24, chapter IV, paragraph 5, Theorem 8].

Corollary 1.14. Theorems 1.1 and 1.6 hold if A is replaced by any ideal (A) : S ∞ where S is any subset of R containing h, provided that (A) : S ∞ is proper.

Proof. The ideal (A) : S ∞ is the intersection of the primary components of (A) : h ∞ which do not meet the multiplicative family generated by S. Since the primary components of (A) : h ∞ do not meet M , the primary components of (A) : S ∞ do not meet M either and, Theorem 1.6 holds for this ideal also. Theorem 1.1 follows from Theorem 1.6 and Proposition 1.4.

Corollary 1.15. Every regular element of R 0 /A 0 is invertible.

Proof. Still a well known theorem. By Theorem 1.6, the ideal A 0 has dimension zero. By [24, chapter VII, paragraph 7], the associated prime ideals of A 0 are maximal. The ideal A 0 is thus contained in finitely many prime ideals. There is a bijection [START_REF] Zariski | Commutative Algebra[END_REF]chapter III,Theorem 7] between the ideals of R 0 which contain A 0 and the ideals of R 0 /A 0 . This bijection maps prime ideals to prime ideals [24, chapter III, Theorem 11]. The ring R 0 /A 0 thus involves only finitely many prime ideals p 1 , . . . , p r which are the associated primes of (0). Assume a ∈ R 0 /A 0 is regular. By [24, chapter IV, paragraph 6, Corollary 3], the element a belongs to none of the ideals p 1 , . . . , p r . The ideal generated by a must contain 1 since it would otherwise have associated prime ideals all different from p 1 , . . . , p r and there are no such prime ideals. Thus there exists some ā ∈ R 0 /A 0 such that a ā = 1 and a is invertible.

Corollary 1.16. Let 1 ≤ i ≤ n be an index. Denote A i = {p 1 , . . . , p i }. If h is the product of the initials of A, denote h i the product of the initials of the elements of A i otherwise, denote h i the product of the separants of the elements of A i . Denote

A i = (A i ) : h ∞ i . Let a ∈ R be any polynomial. If a is regular in R i /A i then a is regular in R/A.
Proof. Denote R 0,i = K(t 1 , . . . , t m )[x 1 , . . . , x i ] and A 0,i = (A i ) : h ∞ i in R 0,i . Assume a is regular in R i /A i . Then, by Theorem 1.1 and Corollary 1.15, there exists some ā ∈ R 0,i such that a ā -1 ∈ A 0,i . Since R 0,i ⊂ R 0 and A 0,i ⊂ A 0 , we have a ā -1 ∈ A 0 and a is invertible in R 0 /A 0 . By Theorem 1.1 again, a is regular in R/A.

Lazard's lemma

In this section, we keep the notations of section 1 but we restrict ourselves to the case of h being the product of the separants of the elements of A. The ideal A = (A) : h ∞ is often denoted (A) : S ∞ A in the literature. It is assumed to be proper.

Theorem 2.1. (Lazard's lemma)

The ideal A is radical. The minimal prime ideals of A have dimension m and do not meet M .

Before proceeding, let us consider the basic case of a system A made of a single polynomial p 1 = t 1 (x 1 -1)3 (x 1 -2). Then the separant h = t 1 (x 1 -1) 2 (4 x 1 -7) involves as a factor the polynomial t 1 which does not depend on x 1 and the multiple factor (x 1 -1) of p 1 . The ideal A is generated by (x 1 -2) and satisfies Theorem 2.1. Observe that the theorem would not hold in the case of h being the product of the initials of A only. In that case, the ideal A, which would be generated by (x 1 -1) 3 (x 1 -2), would not be radical.

Proposition 2.2. To prove Theorem 2.1, it is sufficient to prove that A 0 is radical.

Proof. The second statement of Lazard's lemma follows from Theorem 1.6. Let us assume A 0 is radical. Then R 0 /A 0 does not involve any nilpotent 3 element by [24, chapter IV, Theorem 10 and Corollary]. Thus R/A does not either by Theorem 1.1 (for if a is a non zero element of R/A then its image a/1 in R 0 /A 0 is non zero ; if a power a d of a were zero, then (a/1) d would be zero too and R 0 /A 0 would involve nilpotent elements). Therefore, A is radical and the proposition is proved.

In the rest of this section, we prove that A 0 is radical by proving that R 0 /A 0 is isomorphic to a direct product of fields. Since a direct product of fields does not involve any nilpotent element, the ideal A 0 is radical and the proof of Lazard's lemma is complete.

Indeed, if R 1 , . . . , R k are rings then one denotes In the ring S, zero is equal to (0, . . . , 0) and one is equal to (1, . . . , 1). If the rings R i do not involve any nilpotent element then S does not either. This is the case in particular when the rings R i are fields. See [START_REF] Zariski | Commutative Algebra[END_REF]chapter III,paragraph 13] for an equivalent formulation based on direct sums. The following theorem is a generalization of the Chinese Remainder Theorem. See [24, chapter III, paragraph 13, Theorem 32] or [7, Exercise 2.6, page 79].

S = R 1 × • • • × R k their direct

Theorem 2.3. (Chinese Remainder Theorem)

If A 1 , . . . , A k are ideals of R such that A i + A j = R whenever i = j then the ring

R/(A 1 ∩ • • • ∩ A k ) is isomorphic to the direct product (R/A 1 ) × • • • × (R/A k ).
The proposition below concludes the proof of Theorem 2.1. The scheme of its proof is the original scheme of proof communicated by Daniel Lazard.

Proposition 2.4. The ring R 0 /A 0 is isomorphic to a direct product of fields.

Proof. The ring R 0 /A 0 can be constructed incrementally. It is isomorphic to the ring S n defined by:

S 0 = K(t 1 , . . . , t m ), S i = S i-1 [x i ]/(p i ) : s ∞ i .
The proof is an induction on n.

The basis n = 0 is trivial.

Assume S n-1 is a direct product of fields K 1 × • • • × K r . Then S n is isomorphic to the direct product of the rings K j [x n ]/(p n ) : s ∞
n for all 1 ≤ j ≤ r. In the formula above one assimilates the polynomials p n and s n with their images by the canonical ring homomorphisms, noticing that the image of the separant of p n in each K j [x n ] is the separant of the image of p n in this ring. Therefore, in each K j [x n ], the ideal (p n ) : s ∞ n is generated by the product of the simple irreducible factors of p n . It is thus the intersection of the maximal ideals m ℓ generated by these factors. According to the Chinese Remainder Theorem, each K j [x n ]/(p n ) : s ∞ n is isomorphic to the direct product of the fields K j [x n ]/m ℓ . Since direct products are associative, the ring S n itself is a direct product of fields.

Corollary 2.5. Theorem 2.1 holds if A is replaced by any ideal (A) : S ∞ where S is any subset of R containing the separants of the elements of A, provided that (A) : S ∞ is proper.

Proof. The ideal (A) : S ∞ is the intersection of the primary components of A which do not meet the multiplicative family generated by S. Since A is radical, its primary components are prime ideals [24, chapter IV, Theorem 5]. Thus the primary components of (A) : S ∞ are prime ideals and (A) : S ∞ is radical. The dimension properties shared by all the associated prime ideals of A also hold for all the associated prime ideals of (A) : S ∞ .

Regular chains

We consider the polynomial ring R = K[x 1 , . . . , x n , t 1 , . . . , t m ]. We assume that the m + n indeterminates are ordered according to some total ordering O. Let p be any polynomial of R \ K. The greatest indeterminate w.r.t. O among the indeterminates p depends on is called the main indeterminate of p. We consider a triangular system A = {p 1 , . . . , p n } of R i.e. a polynomial system whose elements have distinct main indeterminates. Renaming the indeterminates if necessary, we assume that the main indeterminate of p i is x i for each 1 ≤ i ≤ n. The multiplicative family M , the initials and the separants of the elements of A are then defined as in section 1.

Fix some 1 ≤ i ≤ n. Denote A i the system {p 1 , . . . , p i }. Denote h i the product of the initials of the elements of

A i . Denote R i the ring K[t 1 , . . . , t m , x 1 , . . . , x i ]. Denote R 0,i the ring K(t 1 , . . . , t m )[x 1 , . . . , x i ]. Denote A i the ideal (A i ): h ∞ i of R and A 0,i the ideal (A i ): h ∞ i of R 0,i . Denote R 0 = R 0,n and A = A n . Definition 3.1. The system A is a regular chain if, for each 2 ≤ i ≤ n, the initial of p i is regular in the ring R i-1 /A i-1 . Assume A is a regular chain. Then A is said to be squarefree if, for each 1 ≤ i ≤ n, the separant of p i is regular in R i /A i .
The above definition is not exactly the same as that of [2, Definition 4.1] but they are strictly equivalent. The difference is that, in [START_REF] Aubry | On the theories of triangular sets[END_REF], the t indeterminates greater than x i would have been withdrawn from the rings R i-1 and R i . This change is not important for the elements of A i do not depend on the t indeterminates greater than x i and, by [24, chapter I, paragraph 16, Theorem 6], if R is a ring, a is one of its elements and x is an indeterminate over it then a is zero (respectively regular) if and only if it is zero (respectively regular) in the ring R[x]. The following results are corollaries to Theorems 1.1 and 2.1.

Corollary 3.2. The system A is a regular chain if, for each 2 ≤ i ≤ n, the initial of p i is invertible in the ring R 0,i-1 /A 0,i-1 . Assume A is a regular chain. Then A is squarefree if, for each 1 ≤ i ≤ n, the separant of p i is invertible in R 0,i /A 0,i .

Proof. It is an immediate corollary of Theorem 1.1 (enlarging the set of the t indeterminates with the x indeterminates which are not needed).

Corollary 3.3. Assume A is a squarefree regular chain. Then A is radical. Its minimal prime ideals have dimension m and do not meet M . Proof. By Corollary 1.16 and the definition of squarefreeness, the separants of the elements of A are regular in R/A. Thus they do not lie in any associated prime ideal of A by [24, chapter IV, paragraph 6, Corollary 3]. Thus, denoting S ∞

A the multiplicative family that they generate, A = A : S ∞ A and the proof follows from Corollary 2.5.

Splittings

In this section, we provide two propositions which permit to justify many algorithms carrying out the "D 5 " principle for triangular systems [START_REF] Della | About a new method for computing in algebraic number fields[END_REF]. We keep the notations of section 3 and we assume that A is a regular chain. Let 1 ≤ i ≤ n be an index. Assume that there exists a factorization

p i = b c in (R 0,i-1 /A 0,i-1 )[x i ] such that 0 < deg(b, x i ), deg(c, x i ) < deg(p i , x i ). For each 1 ≤ j ≤ n, denote B j = A j if j < i otherwise denote B j = (A j \{p i })∪{b}. Denote B = B n .
For each 1 ≤ j ≤ n, denote h b,j the product of the initials of the elements of B j and B j the ideal (B j ) : h ∞ b,j of R and B 0,j the ideal (B j ) : h ∞ b,j of R 0,j . Replacing b by c in the formulas, define C and for each 1 ≤ j ≤ n, define C j , h c,j , C j and C 0,j Denote B 0 = B 0,n and C 0 = C 0,n . Proposition 3.4. The triangular sets B and C are regular chains. For each 1 ≤ j ≤ n we have A j ⊂ B j and A j ⊂ C j .

Proof. We focus on the set B. The arguments for C are similar. Since 0 < deg(b, x i ) < deg(p i , x i ), the set B is triangular. For each 1 ≤ j < i we have A j = B j thus A j = B j and B is a regular chain up to index i -1. In the ring R 0,i-1 , the initial of p i is the product of the initials of b and c. Since A is a regular chain, the initial of p i is invertible in R 0,i-1 /A 0,i-1 . Therefore, the initial of b is invertible in R 0,i-1 /B 0,i-1 . By Corollary 3.2, the set B is a regular chain up to index i and A i ⊂ B i . Let i < j ≤ n be an index. We have A j ⊂ B j . Thus the initial of p j , which is invertible in R 0,j-1 /A 0,j-1 , is also invertible in R 0,j-1 /B 0,j-1 . By Corollary 3.2, the set B is a regular chain up to any index and A j ⊂ B j .

We have proved that A ⊂ B ∩ C. In general the equality does not hold because of possible common factors of b and c. In the particular case of squarefree regular chains, b and c have no common factors and the equality holds, the following proposition shows. Proof. First we prove that B and C are squarefree regular chains. As in the above proof, we focus on B. By Proposition 3.4, the set B is a regular chain. Assume A is squarefree. Let 1 ≤ j < i be an index. Since A j = B j , the separant of p j is regular in R j /B j and B is a squarefree regular chain up to index i -1. Denote s i , s b and s c the separants of p i , b and c. We have s i = s b c + s c b. Let us prove that s b is regular in R i /B i . Since A is squarefree, s i is invertible in R 0,i /A 0,i . By Proposition 3.4, for each 1 ≤ j ≤ n we have A j ⊂ B j thus s i is also invertible in R 0,i /B 0,i . Then, using the fact that b ∈ B i we see that s b c, hence s b , is invertible in R 0,i /B 0,i . By Corollary 3.2, the set B is a squarefree regular chain up to index i. Let i < j ≤ n be an index. Using again the fact that A j ⊂ B j , we see that the separant of p j is regular in R j /B j . Thus B is a squarefree regular chain up to any index.

Similar statements prove that C is a squarefree regular chain. By Corollary 3.3, the ideals B and C are radical. They are equal to the intersections of their minimal prime ideals by [24, chapter IV, Theorem 5]. To conclude the proof of the proposition, it is thus sufficient to prove that the sets of the minimal prime ideals of B and C form a partition of the set of the minimal prime ideals of A. Denote V , V b and V c the sets of zeros of A 0 , B 0 and C 0 in the algebraic closure of K(t 1 , . . . , t m ). Since these ideals have dimension zero, these sets are finite. The minimal prime ideals of A, B and C have dimension m and do not meet M . Therefore, by [24, chapter IV, Theorem 15(d)], the ring homomorphism R → R 0 provides a bijection between the minimal prime ideals of A (respectively B, C) and those of A 0 (respectively B 0 , C 0 ) hence, using [24, chapter VII, paragraph 3, Corollary 2], a bijection between the minimal prime ideals of A (respectively B, C) and the elements of

V (respectively V b , V c ). It is thus sufficient to prove that V b and V c form a partition of V . The cardinal |V | of V is the product n j=1 deg(p j , x j ). Similar statements hold for V b and V c . Since deg(p i , x i ) = deg(b, x i ) + deg(c, x i ) we see first that |V | = |V b | + |V c |. Second, we have V b ⊂ V and V c ⊂ V . Third, V b ∩ V c is

The D 5 principle for triangular systems

In this section, we provide the scheme of many algorithms carrying out the "D 5 " principle for triangular systems. More efficient algorithms can be found in [START_REF] Moreno | Polynomial gcd computations over towers of algebraic extensions[END_REF][START_REF] Moreno | On Triangular Decompositions of Algebraic Varieties[END_REF]. See also [START_REF] Wu Wen Tsün | On the foundation of algebraic differential geometry[END_REF][START_REF] Wang | Decomposing polynomial systems into simple systems[END_REF][START_REF] Bouziane | Unmixed-Dimensional Decomposition of a Finitely Generated Perfect Differential Ideal[END_REF]. The triangular set A is assumed to be a regular chain. The parameter a of algebraic inverse denotes an element of R. The function returns an inverse of a in R 0 /A 0 or fails. If it succeeds then a is proved invertible in R 0 /A 0 hence regular in R/A by Theorem 1.1. The function thus implicitly relies on the equidimensionnality theorem. If it fails by encountering a zero divisor, it exhibits a nontrivial factorization of some element p i of A. The exhibited factorization might allow some calling function to split A as two regular chains by using Proposition 3.4. Observe that the function may fail even if a is regular in R/A for it checks the regularity of many different elements of R/A. function algebraic inverse (a, A) begin if a ∈ K(t 1 , . . . , t m ) then if a = 0 then 1/a else the inverse computation fails (inversion of zero) endif else let x i be the main indeterminate of a (u 1 , u 2 , u 3 ) := extended Euclid (a, p i , x i , A) if u 3 = 1 then the inverse computation fails (inversion of a zero divisor): u 3 is a factor of p i else u 1 endif endif end

Here is the generalization of the extended Euclidean algorithm called by algebraic inverse. The main indeterminate of the two polynomials a and b is x i . The polynomials a and b are viewed as polynomials in (R 0,i-1 /A 0,i-1 )[x i ]. The function fails or returns a triple U = (u 1 , u 2 , u 3 ) of elements of (R 0,i-1 /A 0,i-1 )[x i ] satisfying a Bézout identity in the ring (R 0,i-1 /A 0,i-1 )[x i ] i.e. a relation u 1 a + u 2 b = u 3 . A proof that U satisfies a Bézout identity can be designed by using the two following loop invariants (i.e. properties which hold each time the loop condition is evaluated). These loop invariants are natural generalizations of the very classical loop invariants of the basic extended Euclidean algorithm:

• u 1 a + u 2 b = u 3 and v 1 a + v 2 b = v 3 in (R 0,i-1 /A 0,i-1 )[x i ] ;
• the set of the common divisors of u 3 and v 3 is equal to the set of the common divisors of a and b.

Observe that the second invariant is stated without using the word "gcd" which would be controversial in this context for the ring (R 0,i-1 /A 0,i-1 )[x i ] is not a UFD. A definition of the gcd in this context is however provided in [START_REF] Moreno | Polynomial gcd computations over towers of algebraic extensions[END_REF]. Observe that the function needs to recognize zero in R 0,i-1 /A 0,i-1 in order to evaluate the loop condition and to determine the degree of u 3 after the loop execution. This is achieved by Proposition 3.7. The function also

  chapter I, paragraph 19, Corollary 1]. Let us recall the Lasker-Noether theorem [24, chapter IV, Theorems 4 and 6]. Theorem 1.3. (Lasker-Noether theorem)

  admits a basis made of n + 1 elements in a polynomial ring in n + m + 1 indeterminates. According to the principal ideal theorem, dim A ′ ≥ m. Combining both inequalities, one concludes that dim A ′ = m. Let us recall Macaulay's unmixedness theorem [24, chapter VII, Theorem 26]. Theorem 1.11. (Macaulay's unmixedness theorem) If a proper ideal A of a polynomial ring R = K[x 1 , . . . , x n ] admits a basis made of k elements (1 ≤ k ≤ n) and if dim A = nk then all its associated prime ideals have dimension nk.

  product. Elements of S are tuples with k components. Given any two elements a = (a 1 , . . . , a k ) and b = (b 1 , . . . , b k ) of S one defines a + b as (a 1 + b 1 , . . . , a k + b k ) and a b as (a 1 b 1 , . . . , a k b k ).

Proposition 3 . 5 .

 35 Assume A is squarefree. Then so are B, C and we have A = B ∩ C. Moreover, the sets of the minimal prime ideals of B and C form a partition of the set of the minimal prime ideals of A.

  empty for a common zero of B and C would annihilate s i = s b c + s c b which is invertible. Therefore V = V b ∪ V c , the sets V b and V c form a partition of V and the proposition is proved.

Definition 3 . 6 .Proposition 3 . 7 .

 3637 For every a ∈ R we define the pseudoremainder of a by A asprem(a, A) = def prem(. . . prem(prem(a, p n , x n ), p n-1 , x n-1 ) . . . , p 1 , x 1 ).The pseudoremainder algorithm is based on[24, chapter I, paragraph 16, Theorem 9]. It is defined in [10, volume 2, page 407]. The next proposition is proved in [2, Theorem 6.1]. For every a ∈ R we have a ∈ A if and only if prem(a, A) = 0.

  chapter VII, Theorem 22]. Theorem 1.9. (principal ideal theorem) If a proper ideal A of a ring R = K[x 1 , . . . , x n ] admits a generating family formed of k elements

regular = not a zero divisor.

A nilpotent element of a ring R is a nonzero element of R a power of which is zero.

needs to check the regularity of the leading coefficient of v 3 before performing the Euclidean division. This can be achieved using algebraic inverse.


José Abdelnour-Nocera
email: jose.abdelnour-nocera@uwl.ac.uk

Samia Oussena
email: samia.oussena@uwl.ac.uk

Catherine Burns
email: catherine.burns@uwaterloo.ca

Human Work Interaction Design of the Smart University

Keywords: smart university, human work interaction design, cognitive work analysis, user interface design

In this paper we present human work interaction design challenges and opportunities for the vision of the Smart University as a platform that provides foundational context data to deliver the university of the future. While learning analytics have enabled access to digital footprints of student activities and progress in terms of data such as demographics, grades, recruitment and performance, they cannot provide information about activities and interaction in the physical study and work spaces in a university. The smart university proposes a novel platform that will provide context aware information to students through the integration of learning analytics with data sensed using cyberphysical devices in order to provide a holistic view of the environments that universities offer to students. However, designing the interaction of students and staff in the smart university ecology of information and sensing devices requires an understanding of how they work as individuals, as members of teams and communities. Through two use cases we illustrate how insights obtained from social cognitive work analysis can be used for the design guidelines of the different interfaces part of the smart university ecology.

Introduction

The Smart University is a vision where the university, as a platform, provides foundational context data to deliver the university of the future. As higher education funding in Europe continues decreasing, universities are more reliant on income generated from students. Thus, understanding student progression and identifying ways to improve the students experience are vital to any institution. To improve their students' experience, universities are increasingly reliant on technology to improve and expand their services to students. In this paper we present our proposed version of the smart university and how human work interaction design (HWID) [START_REF] Clemmensen | CHI 2013 human work interaction design (HWID) SIG: past history and future challenges[END_REF] can support the im-plementation of this platform at design level. Although, as part of Smart university ecosystem, some applications have been already implemented [START_REF] Sauer | Approaches to the Use of Sensor Data to Improve Classroom Experience[END_REF][START_REF] Oussena | Exploiting Student Intervention System Using Data Mining[END_REF], the discussions in the paper refer to a conceptual design of the Smart University platform. Smart campus platforms have been researched and reported in the literature and some aspects have been also implemented in some universities. For example, Lei et al., [START_REF] Lei | Building an Intelligent Laboratory Environment via a Cyber-Physical System[END_REF] propose the design of a smart laboratory that measures, analyses and regulates the thermal comfort by use of cyber-physical devices. The University of Southern California implemented smart buildings management in the building. A more holistic vision has also been proposed; iCampus, as envisioned by EBTIC 1 , is an initiative that proposes that the university of the 21st century be composed of six functional areas or pillars, designed to enrich students' experiences throughout their learning lifecycle: iLearning, iGovernance, iGreen, iHealth, iSocial, and iManagement. Although this is inline with our perspective, this initiative is still at a developmental stage and has a very broad scope. Our proposed platform is different in that our emphasis is on the design of the software platform that will allow the delivery of the vision. The implementation of the platform will take a data-oriented architecture approach. The focus is on how to develop a high-quality platform that will allow the use of cyber-physical devices and data analytics for the university of the future. Students' interactions with university systems are leaving an increasing amount of digital footprint which can be harnessed to understand behaviour and activities of students as well as help them become more effective in their studies and preparation of their career. Learning analytics have used these digital footprints left by students to gain insight on the students' progress and to build a personalised learning environment. However, most of Learning analytics projects have been looking at the monitoring of the digital environment that the institution offers to the students. The smart university vision is to provide a novel platform that will provide context aware information to students through the integration of learning analytics with data sensed using cyber-physical devices in order to provide a holistic view of the environments that universities offer to students. Additionally, this will augment the traditional learning analytics with data related to the physical environment and allow the investigation of these intelligent buildings and their effect on the learning processes.

Designing the interaction of students and staff in the smart university ecology of information and sensing devices requires an understanding of how they work as individuals, as members of teams and communities. HWID approach studies how to understand, conceptualize, and design for the complex and emergent contexts in which information and communication technologies (ICT) and work are entangled. In this paper, through two use cases we illustrate how insights obtained from HWID analysis can be used for the design guidelines of the different interfaces part of the smart university platform.

Learning Analytics

Students' interactions with university systems are leaving increasingly large digital footprints which can be harnessed to understand the behaviour and activities of students as well as help them become more effective in their studies and preparation of their career. Learning analytics have used these digital footprints left by students to gain insight on the students' progress and to build a personalised learning environment. However, most of these projects have been looking at the monitoring of the digital environment that the institution offers to the students. Data that is used for these analytics rely on management data, such as student demographics, grades, recruitment figures and the traces left by the students as they use the university IT systems such as virtual learning environment (VLE) or Learning management system (LMS).

Masses of data can be collected from different kinds of student actions, such as solving assignments, taking exams, online social interaction, participating in discussion forums, and extracurricular activities. This data can be used by Learning Analytics to extract valuable information, which might be helpful for lecturers to reflect on their instructional design and management of their courses. Usable Learning Analytics tools for lecturer that support cyclical research activities are still missing in most current VLE or are far from satisfactory [START_REF] Hijon | E-learning platforms analysis and development of students tracking functionality[END_REF]. Data mining tools are usually designed for power and the flexibility of the analytics rather than for the simplicity. Most of the current data mining tools are too complex for educators to use and their features go well beyond the scope of what they might require [START_REF] Romero | Data mining in course management systems: Moodle case study and tutorial[END_REF] If tracking data is provided in a VLE, it is often incomprehensible, poorly organized, and difficult to follow, because of its tabular format. As a result, only skilled and technically savvy users can utilize it [START_REF] Mazza | CourseVis: A graphical student monitoring tool for supporting instructors in web-based distance courses[END_REF]. Many lecturers, using learning analytics are motivated to evaluate their courses and they already have questions related to their teaching in mind.

The Smart University

Over the past decade, innovation in design and manufacturing throughout the industry has enabled the cost, size, power consumption of sensors and the associated networks to improve dramatically. Consequently, sensor-based systems have been proposed for a broad range of monitoring applications; more recently, these technologies have allowed the integration of the cyber world to physical world and effectively blurring the gap between the two. The smart university proposed a novel platform that will provide context aware information to students through the integration of learning analytics with data sensed using cyber-physical devices in order to provide a holistic view of the environments that universities offer to students. Additionally, this will augment the traditional learning analytics with data related to the physical environment and allow the investigation of these intelligent buildings and their effect on the learning processes. The platform aims to combine a responsive architectural environment with an intelligent virtual environment in order to offer a truly personalised learning environment. The responsive architectural buildings will be providing optimal heating, ventilation and lighting based on the requirements of the learning environment (i.e. chemistry lab or ICT lab), the learning models and the behaviour of the occupant of the environment. The behaviour of the occupant (learner or tutor) can be monitored by their interactions with the IT systems as well as some wearable devices. Sensors measuring temperature, humidity, noise and air quality would be used to monitor the behaviour of the building. Figure 1 illustrates the platform that could be used collecting, processing and visualising the data in a smart university. The platform will need to be scalable, data oriented and distributed with a friendly usable interface while, at the same time, being powerful and flexible enough for the repository of data of heterogeneous sources, integration of data sources in real-time, providing real-time exploration and interventions.

Human Work Interaction Design

Human work analysis is focused on user goals, user requirements, tasks and procedures, human factors, cognitive and physical processes, and contexts (organizational, social, cultural). For instance, Hierarchical Task Analysis [START_REF] Annett | Task analysis and training design[END_REF] and Work Domain Analysis [START_REF] Vicente | Cognitive work analysis: Toward safe, productive, and healthy computerbased work[END_REF][START_REF] Salmon | Hierarchical task analysis vs. cognitive work analysis: comparison of theory, methodology and contribution to system design[END_REF] are used to study goal-directed tasks and to map the work environmental constraints and opportunities for behaviour. The study of HCI has historically adapted work analysis methods such as hierarchical task analysis to the design of computer artefacts. Ethnographic methods [START_REF] Button | Studies of Work and the Workplace in HCI: Concepts and Techniques[END_REF] with a sociotechnical perspective have also been used in HCI (e.g., [START_REF] Abdelnour-Nocera | An approach to the evaluation of usefulness as a social construct using technological frames[END_REF]). These approaches focus on work as end-user actions performed collaboratively with other people in a field setting: the worker activity is seen as a social and organisational experience. In this context, human work analysis and HCI are interlinked in such as a way to form a distinct field of knowledge, namely HWID. HWID studies how to understand, conceptualize, and design for the complex and emergent contexts in which information and communication technologies (ICT) and work are entangled. Several aspects influence the way humans work and the work itself. For humans, language, culture, education, skills, knowledge, emotions and cognitive abilities contribute to define the profile of users and their approach to individual and collaborative work. For work, its goals, functions, available tools and content contribute to delineate its characteristics and challenges. In this paper, we illustrate the use of Cognitive Work Analysis (CWA), a well known work analysis technique, to support design considerations for the Smart University system. This technique is driven by a framework that supports and structures the analysis needed when designing a flexible and adaptive system [START_REF] Vicente | Cognitive work analysis: Toward safe, productive, and healthy computerbased work[END_REF][START_REF] Lintern | The Foundations and Pragmatics of Cognitive Work Analysis: A systematic approach to design of large-scale information systems[END_REF]. The framework focuses on analysing the limitations and constraints on workers behaviour; and mapping these constraints is the design of the system that will support the workers.

The CWA framework comprises five different phases; work domain analysis, control task (or activity) analysis, strategies analysis, social organisation and co-operation analysis, and worker competencies analysis. Using CWA has two distinct ad-vantages. First, CWA is a multi-dimensional analysis that incorporates the physical and the social environment to provide a rich description. Secondly, CWA can be paired with Ecological Interface Design (EID) [START_REF] Hajdukiewicz | Strategies for bridging the gap between analysis and design for ecological interface design[END_REF] to generate designs for new information systems. EID has shown success in the design of analytic information displays in power plant displays [START_REF] Burns | Evaluation of ecological interface design for nuclear process control: situation awareness effects[END_REF]; social systems [START_REF] Hajdukiewicz | Work domain analysis for intentional systems[END_REF], healthcare decision support [START_REF] Burns | A cognitive work analysis of cardiac care nurses performing teletriage[END_REF] and community building [START_REF] Euerby | Improving Social Connection Through a Communities of Practice-Inspired Cognitive Work Analysis Approach[END_REF]. For these reasons, CWA may be a promising approach in cyber physical systems like the smart university.

Applying HWID to Smart University Scenarios: two examples

In this paper, HWID concepts and tools will be applied to two types of activities in high education domain; exemplifying two aspects of the smart university platform; the learning analytics aspect and cyber-physical devices. The analysis that is discussed in this paper will shape the guidelines of the user interface of the smart university platform. One of the activities examines a lecturer using learning analytics to provide support to students when preparing for an exam. The lecturer uses Virtual learning platform (VLE) as the main medium for communicating with the students. Past exams, revision notes and other supporting exercises are uploaded on the VLE. The lecturer, might also initiate a special discussion board for exam support. Here we envisage that the learning analytics will allow the lecturer to monitor the effectiveness of the support that is being provided, helping him/her to adapt the materials accordingly.

The other activity is related to the smart campus; i.e. equipping the campus with cyber-physical devices that help provide responsive environment. An example of such environment is an ICT lab. The lab is equipped with a number of sensors; Students usually take a 2 hour practical session in this ICT lab. Each student has a set of exercises that they have to complete using the computer. The session is typically supported by an academic staff and an assistant; typically a PhD student.

In this section we present two possible scenarios for the Smart University platform analysed from a HWID perspective. This involves applying CWA and then translating insights from this process into interaction design guidelines for the different interfaces on this platform.

Scenario A: supporting exam preparation

A lecturer, who offers weekly online exercises has the intention to help his/her students to prepare for an exam. But she is not sure if the currently available exercises are helpful enough for this purpose. Therefore, he/she would like to know if those students who practice with her online exercises on a weekly basis are better in the final exam than students who do not use them. A Learning Analytics toolkit could help him/her to do research on this hypothesis by automatically collecting, analyzing, and visualizing the right data in an appropriate way. The smart university platform should allow for interactive configuration in such a way that its users could easily analyze and interpret available data based on individual interests.

We now look at this scenario through the lens of the different phases of CWA. Since education is a core goal of this scenario, learning needs to be present in the functional purpose and generalized function levels. The scenario indicates that there is a concern that weekly exercises might improve learning, as evaluated through exam results, or might not be helpful. This is why we have chosen to describe at the abstract function level that there must be a balance between evaluation and learning, e.g. you cannot evaluate 100% of the time, but you also need to evaluate at some level. The functional purpose is to find the sweet spot where learning outcomes and student experience are maximised at optimum levels.

WDA will allow us to identify the analytics data needed for designing components of the system. For instance, a key goal derived from this WDA is to enable the instructor to move that sweet spot between evaluation and learning to maximize outcomes and experience. Those are the drivers, i.e. decisions to be made with the analytic system.

Control Task Analysis:

This is done to determine what tasks are being carried within the system and under what conditions. In this learning analytics scenario, control task analysis (ConTA), based Rasmussen's decision ladder [START_REF] Euerby | Improving Social Connection Through a Communities of Practice-Inspired Cognitive Work Analysis Approach[END_REF], the analysis would look like in Figure 3. Is there uncertainty and ambiguity on the possible goal state? Quite possibly, if the instructor is following a new evaluation approach for students, she may move into knowledge based behavior [START_REF] Mazza | CourseVis: A graphical student monitoring tool for supporting instructors in web-based distance courses[END_REF] trying to figure out what is wrong. Analytics could play a role here. Instructors can then 'define a task', i.e. choose to modify their instruction approach. This implies setting a new 'procedure', more or less exercises in this case, which would then be 'executed'.

Strategies, Social and Worker Competencies:

This level of analysis can facilitate the discussion of different teaching strategies (traditional, flipped, blended learning). This could also reveal different evaluation strategies (short quick frequent evaluations, longer midterm/final, or project based evaluation).

The identification and description of social competencies could represent values and intentional constraints being conveyed by the institution. It could also consider the culture and cooperation of the students in this. As a worker, the instructor must have competency in teaching, the material being taught, and the use of the smart learning system. Skills, rules and knowledge is the base for all of these [START_REF] Mazza | CourseVis: A graphical student monitoring tool for supporting instructors in web-based distance courses[END_REF].

Scenario B: monitoring room temperature in the Smart Campus

In this scenario, the ICT lab at the university is equipped with an number of sensors and a display at the lecturer station with dashboard and message board for information about the room. Once the students have entered the room and started working, the information about the room is updated with an estimate of the number of people in the room and the ideal temperature for the ICT lab activity. The lecturer had a quick look at the dashboard and noticed that the room was empty for the morning and he/she understood that it will take about 5 minutes to get the ideal temperature. He/she also noticed that the noise was higher than what is expected for an ICT session and he/she first closed the windows and the door before asking students to work more quietly.

The students with wearable devices capable of giving ambient temperature readings also noticed that the temperature adjusted to the ideal temperature within 5 minutes of being in the lab.

Work Domain Analysis:

As with the scenario A, we now illustrate a possible WDA for the scenario of controlling temperature in the classroom: This is done to determine what tasks, data, and messages are being processed within the system and under what conditions. In this monitoring ICT lab temperature scenario, inspired in Rasmussen's decision ladder [START_REF] Euerby | Improving Social Connection Through a Communities of Practice-Inspired Cognitive Work Analysis Approach[END_REF] the analysis would look like in Figure 4. The objective here is to define and implement contextual, multi-sensory inference strategy services that are able to derive contextual information from aggregating different sources data. This will allow us to model user-based energy profiles and user behaviours in the ICT sessions. Based on the contextual models defined and considering the constraints related to comfort, it will be possible to identify diverse energy awareness rendering messages providing adequate feedback on various personalized display (wearables) or the instructor dashboard.

WDA:
Strategies, Social and Worker Competencies:

Two strategies are apparent; first, to be energy efficient (i.e. suggestion of taking piece of clothing, opening a window or closing a door) or second, to emphasis on the comfort and make more use of the heating or cooling system. Ultimately, a smart university system will aim to use thermal comfort to change the expectation from largely invisible centralized control of the environment into a more active and respon-sive approach. Furthermore, the system will implement a 2-way information exchange between occupants and buildings.

In terms of social competencies, students and staff awareness of, and responsibility for, environmental issues is variable. How can we facilitate and encourage sharing of thermal comfort strategies and learning from others? For example what are the alternative ways to keep cool or warm, or how to generate reflection on clothing and its role in thermal comfort? As a worker, the instructor must have the competency to understand the correlations between climatic comfort and student performance and make the right type of decisions about the required behavior.

CWA Moderated Interface Design

A Looking at two scenarios through the CWA approach facilitates the decision on which usability and user experience goals that should drive the interaction design of the different interface components of the smart university sociotechnical system. This also applies for the type of user interface design guidelines and levels of representation required at conceptual design level.

The CWA of the learning analytics scenario clearly points to importance of effectiveness and utility [START_REF] Rogers | Interaction design: beyond human-computer interaction[END_REF] as main usability goals driving the design of the user interface of the lecturer trying to establish the optimum level of exercises that should be set for students to meet learning goals in a satisfactory way.

In terms of design guidelines feedback and mapping become core objectives in the presentation of the student performance data. Good user interface design meeting these goals and principles will support the overall functional purpose of the learning analytics systems, i.e. Maximize learning outcomes, Maximize student experience. Similarly, in the ICT lab monitoring scenario, the effectiveness of the adaptive comfort messages is an essential goal since the underlying objective is to change the students environmental behaviour.

It can be illustrated how specific interface design decisions on relevant guidelines and heuristics can be derived from looking at the WDA and ConTA for both scenarios: Table 3 links key tasks to user behaviour, user interface components, design concepts and design principles and heuristics.

Conclusions

This paper introduces a proof of concept attempt to illustrate how HWID can be a useful framework in the design of the smart university platform as sociotechnical ecology of information sharing actors and devices. Through the application of CWA to two different scenarios we have identified the nature of work, artefacts and interactions in which smart university users will engage. The smart university scenarios have been focused on the common functional purpose of maximizing learning outcomes while maximizing the student experience. CWA has enabled an understanding of the different task requirements in each scenario in order to achieve this: in the first scenario on learning analytics, it has been uncovered how finding the right balance between learning and evaluation is a critical goals; on the second scenario on the use of temperature sensors in the smart campus it is clear that the visualization of relations between climatic comfort and learning experience is a critical goal. Looking at these scenarios through ConTA provides an understanding of behaviours according to skills, rules and knowledge in the context of task goals. Identifying the type of behaviour the user is engaged in will provide useful information on cognitive and material elements of the tasks that should be supported. For instance, it is evident to see how in the learning analytics scenario the teacher is likely to be engaged in knowledge based behaviour more often trying to establish the optimum level of exercises for a particular group, while in the smart campus scenario a rule based behaviour is likely to be more frequent as the relation between climatic comfort and student experience will tend to be more stable.

We were then able to illustrate how this analysis of work in these two smart university scenarios can feed the interaction design of user interface components in the different points of interaction with the platform. There will be a need to prioritize different types of usability and user experience goals in terms of the functional purpose and desired goal states in identified in CWA. Even in the present examples, it is easy to predict that supporting the instructor with analytics on student performance would be a more extensive design challenge than providing monitoring of the thermal conditions of the classroom. This also has implications on data visibility, information accessibility, and information architecture. In the case of monitoring student performance, the instructor needs a deeper architecture, more data accessibility and more control latitude to develop the view he or she may want. In contrast, the thermal comfort situation may require quite straightforward information display and limited control to the instructor and students.

General

HWID models also provide considerations for nation, geographic, cultural, social and organizational factors shaping the activities being supported through design [START_REF] Katre | Human Work Interaction Design for e-Government and Public Information Systems[END_REF]. The smart university does not escape these considerations and any of the models and design principles and heuristics shaping the interactive points in these platforms will have to be moderated by them. For instance, Northern European universities will have challenges for design very different from those in the Southern Europe due to cultural, political and climatic factors.

In summary, we have illustrated a case for HWID in the context of the design of the smart university. Work analysis and interaction design can be integrated to support important design decisions affecting the ecology of devices and information repositories in the smart university with a clear focus on its users, their contexts and interactions.

Fig 1 .

 1 Fig 1. Example of Smart University platform

Fig. 3 .

 3 Fig. 3. Decision ladder for learning analytics scenario.

Fig. 4 .

 4 Fig. 4. Monitoring room temperature in the Smart Campus scenario.

Table 1 .

 1 Work Domain Analysis (WDA) provides an overview model of the work environment with a view to understanding what kinds of information should be included in the user interface and how this should be presented. The learning analytics toolkit is part of a sociotechnical system whose main goal is maximising learning outcomes and the learning experience for students. The following presents an Abstraction Hierarchy (AH) typically used for WDA[START_REF] Burns | A cognitive work analysis of cardiac care nurses performing teletriage[END_REF]. This is made of five levels, which are now described in terms of the learning analytics scenario: Work domain analysis for learning analytics scenario

	Work Domain Analysis:

WDA: supporting exam preparation

Physical form for student (type, program, year of admission, status, performance level); for learning material (type, date available); for evaluation material (type, date of evaluation, grades achieved), for lecturer (level, name, availability); for student record system (type, data available, dates accessed)

Monitoring Room Temperature in the Smart Campus

	Physical form student (type of clothing, course studying ); for ICT lab (size, nb of machine,
		nb of windows, ideal climate); sensors(type), for lecturer (level, name); ses-
		sion (activity, nb of students, duration), for display(type (dashboard, weara-
		ble), data available, messages/alert)
	Physical	Student , ICT lab, sensors, snapshot of climate , ICT session, display
	function
	Generalized	Student attending the ICT session, lecturer receiving messages about the room,
	function	adjusting the room, adjusting own clothing/noise, student receiving personal-
		ized message , evaluation process
	Abstract	Balance the ratio of climatic comfort to learning
	function
	Functional	Maximize learning outcomes, Maximize student experience
	Purpose

Table 2 .

 2 WDA Monitoring Room Temperature in the Smart Campus (the Smart Campus)

	Control Task Analysis:

Table 1 .

 1 Mapping CWA to choose relevant user interface design concepts, principles and heuristics.

	-	Behaviour	User Inter-	Design Con-	Design Principle and
	ised Func-	Type Re-	face Compo-	cepts	Heuristic
	tion Task:	quired	nents
	Monitoring	Skill-based	Learning	Visualise infor-	Feedback: lecturer
	and evalua-	behaviour	Analytics	mation on stu-	should receive immedi-
	tion of		Dashboard:	dent perfor-	ate, intelligible alert if
	student's		Messag-	mance as well	performance falls below
	progress		es/alerts	as the level of	expected levels.
				instructor sup-
				port.	Mapping: data visualised
					should map naturally to
					student's activity record.
					Any non-technical user
					should be able to under-
					stand the student's posi-
					tion in relation to her
					cohort.
	Lecturer	Knowledge-	VLE content	Supports the	Visibility: lecturer should
	creating and	based behav-	creation mod-	creation and	be able to view historical
	uploading	iour	ule	upload of new	performance data on
	new materi-			exercises	exercises attempted and
	al				overall module perfor-
					mance while setting
					exercise levels.
					User Control and Free-
					dom: enable lecturers
					maximum control of
					creation and uploading of
					as many exercises as
					required.
	Lecturer	Rule-based	Temperature	Visualising the	Throughput: monitoring
	monitors	behaviour	and noise	required infor-	temperature and noise
	temperature		charts in	mation quickly	levels should not disrupt
	and noise		classroom	and in a non-	the core teaching tasks
	levels		based control	disruptive form	and should be done as
			panel		quickly as possible.
					Feedback: lecturer
					should receive clear
					indication of temperature
					and noise levels, with
					clear indication of ac-
					ceptable thresholds.

http://www.ebtic.org/pages/the-intelligent-campus


Åsa Cajander
email: asa.cajander@it.uu.se

Marta Larusdottir

Elina Eriksson

Gerolf Nauwerck
email: gerolf.nauweck@uadm.uu.se

Contextual Personas as a Method for Understanding Digital Work Environments

Keywords: Personas, digital work environment, usability

The role of IT at the workplace has changed dramatically from being a tool within the work environment to include all aspects of social and private life. New workplaces emerge where IT becomes more and more divergent, embedded and pervasive. These new aspects of IT at work need to be addressed with new or adapted human centred activities. This paper present and discuss a modified version of personas called contextual personas developed to better address the new working life. The contextual personas were developed using contextual inquiry, and focus groups as well as argumentative design. From the process of developing the contextual personas we learned that they are indeed a promising tool to understand the new work situations, and especially the holistic view of IT at work as they bring the whole working-life of the personas into focus. Finally, we discuss in what way the contextual personas could give developers extended understanding of the users' future office work environment.

Introduction

Working life has been changing for many with a move towards goal oriented tasks, informal communication and virtual collaboration. Moreover, the use of IT has moved from being an isolated tool within the work environment to becoming pervasive through for example mobile apps and social media. Hence this shift has led to work that blurs the borders to social and private life, which is related to definition of one's identity as a digital person in a technical world. Often these changes of working life are enhanced through technology with the technical solutions available. It is common that people use various computer systems, including various equipment to use those such as mobile phones, tablets and laptops. Moreover, they are at different places while using the software systems. This creates complexities that are new and challenging. Other new ways of managing private life and working life has emerged, and one can note trends like for example BYOD (Bring Your Own Device) where the company policy allows employees to bring their own technology to the workplace. This shift makes systems development ever more complicated and there is an increasing need for illustrating this new work situation in order to better understand all the variations in the work environment that has emerged.

In this study we made extensive contextual interviews with administrative staff at university offices to gather information on their current work situation from a computer supported work environment perspective. In the paper, we explain a new method for describing results of such contextual interviews called contextual personas. These personas are developed to explain the various work contexts of the workers.

Our results show that the employees are willing to give rich information about their work context, when prompted through the persona descriptions. Through the focus group discussions of the contextual personas real life was brought into the conversation.

In this paper we describe how contextual personas can be used to elicit IT based administrative work as one method to understand the digital work environment. Firstly the personas can be used as a tool to describe the work situation and the users within this work situation, which also extends to social life. However, we also argue that the contextual personas can be used as a reflexive tool that lets the respondents further reflect on their own work situation. Hence in our study, the contextual personas gave us information on the work environment and potential health hazards when looking at the holistic work situation including IT based tools and the overarching workload. On concrete finding from our interviews has implications for pervasive and smart workplaces, since the perceived connectedness of these administrators led to an always on status which made it hard for them to feel that they were ever free from work. Furthermore we discuss how the personas can be used in pervasive and smart workplaces such as future office spaces presented below.

Background

When creating the contextual personas, the healthy work model by Karasek and Theorell [START_REF] Karasek | Healthy Work. Stress, Productivity and the Reconstruction of Working Life[END_REF] was used. Hence the first section below presents this model. This is followed by a presentation of future offices and the emerging trends in this area. Finally, we present a short description of research on the personas method.

Digital Work Environment and Healthy Work

As work has become more digitalised the digital work environment becomes more central for employees productivity and also wellbeing. The digital work environment is here defined broadly as the work environment that is the result of digital tools and work support systems. The digital work environment consists of physical, psychosocial and cognitive problems and challenges. It encompasses all cognitive work situations where people interact or are dependent on computer systems. In this definition of the digital work environment we also include aspects of integrity, control and surveillance.

When analysing the data from the contextual interviews, the Demand-Control-Support model is used. In the 1970's Robert Karasek developed a model for analysing work-related stressors associated with cardiovascular illness. His demand and control model was thereafter further developed together with Töres Theorell [START_REF] Karasek | Healthy Work. Stress, Productivity and the Reconstruction of Working Life[END_REF] and is now one of the most widely used models for explaining psycho-social work conditions and their effects on health. This model suggests that the combination of perceived demands and perceived control at work is a determining factor for stress. This model was used since our previous research, see for example [START_REF] Cajander | Usability-Who Cares?: The Introduction of User-Centred Systems Design in Organisations[END_REF] and [START_REF] Sandblad | Work environment and computer systems development[END_REF] that have shown that the model is easily understood and applied in organisations when the digital work environment has been discusses. The figure below illustrates the Demand-Control-Support Model. The figure above illustrates how healthy and sustainable a work is, in relation to the experienced demands, control and social support. High demands are normally not a problem, if combined with high self-control over work situation and tools and strong social support from management and colleagues. A skilled worker can experience this as a challenging situation. She has full control over the work conditions and planning and gets full support when needed. The work is efficient and sustainable. On the other hand, if high demands are not met by strong self-control and social support, the situation will soon become dangerous health wise. If the worker does not have control over work conditions and planning, does not have usable tools and feels totally exposed if things go wrong, the work will be very unhealthy. Such work situations are associated with high stress. In this extreme, different health risks are common and people do not withstand the situation for long. Research shows that subjective control and support factors often decrease when new IT systems are introduced, as did Åborg, [START_REF] Åborg | How does IT feel@ work? And how to make IT better: Computer use, stress and health in office work[END_REF][START_REF] Åborg | Integrating work environment considerations into usability evaluation methods-the ADA approach[END_REF].

In this research project we have focused on administrative work. The main health problems in such computer-supported work situations are: Users are bound to use the computer for a major part of their working hours entailing constrained, static work postures for long periods. The computer controls the work pace and task order, leaving the users little or no control over their work. Users suffer from stress, caused by excessive workload, time pressure and poorly designed computer support [START_REF] Bergqvist | Visual display terminal work-a perspective on long-term changes and discomforts[END_REF][START_REF] Punnett | Visual display unit work and upper extremity musculoskeletal disordersErgonomic expert committee document/National institute for working life[END_REF]. Traditionally, occupational health experts work in isolation from the software development process. They evaluate and suggest improvements to existing workplaces and tools [START_REF] Sandblad | Work environment and computer systems development[END_REF]. It is, however, often too late to do something about poorly designed software tools once they have been installed and are running. Thus, poor and inadequate design leading to health problems cannot be sufficiently modified post-hoc of systems development when the computer system is in use. Instead, occupational health and ergonomics experts must be involved in the actual software development process. Work-related stress has increased in the past years and since long it is well known that it is a growing health problem [START_REF] Marklund | Worklife and health in Sweden[END_REF]. Organisation of work organisation and work content are important factors underlying stress problems, and in work, IT support systems, especially computer software, play a major role. The mental workload tends to increase when new IT systems are introduced [START_REF] Aronsson | Dimensions of control as related to work organization, stress, and health[END_REF], and the decision latitude is lower for extensive computer users than for others [START_REF] Tornqvist | Risks factors at computer and office workplaces[END_REF].

Future Offices and its Relevance to HCI

The physical work environment in all its variety is of course another factor influencing office workers [START_REF] Bodin Danielsson | The Office-an explorative study. Architectural design's impact on health, job satisfaction and well-being[END_REF]. Recently, flexible offices and more specifically management philosophies such as the idea of activity-based working (ABW) has attracted the interest of both organisations and researchers [START_REF] Appel-Meulenbroek | An end-user's perspective on activity-based office concepts[END_REF]. Place is here seen as a mediating factor between people and IT. Indeed, the strategic use of corporate space is seen as the necessary, though not sufficient, factor in empowering the workforce and ameliorating many of the downsides of computer-supported work [START_REF] Van Koetsveld | How flexible workplace strategies can be made successful at the operational level[END_REF]. A part of the concept is increased use of IT in support of both mobility and monitoring, and not surprisingly, the IT industry itself is one of the major proponents of this new way of work [START_REF] Gates | The New World of Work[END_REF]. Yet, while ABW is proposed as a solution to the problems associated with open plan offices, not least cognitive stress, research is inconclusive [START_REF] Too | TOXIC" workplaces: the negative interface between the physical and social environments[END_REF]. In a seminal paper [START_REF] Humphry | Visualising the future of work: myth, media and mobilities[END_REF] traces the origins of this seemingly new-anywhere, anyplace-work rhetoric and exposes some its inherent paradoxes, not least how these images of newness contribute to the conservation of old work patterns.

As IT in the workplace thus becomes more and more embedded and pervasive the scope for HCI broadens. The pioneering works of [START_REF] Alexander | The application of usability concepts in the built environment[END_REF] have already argued for the application of usability concepts in the field of facility management, yet the field seems to have attracted limited attention so far [START_REF] Rasila | Dimensions of usability assessment in built environments[END_REF]. Instead, the most common tool seems to be variations on the model developed by [START_REF] Laing | New Environments for Working: The Re-Design of Offices and Environmental Systems for New Ways of Working[END_REF]), where the amount of face-to-face interaction is contrasted with the amount of job autonomy (resulting in a matrix of four basic office types: the hive, the cell, the den and the club). While this broad categorisation of work can aid planning, other techniques, such as personas, could provide a deeper understanding on how to improve the quality of work and the work environment.

Personas

Within development of IT-based systems, the persona method has become frequently applied tool and is used extensively in both industry and in research. The persona method is a user-centred way of representing users in situations where users cannot be available; the idea is that the overall focus and awareness of the users in development projects are heightened when working with personas [START_REF] Grudin | Personas, participatory design and product development: An infrastructure for engagement[END_REF]. However the use of the method can be manifold for example, the personas are described both as a communication tool and as a design aid. However, [START_REF] Guðjónsdóttir | Personas and Scenarios in Use[END_REF] argues that by trying to separate the different ways to use the method will help the usability practitioner to more skillfully use the personas, as well as being open for alternative applications.

Within the field of Human-Computer Interaction, the persona method was originally introduced by Cooper [START_REF] Cooper | The inmates are running the asylum[END_REF], and he argued for hypothetical archetypes of real users in order to avoid designing systems that become too generic and in the end does not fit anyone. According to Cooper [START_REF] Cooper | The inmates are running the asylum[END_REF], the personas should be based on actual users and the personas should be precise and specific since it is more difficult to ignore a detailed persona than aggregated user data. The ideas is that numerous personas initially are created through an iterative process, and then these are condensed, according to their goals, into fewer but more precise, personas. One more important claim from Cooper is that even though multiple personas can be created, the developers should focus on one primary persona [START_REF] Cooper | The inmates are running the asylum[END_REF].

The extensive use of personas result in different views on what should be the basis of the personas. The most common argument is for collecting qualitative data through for example interviews and observations of real users [START_REF] Guðjónsdóttir | Personas and Scenarios in Use[END_REF][START_REF] Cooper | The inmates are running the asylum[END_REF][START_REF] Pruitt | Personas: practice and theory[END_REF]. However, the data underpinning the personas does not need to be based on ethnographic studies of real users; for example [START_REF] Faily | The secret lives of assumptions: Developing and refining assumption personas for secure system design[END_REF] describe personas within secure systems design based on assumptions. Quantitative data from for example surveys can also be used to statistically render personas, although these can later be refined by interviews and observations [START_REF] Sinha | Persona development for information-rich domains[END_REF]. Moreover, the widespread usage of the method in disparate settings and contexts has made the resulting personas in different shapes and forms. [START_REF] Floyd | Resolving incommensurable debates: a preliminary identification of persona kinds, attributes, and characteristics[END_REF] outline in their paper a loose typology of personas, however, the persona kinds that are described in their paper are not exhaustive; for example, other possibilities are assumption personas [START_REF] Faily | The secret lives of assumptions: Developing and refining assumption personas for secure system design[END_REF] or a collaboration persona [START_REF] Sinha | Persona development for information-rich domains[END_REF]. The persona method is also criticised, where the most alarming critique is that the personas are being misused and that this leads to designers distancing themselves from real users [START_REF] Portigal | Persona non grata[END_REF]. [START_REF] Portigal | Persona non grata[END_REF] argues that it is better to engage with users directly than to create a façade of user-centeredness [START_REF] Portigal | Persona non grata[END_REF]. Other types of critique are that the method is difficult to verify as (more) beneficial compared to other method [START_REF] Chapman | The personas' new clothes: methodological and practical arguments against a popular method[END_REF] or that it is inevitable that designers will create stereotypes [START_REF] Turner | Is stereotyping inevitable when designing with personas?[END_REF]. Furthermore there is literature that shows when personas have failed to work, such as the case that [START_REF] Blomquist | Personas in action: ethnography in an interaction design team[END_REF] present. The reasons for this was mainly because the design team was not familiar enough with the method and the interaction designers were not involved in the creation of the personas. [START_REF] Rönkkö | Personas is not applicable: local remedies interpreted in a wider context[END_REF] present a case where the persona method was abandoned in the context of developing software for mass-market mobile devices, although this mainly was because of the power and dominion of stakeholders outside the development organisation. Some recent studies have shown that the probability of personas being used is higher if the designer has participated in the creation of them [START_REF] Matthews | Collaboration personas: A new approach to designing workplace collaboration tools[END_REF][START_REF] Friess | Personas and decision making in the design process: an ethnographic case study[END_REF](. Furthermore, [START_REF] Markensten | Procuring a usable system using unemployed personas[END_REF] presents a case where personas were used outside the development project in which they were developed. In their case the educational department adopted the personas as a way to introduce newly employed to different clusters of customers.

Using Contextual Personas to Define Current Work Situation

In this section the background of the study is presented as well as the data gathering method used.

The Case Uppsala University

Uppsala University is a large Scandinavian research university with about 40 000 students and 6000 employees. The business administration department of the university handles the overall strategic economy at the university, whereas the different business administrators at the departments work with the daily economical work. The majority of the business administrators at the departments are women. The work of the business administrators at the different departments at the university are however organised in many different ways. Some work both with economy, human resource matters and study administration since they belong to very small departments, whereas others are very specialized in one area such as EU project economy and work with that full time with the support of a larger group of business administrators. Today, computers constitute the primary working tool for the business administrators at the universities, and hence comprise a major part of the work environment and procedures.

The study presented in this paper was a part of a collaborative project (KiA) between a research group of Human-Computer Interaction and the university administration [START_REF] Cajander | Things take time: Establishing usability work in a university context[END_REF]. The KiA project ran for two years, 2012-2013, and was coordinated by the university administration whereas the researchers did most intellectual contributions. It should be noted that the project was not a research project as such but the researchers were allowed to use the findings for scientific work beyond the project. The researchers worked within a participatory action research tradition [START_REF] Heron | The practice of co-operative inquiry: Research 'with'rather than 'on'people[END_REF][START_REF] Mckay | The dual imperatives of action research[END_REF] meaning that they were used to work closely with organisations, rather than observing without interfering.

Data Gathering

Contextual Inquiry

The data gathering was conducted from May to late August 2013. Field studies and interviews were conducted with 12 economics administrators at four different institutions in Uppsala University lasted approximately 2-3 hours. The field studies followed the method Contextual Inquiry and its four principles [START_REF] Holtzblatt | Rapid contextual design: a how-to guide to key techniques for user-centered design[END_REF]:

Studies of the work in its context. Field studies were conducted on site. Researchers are studying the users who do their tasks and discuss the systems used to solve them.

Cooperation. The user and researcher work together to understand the user's work. The researcher alternate between observing the user when they work and discuss what the user was doing and why.

Interpretation. The researcher share their interpretations and insights with the user during the interview. The user can expand or correct the researcher's understanding.

Focus. The researcher focus the conversation on topics that are relevant to the survey.

During the field studies the researcher took notes using pen and paper. The first field studies were made with an open mind to understand the work and the situation, but eventually the researcher asked the business administrator more specific questions as for example to show some particular parts of the systems or tasks.

Data Analysis and Sketches of Contextual Personas

The collected data was analyzed based on four categories from Karasek and Theorell's model of work [START_REF] Karasek | Healthy Work. Stress, Productivity and the Reconstruction of Working Life[END_REF], and the fourth category was the general working environment problems. Three researchers (three of the authors), analysed the data together in a workshop in August 2013 and wrote the first descriptions of the three personas together during the workshop which resulted in rough contextual personas. The researchers worked visually with the personas and made use of large white boards where the dimensions of healthy work were visualised together with descriptions of personas. After the workshop, the descriptions were reviewed individually by the researchers and modified. The illustrations of the personas were just modified images from Clip Art at this point.

Focus Groups

One researcher revisited two of the business administration departments, and arranged focus groups with four and three business administrators to discuss and reflect upon the presentations made in the personas. After the focus groups the personas were revised according to the reflections made by the business administrators. An illustra-tor was asked to draw the faces of each persona according to the descriptions at this point.

Results

Three Contextual Personas

Three personas were made based on the interviews and analysis of those. Each description was about one page of text describing the personal life, one day at work and the goals of the persona. Additionally, the need for control, support and the demands that the persona has are described. Each persona had a figure, illustrated by a professional illustrator. One example of a persona can be seen in figure 2.

Fig. 2. An example of a persona that was created using the contextual personas method

4.2

Reflections on the Contextual Personas Some really interesting points emerged during the focus group discussions that did not occur in the earlier interviews. The business administrators really liked the job descriptions made, and could identify themselves with the personas. Someone said, "It really feels like this person is working here". One of the things that one of the groups wanted to add was how life was affected by their work. In their mind private life and work forms one whole, and they wanted that to be better represented in the personas. They also gave concrete examples of situations where they need to adapt their working hours, their vacation and their weekends to meet the goals of different business administration deadlines. As a business administrator at the university in their opinion, you need to adapt life to work, and they often worked long hours before deadlines. One person described that new deadlines might occur in the calendar if researchers receive new research projects, and that these have individual deadlines for reporting. Especially EU projects mean a lot of work for the business administrators, who adapt their working hours according to the schedule of the projects. Some business administrators also complained about the current systems at the university that does not support flexible work, and some systems require that you are on the university network to be able to work. They also wanted more possibilities to work using their mobile devices.

Another interesting discussion that emerged was the lack of a persona that would represent the elderly generation of business administrators. This persona would describe the situation that technology has changed work so much that it is not the same at all. The persona would illustrate an elderly lady who has had problems understanding the new technology, and being terrified with the changes even though they are at the very core of their work. This persona has the feeling that her knowledge is not valid any more.

Discussion

The contextual personas were based on the theory of healthy work by Karasek and Theorell [START_REF] Karasek | Healthy Work. Stress, Productivity and the Reconstruction of Working Life[END_REF]. The theory was used in a very concrete way when designing the personas, and each aspect (control, support, demand) were discussed separately to ensure the quality of the descriptions. However, during the discussions it became clear that some aspect of the workplaces were not fully covered by the theory such as for example the aspect of interruptions and doing things in parallel. Other aspects that are not covered by the model that emerged in the discussions were the personality of the persona and how that affects the perceived stress. The contextual personas method could hence be further improved in the future to cover all aspects of working life through an conceptual development of the theoretical foundations of the process. One possible theoretical model that could be relevant to use to enhance the personas is the Effort-Reward Imbalance theory that was developed by Johannes Siegrist [START_REF] Siegrist | The measurement of effort-reward imbalance at work: European comparisons[END_REF]. This model is widely used in work environment research and would add aspects of intrinsic and extrinsic motivation, and overcommitment which is highly relevant in the context of new ways working.

In our contextual personas we describe the holistic work environment as it is today for the business administrators. The main difference in our way of describing personas compared to how Cooper defined personas [START_REF] Cooper | The inmates are running the asylum[END_REF] is that Coopers personas are used for improving one particular software system. The descriptions of these personas are aimed to describe their way of working solving the goals that the software system being developed will support in solving. Contextual personas are not focused on the usage of one system, they are focused on describing the whole context of work, so contextual personas could typically be using 20 software systems for solving various tasks at work. The usage of personas in new contexts is also argued in [START_REF] Eriksson | Situated Reflexive Change : User-Centred Design in(to) Practice[END_REF] where the author elaborates on how the users in the study started to more readily talk about their work situation when confronted with the personas that depicted themselves. This could be interpreted as the personas as a reflexive tool to be used as trigger material when talking with users, which in our cased highlighted the diverse and multifaceted work of the business administrators.

Contextual personas describe the current work situation, similar to Cooper's personas [START_REF] Cooper | The inmates are running the asylum[END_REF] and to Hackos and Redish's [START_REF] Hackos | User and task analysis for interface design[END_REF] scenarios. [START_REF] Hackos | User and task analysis for interface design[END_REF] describe two types of scenarios, "task scenarios" that describe the current situation for a persona and "use scenarios", that describe: "the future use of a computer system". Contextual personas could also be used in a similar way, using the contextual personas to give insights into future work environments. Previous research has show that user feedback is often informal and limited [START_REF] Hackos | User and task analysis for interface design[END_REF]. The contextual personas could work as a new human centred activity that would improve the quality of usability work for developers. In that case, the personas should be based on data from brainstorming sessions with users, so the descriptions will not be too hypothetical and superficial. [START_REF] Cooper | The inmates are running the asylum[END_REF] and [START_REF] Holtzblatt | Rapid contextual design: a how-to guide to key techniques for user-centered design[END_REF] have emphasised that personas are grounded in interview data from users and are not based on designers imagination. There is a need to develop and adapt current human centred activities to better address the future workplaces, and contextual personas shows promising results in this context. One should note that contextual personas are not recommended to be used in isolation, but together with other human centred activities to fully incorporate the essence of future work places. We conclude that the personas need to be complemented with a vision work concerning the future needs and visions of the users as well as other user centred activities 6

Fig. 1 .

 1 Fig. 1. The Demand-Control-Support Model of stress in a work situation. High job strain, i.e. high demands in combination with low decision latitude, and low social support are associated with the highest risks for health problems.

Acknowledgments

This study was made with financial support from Uppsala University Administration through the KiA project (Quality in Use). We would like to thank everyone that participated in the interview study, and those who have given valuable input to this paper. We would also like to thank the illustrator, Maria Osk Jonsdottir, for her professional work on illustrating the faces of the personas.