Philippe Moustrou
email: philippe.moustrou@uit.no

AND Cordian Riener
email: cordian.riener@uit.no

Hugues Verdure
email: hugues.verdure@uit.no

SYMMETRIC IDEALS, SPECHT POLYNOMIALS AND SOLUTIONS TO SYMMETRIC SYSTEMS OF EQUATIONS

Keywords: Symmetric group, Specht polynomials, polynomial equations

An ideal of polynomials is symmetric if it is closed under permutations of variables. We relate general symmetric ideals to the so called Specht ideals generated by all Specht polynomials of a given shape. We show a connection between the leading monomials of polynomials in the ideal and the Specht polynomials contained in the ideal. This provides applications in several contexts. Most notably, this connection gives information about the solutions of the corresponding set of equations. From another perspective, it restricts the isotypic decomposition of the ideal viewed as a representation of the symmetric group.

Introduction

Let S n denote the symmetric group on n-elements, and let K be a field. Then S n acts naturally on an n-dimensional space K n by permuting coordinates. This linear action then gives rise to an action on the corresponding polynomial ring by permuting coordinates. In this article we consider ideals I ⊂ K[X 1 , . . . X n ] which are stable under this action. The study of such ideals appears quite naturally in different contexts (see for example [21,[START_REF] Faugère | Solving polynomial systems globally invariant under an action of the symmetric group and application to the equilibria of n vortices in the plane[END_REF][START_REF] Krone | Equivariant gröbner bases of symmetric toric ideals[END_REF]4]).

Our interest in such ideals stems from algorithmic purposes: the symmetry on a set of equations often can be used to simplify its resolution. In this flavour, a fundamental result by Timofte (see [START_REF] Timofte | On the positivity of symmetric polynomial functions.: Part i: General results[END_REF][START_REF] Riener | On the degree and half-degree principle for symmetric polynomials[END_REF]) yields that every symmetric variety defined by polynomials of degree d is non-empty over R if and only if it contains a real point with at most d distinct coordinates. Here we aim at generalising this aspect of Timofte's result in various ways. We are able to show that -under the assumption that the number of variables is sufficiently large -the variety corresponding to the symmetric ideal will not contain any point with strictly more than d distinct coordinates. Moreover, our result yields that the set of possible configurations of these d distinct coordinates can be further restricted by the shape of the monomials of highest degree amongst the generators of I, see Section 5.1. The arguments put forward to establish this result are purely algebraic and work with no requirements on K. In fact, this result will follow from a study of Specht polynomials contained in symmetric ideals. More precisely, we assign a partition of n to these monomials and we relate these partitions to specific Specht polynomials which belong to the ideal, see Section 4.

In characteristic 0, this property also has an application to the structure of the decomposition of I in terms of S n -representations. The action of S n on I turns I and K[X 1 , . . . , X N ] I into K[S n ] modules. Over a field of characteristic zero this modules can be decomposed into irreducible K[S n ] modules, which are usually called Specht modules and we show in Section 5.2 that the possible Specht modules appearing in this decomposition are also very restricted (see [START_REF] Basu | On the isotypic decomposition of cohomology modules of symmetric semi-algebraic sets: polynomial bounds on multiplicities[END_REF] for a result in a similar spirit in the real setting). One application of such a decomposition concerns sums of squares representations of positive symmetric polynomials modulo symmetric ideals. The understanding of the irreducible representations in I gives a control on the complexity of sums of squares decomposition in this setup, see Section 5. 3. This article is structured as follows. Section 2 collects necessary standard notations and definitions. Then, Section 3 focuses on varieties defined by Specht polynomials and their properties. This is used in Section 4 to describe the Specht polynomials contained in symmetric ideals. Finally, Section 5 is devoted to applications.

Preliminary notations and definitions

Partitions and Young tableaux. For any natural number n, one can consider its partitions:

Definition 1. Let n ∈ N. A partition λ of n, (denoted λ ⊢ n) is a sequence λ = (λ 1 , ⋯, λ l ) of positive natural numbers ordered such that λ 1 ⩾ λ 2 ⩾ ⋯ ⩾ λ l ⩾ 0 with the property λ 1 + . . . + λ l = n. The length of λ is len(λ) = max{i ∶ λ i ≠ 0}.
We will allow ourselves to identify partitions that only differ by 0 terms. Definition 2. For a given partition λ, its dual partition λ ⊥ is defined by

(λ ⊥ ) i = {j, λ j ⩾ i} .
Partitions are very well known and are closely related to Young tableaux (see e.g. [START_REF] Sagan | The symmetric group: representations, combinatorial algorithms, and symmetric functions[END_REF]). Definition 3. Given λ ⊢ n, a Young tableau of shape λ ⊢ n, or a λ-tableau consists of len(λ) rows, with λ i entries in the i-th row. Each entry is an element in {1, . . . , n}, and each of these numbers occurs exactly once. Definition 4. Let n ∈ N. Let λ = (λ 1 , ⋯, λ l ) ⊢ n and µ = (µ 1 , ⋯, µ m ) ⊢ n be two partitions. We say that λ dominates µ if i j=1 λ j ⩾ i j=1 µ j holds for all 1 ⩽ i ⩽ min{len(λ), len(µ)}.

We will write λ ⊵ µ in this case.

Equipped with the dominance order, the set of all partitions of a given n ∈ N is a partially ordered set. We will further use λ ⊵ ν to denote the case in which λ does not dominate ν. Note that the order is only partial and hence this does not entail that ν dominates λ, since they also might be not comparable. Furthermore, note that (λ ⊥ ) 1 = len(λ), (λ ⊥ ) ⊥ = λ, and

λ ⊵ µ ⇔ µ ⊥ ⊵ λ ⊥ .
Orbit types and partitions. The action of S n on K n naturally decomposes the space into orbits.

Definition 5. Let x ∈ K n . The orbit type of x is the conjugacy class of the associated stabilizer subgroup Stab(x) ⊆ S n . Since, up to conjugation,

Stab(x) = S 1 × S 2 × ⋯ × S k with 1 ⩾ 2 ⩾ . . . ⩾ k , the orbit type of x is characterized by Λ(x) ∶= ( 1 , 2 , ⋯, k ).
Then, for a given λ ⊢ n we can define

H λ ∶= {x ∈ K n ∶ Λ(x) = λ} .
Remark 1. Note that we have

K n = ⊍ λ⊢n H λ .
Polynomials, varieties, and symmetric ideals. Let K be a field, and consider the polynomial ring in n variables K[X 1 , ⋯, X n ]. For any P ∈ K[X 1 , ⋯, X n ], we denote by Mon(P ) the set of monomials appearing in P , and by P h its homogenous component of degree h. Given a monomial

m = n j=1 X kj j ,
its support is Supp(m) = {j, k j ≠ 0} and its weight wt(m) is the cardinality of its support. By taking the union over all the monomials in Mon(P ), we generalize these notions to P to define Supp(P ) and wt(P ).

To every ideal in K[X 1 , . . . , X n ] one can associate a variety: Definition 6. Let I be an ideal in K[X 1 , . . . , X n ]. The variety V (I) associated with I is the subset of K n made by the common zeros of all the polynomials in I, namely V (I) = {x ∈ K n , P (x) = 0 for every P ∈ I}.

The action of S n on K n induces an action on K[X 1 , . . . , X n ] by permuting the variables. We denote by σP the image of a polynomial P under the action of a permutation σ. Definition 7. An ideal I ⊂ K[X 1 , . . . , X n ] is called a symmetric ideal if for every P in I and for every σ ∈ S n , σP belongs to I.

Note that if I is a symmetric ideal, then the variety V (I) is closed by the action of S n on the coordinates. Specht Polynomials. The so-called Specht polynomials will play a central role in our proofs. Those polynomials were originally designed by Specht [START_REF] Specht | Die irreduziblen darstellungen der symmetrischen gruppe[END_REF] to construct the different irreducible representations of S n . Definition 8. Let n ∈ N.

(1) For a set S = {i 1 , . . . , i r } ⊂ {1, . . . , n}, we define the Vandermonde determinant ∆(S) of the variables X i , for i ∈ S:

∆(S) ∶= 1⩽j<k⩽r (X ij -X i k ).
(2) Let λ ⊢ n and T be a λ-tableau. Then the Specht polynomial associated with T is the polynomial

sp T ∶= c ∆(T ⋅,c ),
where c runs through the columns of T , and T ⋅,c denotes the entries in the cth column. We will say that sp T is a Specht polynomial of shape λ.

Example 1. The Specht polynomial associated with the Young tableau

4 2 6 1 8 7 5 3 is (X 4 -X 8 )(X 4 -X 3 )(X 8 -X 3 )(X 2 -X 7 )(X 6 -X 5 ).

Zeros of Specht polynomials

Throughout the paper, we will be interested in the ideal generated by all the Specht polynomials of a given shape, as well as in the corresponding variety. Definition 9. Let K be a field, n be an integer, and µ ⊢ n.

• The µ-Specht ideal, denoted by I sp µ , is the ideal of K[X 1 , . . . , X n ] generated by all the Specht polynomials of shape µ.

• We denote by V µ the set of common zeros of all Specht polynomials of shape µ, that is

V µ ∶= V (I sp µ ) = ⋂ sh(T )=µ V (sp T ).
We aim at describing more precisely those varieties. Particular cases of these varieties have been studied for example in [START_REF] Yanagawa | When is a specht ideal cohen-macaulay?[END_REF][START_REF] Watanabe | Vandermonde determinantal ideals[END_REF]7]. Let us start with a few remarks. Let T be a Young tableau, of shape µ ⊢ n. A point x = (x 1 , . . . , x n ) is a zero of sp T if and only if there exists a column of T containing two indexes i ≠ j such that x i = x j . Following this easy observation, we get a characterization of V µ that will be useful later: a point x ∈ K n does not belong to V µ if and only if one can fill in a Young tableau of shape µ with the coordinates of x in such a way that in every column, all the values are distinct. Now we want to prove that V λ contains all the V µ such that µ ⊵ λ. This is a consequence of the corresponding inclusion of ideals:

Theorem 1. Let λ ⊢ n. Then for any µ ⊢ n such that µ ⊵ λ, I sp λ ⊂ I sp µ . In particular V µ ⊂ V λ .
Remark 2. Note that the inclusion of ideals is a little bit stronger statement than the inclusion of varieties, since it is not known whether Specht ideals are radical, see [START_REF] Yanagawa | When is a specht ideal cohen-macaulay?[END_REF].

Proof. Let λ = (λ 1 , . . . , λ t ). We only need to consider the particular case where µ is of the form

µ = (λ 1 , . . . , λ i-1 , λ i + 1, λ i+1 , . . . , λ j-1 , λ j -1, λ j+1 , . . . , λ t ),
where λ i-1 > λ i and λ j > λ j+1 , in order to ensure that µ is a partition. Indeed, it is known that we can go from λ to any µ ⊵ λ by a finite number of such steps, see e.g. [3, Prop.

2.3].

Let T be a Young tableau of shape λ. We need to show that sp T belongs to the ideal generated by all the sp U with sh(U ) = µ. If U is a Young Tableau of shape µ, then its columns have the same number of elements than T , except for two of them. Let U 1 and U 2 be these columns, and let T 1 and T 2 be the corresponding columns in Consider the polynomial

T . If a = T 1 and b = T 2 , then U 1 = a -1, U 2 = b + 1 and a -b ⩾ 2.
Q = ∆({2, . . . , a})∆({1, a + 1, . . . , a + b})X a-b-1 1 ,
and the polynomial

R = a i=1 ((1, i))(1, i)Q,
where denotes the signature. By construction, R is in the ideal I. We need to prove that for any pair 1 ⩽ α < β ⩽ a and a

+ 1 ⩽ α < β ⩽ a + b, R α,β = R(X 1 . . . , X β-1 , X α , X β+1 , . . . , X n ) = 0.
The latter case is obvious by definition of Q and R. Suppose 1 ⩽ α < β ⩽ a. Then for every i ≠ α, β we have

((1, i)Q) α,β = 0, so that R α,β = ((1, α))((1, α)Q) α,β + ((1, β))((1, β)Q) α,β .
Now, we have do distinguish two cases, namely α = 1 and α ≠ 1. In the first case, we have

R 1,β = Q 1,β -((1, β)Q) 1,β =
0 while in the second case, the signatures are both negative, but the polynomial differ from each other by interchanging the variables X 1 and X α in places α and β, and by definition of Q, these polynomials are opposite of each other, that is,

R α,β = 0. This implies that P = ∆({1, . . . , a})∆({a + 1, . . . , a + b}) divides R. Since the degree of R satisfies deg(R) ⩽ deg(Q) = (a -1)(a -2) 2 + (b + 1)b 2 + a -b -1 = a(a -1) 2 + b(b -1) 2 = deg(P ),
it implies that R = cP with c ∈ K, and we need to check that c is not zero. The degree of Q in the variable

X 1 is b + a -b -1 = a -1 while its degree in the variable X i for 2 ⩽ i ⩽ a is a -2.
Thus, the degree of R in the variable X 1 is exactly a -1 and the corresponding coefficient is ∆({2, . . . , a})∆({a + 1, . . . , a + b}), which is also the coefficient of X a-b 1 in P . Hence R = P . Finally, we get a characterization of V µ in terms of orbit types: Theorem 2. The set of common zeros of Specht polynomials associated with Young tableaux of given shape µ can be characterised as

V µ = ⎛ ⎝ ⋃ λ⊴µ H λ ⎞ ⎠ c = ⋃ ν ⊴µ H ν
Proof. We want to show V c µ = ⋃ λ⊴µ H λ . Take x not in V µ . This means that there exists a Young tableau T of shape µ such that sp T (x) ≠ 0. Let λ = Λ(x) = ( 1 , . . . , r ). Let u 1 , . . . , u r be the distinct coordinates of x, where for each 1 ⩽ i ⩽ rr, u i appears l i times. Since sp T (x) ≠ 0, we can fill T with u 1 , . . . , u r in such a way that for every 1 ⩽ i ⩽ r, u i appears at most once per column, and we may assume without loss of generality that in every column, if u j is below u i , then i < j. As a consequence, for every 1 ⩽ k ⩽ r, the u i for i ⩽ k are contained in the first k rows of T . This means that Conversely, let x ∈ K n , let λ = Λ(x) = ( 1 , . . . , r ), and let u 1 , . . . , u r be the distinct coordinates of x, where for each 1 ⩽ i ⩽ r, u i appears l i times. Let T be any Young tableau of shape λ such that the indexes in the ith row are the indexes j such that x j = u i . Then sp T (x) ≠ 0, and hence x is outside V λ . As a consequence, if λ ⊴ µ, then according to Theorem 1, V µ ⊂ V λ , so that x ∈ V c µ .

Specht polynomials in symmetric ideals

In this section we show that if a symmetric ideal contains polynomials with sparse leading component, then this ideal will contain many Specht polynomials. Let us be more precise: First, to every monomial we associate a partition: Definition 10. Let be a monomial of weight l and degree d in K[X 1 , . . . , X n ]. The partial degrees of m induce a partition of d of length l, say (λ 1 , . . . , λ l ).

If moreover we assume that l + d ⩽ n, we can define a partition µ(m) of n by

µ(m) = (λ 1 + 1, λ 2 + 1, . . . , λ l + 1, 1, . . . , 1 n-d-l
).

Example 2. Let n = 12, and

m = X 2 X 4 4 X 2 5 . Then µ(m) = (5, 3, 2, 1, 1).
Then we will show:

Theorem 3. Let I ⊂ K[X 1 , .
. . , X n ] be a symmetric ideal. Assume that there exists P ∈ I of degree d, such that d+wt(P d ) ⩽ n. Then, for every monomial m ∈ Mon(P d ), the ideal I contains every sp T for which sh(T ) ⊴ µ(m) ⊥ .

According to Theorem 1, it is enough to prove that I contains the Specht polynomials of shape µ(m) ⊥ . Hence we only need to prove: Theorem 4. Let I ⊂ K[X 1 , . . . , X n ] be a symmetric ideal. Assume that there exists P ∈ I of degree d, such that d+wt(P d ) ⩽ n. Then, for every monomial m ∈ Mon(P d ), the ideal I contains every sp T for which sh(T ) = µ(m) ⊥ .

In the following proof, we will assume that the characteristic of K is zero. This allows for more conceptual proof. We provide a general proof in the Appendix.

Proof. Let K be a field of characteristic 0. Since the ideal I is symmetric, we may assume that m = X k1 1 X k2 2 ⋯X k l l . and that Supp(P d ) = {1, . . . , wt(P d )}. Its associated partition is

µ ∶= µ(m) = (k 1 + 1, k 2 + 1, . . . , k l + 1, 1, . . . , 1 n-d-l
).

The statement says that the ideal I contains any Specht polynomial of the form

∆ 1 ∆ 2 ⋯∆ l ,
where the ∆ i are Vandermonde polynomials in disjoint sets of variables, each of size k i + 1 = µ i . Thanks to its symmetry, it is enough to show that I contains one such polynomial.

Our strategy consists in using X ki i to build a Vandermonde polynomial involving X i and k i variables that do not appear in P d . Our assumption on P d guarantees that there are enough free variables to do so.

More precisely, we can take I 1 , . . . , I l , disjoint subsets of {1, . . . , n} such that for any 1 ⩽ i ⩽ l, there are k i elements in I i , and none of them appears in P d . Let, for 1 ⩽ i ⩽ l, J i = {i} ∪ I i .

We will show that there exist polynomials R σ ∈ K[X 1 , . . . , X n ], for σ ∈ S n such that: ∆(J 1 )⋯∆(J l ) = σ∈Sn R σ σP.

Here, applying the strategy used to prove Theorem 1 we give explicit polynomials R σ when the characteristic of K is 0. In the general case, we can give a recursive construction of these polynomials; we postpone this construction to Appendix A. Consider the polynomials

Q = ∆(I 1 )⋯∆(I l )P and R = σ∈S J 1 ×⋯×S J l (σ)σQ,
where S J1 × ⋯ × S J l is seen as a subgroup of S n . By construction, for every ρ ∈

S J1 × ⋯ × S J l , ρR = (ρ)R so that ∆(J 1 )⋯∆(J l ) divides R. Furthermore, since deg(Q) = d + l i=1 k i (k i -1) 2 = l i=1 k i (k i -1) 2 + k i = l i=1 k i (k i + 1) 2 = deg(∆(J 1 )⋯∆(J l )), we get R = c∆(J 1 )⋯∆(J l )
with c ∈ K. In order to check that c is not 0, we look at the coefficent of R corresponding with m = X k1 1 ⋯X k l l , seen as an element of (K[X l+1 , . . . , X n ])[X 1 , . . . , X l ]. In Q, this coefficient is ∆(I 1 )⋯∆(I l ). If, for 1 ⩽ i ⩽ l, the permutation σ ∈ S J1 × ⋯ × S J l does not let i invariant, the assumption on P d ensures that the coefficient of m in σQ will be 0. Therefore, the coefficient of R corresponding with m is

σ∈S I 1 ×⋯×S I l (σ)σ∆(I 1 )⋯∆(I l ) = k 1 !⋯k l !∆(I 1 )⋯∆(I l )
and hence R = k 1 !⋯k l !∆(J 1 )⋯∆(J l ).

Applications

5.1.

Computing points in symmetric varieties. Let n be an integer and I be a symmetric ideal in K[X 1 , . . . , X n ]. What can we say about the variety V (I)? For instance, can we algorithmically decide if V (I) is empty in an efficient manner making use of the structure of I?

Over any real closed field R the so-called half-degree principle [START_REF] Timofte | On the positivity of symmetric polynomial functions.: Part i: General results[END_REF] can be used to simplify the algorithmical task of root finding. This statement says ([16, Corollary 1.3]): Theorem 5. Let K be a real closed field, and let P be a symmetric polynomial in K[X 1 , . . . , X n ] of degree d, and let k = max(2, ⌊ d 2 ⌋). Then there exists x ∈ K n such that P (x) = 0 if and only if there exists y ∈ K n with at most k distinct coordinates such that P (y) = 0. This implies the following result on symmetric ideals: Corollary 1. Let K be a real closed field, and let I be a symmetric ideal of K[X 1 , . . . , X n ], generated by P 1 , . . . , P l . Let d = max(deg(P 1 ), . . . , deg(P l )). Then V (I) is non empty if and only if it contains a point x ∈ K n with at most d distinct coordinates.

Proof. Over a real closed field the variety V (I) is exactly the variety defined by

Q = l i=1 σ∈Sn σ(P i ) 2
and we can apply Theorem 5.

The algorithmic implications of this result are the following. Consider µ ⊢ n of length d. For every polynomial P ∈ K[X 1 , . . . , X n ] we consider ).

This map is clearly surjective and thus Corollary 1 says precisely that V (I) is empty if and only if V (I µ ) is empty for every µ partition of n with len(µ) ⩽ d. Since the number of d-partitions of n is bounded by (n + 1) d the original problem in n variables reduces to a polynomial number of problems in d variables.

Our result yields a stronger version of this principle, under additional assumption on the support of the polynomials: On the one hand our results are valid for any field, and on the other hand, not only our varieties contain points with few distinct coordinates, but they contain only points with few distinct coordinates. More precisely, Theorem 3 gives, in this context: Theorem 6. Let I ⊂ K[X 1 , . . . X n ] be a symmetric ideal. Assume that there exists P ∈ I of degree d such that wt(P d ) + d ⩽ n and let m ∈ M on(P d ). Then

V (I) ∩ H λ = ∅ for all λ ⊴ µ(m) ⊥ .
In other words,

V (I) S n = ⋃ ν ⊴µ(m) ⊥ Φ ν (V (I ν )).
Hence, if one is able to compute the points in the variety V (I ν ), one gets all the points of V (I). Also note that the length of the partitions ν is at most d, this comes from the following observation: Lemma 1. Let n be an integer, and m be a monomial of degree d, with d+wt(m) ⩽ n. Then for every λ partition of n such that len(λ) > d, µ(m) ⊥ ⊵ λ.

Proof. The proof consists in two steps. First we prove that

µ(m) ⊥ ⊵ (n -d, 1, . . . , 1 d ). Indeed, if m = X k1 1 ⋯X k l l , µ(m) = (k 1 + 1, k 2 + 1, . . . , k l + 1, 1, . . . , 1 n-d-l
)

has length n -d, so that µ(m) ⊥ = (n -d, . . .) ⊵ (n -d, 1, . . . , 1 d ). Second, if len(λ) > d, then (n -d, 1, . . . , 1 d ) ⊵ λ.
Indeed, if not, there exists j such that

j i=1 λ i > (n -d) + (j -1).
In this case, since len(λ)

⩾ d + 1, n = len(λ) i=1 λ i ⩾ j i=1 λ i + len(λ) -j > n -d + j -1 + len(λ) -j > n.
Remark 3. The above Lemma shows that we can always ensure that every point of the variety V (I) has at most d distinct points. If the monomial m is X d i then µ(m) ⊥ = (nd, 1, . . . , 1) and we need to consider all d-partitions of n. However, depending on the actual monomials of highest degree in the generators of I a finer analysis can even reduce the number of partitions that need to be considered. Hence we get a stronger version of the degree principle.

One further natural consequence is that our variety is contained in a union of d-dimensional subspaces, hence: Corollary 2. Let I ⊂ K[X 1 , . . . , X n ] be a symmetric ideal. Assume that there exists P ∈ I of degree d, such that d + wt(P d ) ⩽ n. Then the dimension of the variety V (I) is at most d.

In a more general setup, Nagel and Römer [START_REF] Nagel | Equivariant hilbert series in non-noetherian polynomial rings[END_REF] study sequences of symmetric ideals. They show in particular that the dimension of the ideals they study is a linear function in n. In our more restricted framework, we thus obtain a stabilization of the dimension of such sequences. 5.2. Isotypic components of symmetric ideals. The action of the symmetric group S n on K[X 1 , . . . , X n ] is linear, giving the polynomial ring the structure of a K[S n ]-module. If we assume that the characteristic of K is 0, then every K[S n ]module can be decomposed as a direct sum of irreducible submodules. It is well known (see e.g. [START_REF] Sagan | The symmetric group: representations, combinatorial algorithms, and symmetric functions[END_REF]) that the irreducible K[S n ]-modules are in correspondence with the partitions of n. These modules are called Specht modules, denoted by S λ . It follows that every K[S n ]-module U can be uniquely written as

U ≃ ⊕ λ⊢n U λ ,
where for every partition λ of n, U λ is a direct sum of irreducible submodules isomorphic to S λ . The submodule U λ is called the λ-isotypic component of U . Now let I ⊂ K[X 1 , . . . , X n ] be a symmetric ideal. It is also a K[S n ]-module, and we have, for every partition λ of n,

I λ = K[X 1 , . . . , X n ] λ ∩ I.
Let λ be a partition of n, then the linear subspace W λ of K[X 1 , . . . , X n ] generated by all the Specht polynomials of shape λ is an irreducible submodule of K[X 1 , . . . , X n ] λ isomorphic to S λ . For any other irreducible submodule W λ in K[X 1 , . . . , X n ] λ , we have an isomorphism ϕ between W λ and W λ . Let T be a Young tableau of shape λ. Since ϕ respects the action of S n , for any τ transposition of two elements in a same column of T , τ ϕ(sp T ) = -ϕ(sp T ), so that ϕ(sp T ) has to be divisible by sp T . It follows that K[X 1 , . . . , X n ] λ is included in the Specht ideal I sp λ , and therefore Theorem 3 gives: Theorem 7. Let K be a field of characteristic 0 and I ⊂ K[X 1 , . . . , X n ] be a symmetric ideal. Assume that there exists P ∈ I of degree d, such that d+wt(P d ) ⩽ n. Then for every m ∈ Mon(P d ), for every λ ⊴ µ(m) ⊥ , the ideal I contains the isotypic component K[X 1 , . . . , X n ] λ . In other words

I λ = K[X 1 , . . . , X n ] λ , or equivalently (K[X 1 , . . . , X n ] I) λ = {0}.
Given polynomials P 1 , . . . , P l ∈ K[X 1 , . . . , X n0 ], they naturally induce a symmetric ideal in K[X 1 , . . . , X n ] for any n ⩾ n 0 . We get an increasing sequence of symmetric ideals (I) n⩾n0 , and one can study stabilization properties in terms of representations (see for instance [START_REF] Sam | Stability patterns in representation theory[END_REF]5]).

We remark that when n is large enough, the condition on the support of the leading component of P is automatically fulfilled and we can immediately deduce: Theorem 8. Let K be of characteristic 0 and n 0 be an integer. Given Q 1 , . . . , Q l in K[X 1 , . . . , X n0 ], consider for any n ⩾ n 0 the ideal

I n = ⟨σ(Q i ), σ ∈ S n , 1 ⩽ i ⩽ l⟩.
Then if n is large enough, for every 1 ⩽ i ⩽ l, every monomial m of Q i of maximal degree, and any λ partition of n such that λ ⊴ µ(m) ⊥ , (K[X 1 , . . . , X n ] I n ) λ = {0}. 5.3. Symmetric sums of squares on symmetric varieties. As a final application we consider sums of squares of real polynomials. Let P ∈ K[X 1 , . . . , X n ]. Then P is called a sum of squares, if there exists polynomials P 1 , . . . , P k with P = P 2 1 + . . . + P 2 k . Sum of squares are the cornerstone in the so called moment approach to polynomial optimization [START_REF] Lasserre | Global optimization with polynomials and the problem of moments[END_REF]: In general, it can be decided by semidefinite programming if a given polynomial affords a decomposition as a sum of squares. The case of symmetric sums of squares has received some interest by different authors [START_REF] Blekherman | Symmetric nonnegative forms and sums of squares[END_REF]8,15,17,11].

In [START_REF] Blekherman | Symmetric nonnegative forms and sums of squares[END_REF], Blekherman and the second author described how to characterize symmetric sums of squares through representation theory. More precisely, they use the theory of higher Specht polynomials [22] to construct, for every λ ⊢ n, a square matrix polynomial Q λ of size s λ = dim(S λ ), whose entries are symmetric polynomials. Furthermore, these entries are products and sums of elements in R[X 1 , . . . , X n ] λ .

So even though they are symmetric, they belong to the ideal generated by the Specht polynomials of shape λ. These matrices can be used to show that every symmetric polynomial P that is a sum of squares can be written in the form

P = λ⊢n Tr(P λ ⋅ Q λ ),
where each P λ ∈ R[X 1 , . . . , X n ] s λ ×s λ is a sum of symmetric squares matrix polynomial, i.e.

P λ = L t L for some matrix L whose entries are symmetric polynomials.

Since the λ-Specht ideal contains all the coefficients of Q λ we can apply Theorem 7 to obtain the following result on representations of a symmetric polynomial modulo a symmetric ideal.

Theorem 9. Let P ∈ R[X 1 , . . . , X n ] Sn be a symmetric sum of squares polynomial and I be a symmetric ideal in R[X 1 , . . . , X n ]. Further, we assume that there exists F ∈ I of degree d, such that d + wt(F d ) ⩽ n. Then P can be written as

P = ν ⊴µ(m) ⊥ Tr(P λ ⋅ Q λ ) mod I,
where again each P λ = L t L for some matrix polynomial L whose entries are symmetric polynomials.

A case of special interest is the case when I is the gradient ideal I grad (P ) of a given polynomial P of even degree 2d. Sturmfels and Nie [START_REF] Nie | Minimizing polynomials via sum of squares over the gradient ideal[END_REF] showed that a polynomial that is positive on its gradient variety V (I grad ) can always be written as a sum of squares modulo its gradient ideal. When P is a symmetric polynomial our results can be applied to reduce the problem size. Building on the ideas of considering the gradient ideal, Jibetean and Laurent [START_REF] Jibetean | Semidefinite approximations for global unconstrained polynomial optimization[END_REF] considered the following perturbation of a polynomial:

P ε ∶= ε ⋅ (X 2d+2 1 + . . . + X 2d+2 n ) + P.
Since the perturbation term is positive definite and of higher degree, a global minimum of P converges to a global minimum of P , when ε → 0. Furthermore, in this case the quotient R[X 1 , . . . , X n ] I grad (P ε ) is generated by the polynomials 2dX 2d-1 i + ∂P ∂Xi , for 1 ⩽ i ⩽ n. Thus the quotient is a finite dimensional vector space and contained in the residues of all polynomials of degree at most 2d -1. This implies in particular, that both the number and the sizes of the matrices in Theorem 9 can be restricted further (see [START_REF] Blekherman | Symmetric nonnegative forms and sums of squares[END_REF]Proposition 4.1] and are in fact independent of n whenever n > 2d + 1. In order to use Theorem 9 practically to decide if a polynomial is a sum of squares of polynomials, one uses semi-definite programming. The complexity of such a program is mainly determined by the size of the matrices used to define it. Therefore our discussion actually shows that checking if a symmetric polynomial P ε ⩾ 0 for an ε > 0 can be done in a complexity which does not depend on n.

We will show the existence of polynomials R σ , only involving the variables Z i,s for 1 ⩽ i ⩽ l, such that ∆(J 1 )⋯∆(J l ) = and each factor acts on the corresponding subset of variables, namely:

σZ i,s = Z i,σi(s) .
Let us show this by induction on the degree d of P . If d = 0, there is nothing to prove. Now assume d > 1. Up to a rescaling, we may assume that

P = m + S,
where S is a polynomial of degree at most d such that m ∈ Mon(S) and S d does not contain any variable Z i,s with s > 1.

Let τ be the transposition exchanging Z l,1 and Z l,µ l . Then

P -τ P = (Z l,1 -Z l,µ l )Q
where Q is a polynomial of degree d -1. Because Z l,µ l does not appear in S d , we can write

Q = m ′ + S ′ with m ′ = ∏ l-1 i=1 Z ki i,1 Z k l -1 l,1
and S ′ is a polynomial of degree at most d -1. Since the only new variable appearing in Q d is Z l,µ l , we have d -1 + wt(Q d ) ⩽ d + wt(P d ) ⩽ n, in such a way that we can apply the induction hypothesis on Q. This provides polynomials R ′ ρ , only involving the variables Z i,s for 1 ⩽ i ⩽ l, except Z l,µ l , such that ⎛ ⎝ l-1 i=1 1⩽s<t⩽µi

(Z i,s -Z i,t ) ⎞ ⎠ 1⩽s<t⩽µ l -1 (Z l,s -Z l,t ) = ρ∈S ′ R ′ ρ ρQ,
where

S ′ = S µ1 × S µ2 × ⋯ × (S µ l -1 × S 1 ) × S 1 × ⋯ × S 1 n-d-l
can be seen as a subgroup of S µ . Since any ρ ∈ S ′ leaves the product ∏ µ l -1 s=1 (Z l,s -Z l,µ l ) unchanged, we have

l i=1 1⩽s<t⩽µi (Z i,s -Z i,t ) = µ l -1 s=1 (Z l,s -Z l,µ l ) ρ∈S ′ R ′ ρ ρQ = ρ∈S ′ R ′ ρ ρ µ l -1 s=1 (Z l,s -Z l,µ l ) Q = ρ∈S ′ R ′ ρ µ l -1 s=2
(Z l,ρ l (s) -Z l,µ l )ρ (Pτ P ) .

  By restricting our attention to these two columns, it is enough to prove, up to permutation, that the polynomial P = ∆({1, . . . , a})∆({a + 1, . . . , a + b}) is in the ideal I of K[X 1 , . . . , X a+b ] generated by all the polynomials of the form ∆(S 1 )∆(S 2 ), with S 1 = a -1, S 2 = b + 1, and S 1 ∪ S 2 = {1, . . . , a + b}.

P

  µ ∶= P (Z 1 , . . . , Z 1 µ1 , Z 2 , . . . , Z 2 µ2 , . . . , Z d , . . . , Z d µ d ) and denote by I µ ⊂ K[Z 1 , . . . , Z d ] the resulting ideal in d variables. In other words, consider the topological closure H µ of H µ and the map Φ µ ∶ V (I µ ) → (V (I) ∩ H µ ) S n which associates to a point x = (x 1 , . . . , x d ) ∈ V (I µ ) the S n -orbit of the point x = (x 1 , . . . , x 1 µ1 , x 2 , . . . , x 2 µ2 , . . . , x d , . . . , x d µ d

(

  Z i,s -Z i,t ) = σ∈Sµ R σ σP,whereS µ = S µ1 × S µ2 × ⋯ × S µ l × S 1 × ⋯ × S 1 n-d-l

+ . . . + k ⩽ µ 1 + . . . + µ k , which implies λ ⊴ µ.

This work has been supported by European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement 813211 (POEMA) and the Tromsø Research foundation grant agreement 17matteCR. .

Appendix A. Proof of Theorem 4 in general characteristic

Recall that we want to show that there exist polynomials R σ ∈ K[X 1 , . . . , X n ], for σ ∈ S n such that:

In order to avoid an overload of notation we rename the variables in the following way:

where we identify Z i,1 with X i for 1 ⩽ i ⩽ len(µ). In this way whenever s > 1 then Z i,s does not appear in P d . In this setting, the monomial m is written as

Because ρτ ∈ S µ , we can rewrite this expression as ∑ σ∈Sµ R σ σP with the desired properties.


Morel

Q Molin

M Kanoun

C Raynaud

H Morel

Measurement and Analysis of SiC-MOSFET threshold voltage shift

Measurement and analysis of SiC-MOSFET threshold voltage shift

Introduction

Thanks to its superior electrical properties compared to Silicon, SiC is a promising material for high voltage and high temperature devices. However there are still a lot of reliability issues that remain to be understood such as oxide degradation [START_REF] Ouaida | Gate Oxide Degradation of SiC MOSFET in Switching Conditions[END_REF], threshold voltage instability [START_REF] Lelis | Basic Mechanisms of Threshold-Voltage Instability and Implications for Reliability Testing of SiC MOSFETs[END_REF], [START_REF] Kikuchi | Modeling the threshold voltage instability in SiC MOSFETs at high operating temperature[END_REF] and short-circuit behaviours [START_REF] Berthou | Repetitive Short-Circuit tests on SiC VMOS devices[END_REF], [START_REF] Castellazzi | Short-circuit tests on SiC power MOSFETs[END_REF] and [START_REF] Chen | Robustness in short-circuit Mode of SiC MOSFETs[END_REF]. Some of these key points are crucial to develop reliable power devices for industrial applications [START_REF] Maneiro | Design of a SiC based triple active bridge ceil for a multimegawatt DC-DC converter[END_REF].

Reliability standards regarding ageing such components [START_REF] Jedec | Temperature, Bias, and Operating Life -22A108D[END_REF] have been written for Si power switches. But SiC transistors have not the same behaviour than Si devices during ageing [START_REF] Yu | Reliability Issues of SiC MOSFETs: A Technology for High-Temperature Environments[END_REF]. So the scientific community is forced to question whether these standards are suitable for SiC devices or not and how they should be updated [START_REF]ECPE SiC & GaN User Forum: Potential of Wide Bandgap Semiconductors in Power Electronic Applications[END_REF]. Meanwhile, standard tests are still a known base to provide more appropriate ageing test in a near future for SiC devices. This paper studies the gate instability behaviour in several ageing conditions: static HTGB (High Temperature Gate Bias) [START_REF] Lelis | SiC MOSFET reliability and implications for qualification testing[END_REF] and dynamic HTGS (High Temperature Gate switching) tests [START_REF] Ouaida | Gate Oxide Degradation of SiC MOSFET in Switching Conditions[END_REF], [START_REF] Fayyaz | High temperature pulsed-gate robustness testing of SiC power MOSFETs[END_REF]. The test bench designed for such reliability tests is also presented, along with the conditions of measurements of the threshold voltage VTH, which is a key parameter. Important findings about its instability have been already discussed [START_REF] Lelis | Basic Mechanisms of Threshold-Voltage Instability and Implications for Reliability Testing of SiC MOSFETs[END_REF], [START_REF] Lelis | SiC MOSFET reliability and implications for qualification testing[END_REF]. This article proposes a protocol to get rid of the temporary To the best of our knowledge, no ageing data on industrial 1700V 45 m SiC MOSFET (Metal Oxide Semiconductor Field effect Transistor) have been reported so far.

Experimental set-up

Test bench and ageing test definition

The test bench presented on Fig. 1 (a) has been designed to run both static (HTGB) and dynamic (HTGS) ageing tests on up to 18 MOSFETs at the same time. The hotplate enables a maximal temperature spreading of 2 °C. HTGB+ (resp. -) consists in ageing the device by applying a positive (resp. negative) bias on the gate. Because this test is the most used in the industrial world, we decided to start with it in this study.

HTGS consists in ageing the device by applying a pulsed bias on the gate (switching alternatively from V GS Low to V GS High ), with a given duty cycle. This test has become more and more common in recent reliability studies papers [START_REF] Fayyaz | High temperature pulsed-gate robustness testing of SiC power MOSFETs[END_REF], [START_REF] Puschkarsky | Understanding and modeling transient threshold voltage instabilities in SiC MOSFETs[END_REF].

During ageing devices were characterized using the Keysight B1506A for measuring datasheet-like characteristics and the B1500 for capacitance-voltage and charge pumping measurement.

Measurement protocols.

Ageing results on C2M0045170D (part number from Wolfspeed) devices will be presented. The main monitored parameters are the threshold voltage VTH, defined as the gate voltage for which IDS = 18 mA at VDS = VGS, the gate leakage current IGSS measured at VGS = 20 V and the on-state resistance RDSON which is defined for VGS = 20 V at the nominal current value IDS = 50 A of the device. The IGSS value measured at 20 V is used to check the integrity of the gate oxide. When IGSS goes above a maximum value (fixed at 600 nA as stated on the electrical characteristics of the datasheet) the device is considered as a failed one.

The stress test was interrupted on a regular basis and the devices went through a full characterization i.e classic direct and reverse I-V characteristics, C-V and charge pumping measurements to monitor the variations of the main parameters VTH, IGSS and RDSON.

Fig. 2 shows the degradation kinetic of VTH recorded during the first measurement batch. As we can notice, the variation of VTH as function of time shows instabilities. For instance, the VTH value after 86 h of ageing time is not consistent with the previous characterization step. We realized later that VTH is recovering as shown on Fig. 3 after a stress on the gate of the devices. This relaxation phenomenon has a critical impact on characterization since it has the same order of magnitude of VTH shift than the one induced by the ageing itself. Only the definitive degradation of the VTH is interesting here. For example, even after a 20 min bias with VGS = 20 V at ambient temperature on a brand new component, the threshold voltage is shifting back to pre-stress value with a very slow dynamic as shown on the Fig. 3. There is a timedependent shift on the threshold voltage which seems to be related to the initial conditions after a positive bias. No stabilization has been observed up to  60 min. To avoid this phenomenon we applied the following protocol, presented in [START_REF] Molin | Robustness study of 1700 V 45 m #x03A9; SiC MOSFETs[END_REF] and shown on the Fig. 3 : 1. Negative bias of -8 V during 1 s is applied.

2. Threshold voltage is measured. All the following measurements (fig. 4) have been performed with this protocol.

Reliability testing plan.

Studied devices are 1700V-45 m MOSFET that have been purchased at Wolfspeed and went under a stress plan presented in TABLE I.

Experimental results and discussion

IGSS and RDSON data have been monitored but as they do not show any significant shifts, only VTH parameter variations are described in the following parts.

HTGB+ static test.

All the presented DUTs went under a full characterization at t = 0 h (before any stress). The spreading of the VTH between each device reached 1.05 V, which is very high. Fig. 2 shows the variation of the normalized VTH related to the initial threshold voltage value. A positive shift on the threshold voltage occurs and ranges from 5% to a maximum of 8% (270mV) for D36. This can be explained by negative charge trapping either in the SiO2/SiC interface states or in deeper traps in the oxide.

In the literature, authors worked on reliability of lower nominal blocking voltage of 1.2 kV SiC MOSFET [START_REF] Fayyaz | High temperature pulsed-gate robustness testing of SiC power MOSFETs[END_REF]. Earlier results [START_REF] Lelis | SiC MOSFET reliability and implications for qualification testing[END_REF] obtained on 1.2 kV SiC MOSFET for similar ageing stress conditions indicates a positive VTH shift of ~2.5 V for VGS = 20 V during a 1000h static test at 150 °C. First conclusion is therefore that 1.7 kV MOSFET are more reliable in terms of VTH shift after HTGB ageing for 1000h.

HTGS dynamic test.

Working only on static ageing test is not enough to study reliability of the power devices. In this matter HTGS is closer to the real operation of MOSFET devices in power applications. Therefore V GS High and V GS Low , high commutation frequency and duty cycle are all parameters that companies are interested in. In this part, the influence of several parameters such as VGS values, commutation frequency and duty cycle were investigated on 9 similar devices (see Fig. 4a). The frequency of 20 kHz was chosen because it is representative of the industrial application reality [START_REF] Maneiro | Design of a SiC based triple active bridge ceil for a multimegawatt DC-DC converter[END_REF]. So there is an increase of 1 % on the threshold voltage shift between devices aged with α = 20 % and 80 %. The impact of the duty cycle was already pointed out in the literature [START_REF] Fayyaz | High temperature pulsed-gate robustness testing of SiC power MOSFETs[END_REF] with a higher shift. This could be a consequence of the higher frequency of 20 kHz used in our experiments in regards to the frequency of 10 kHz used in [START_REF] Fayyaz | High temperature pulsed-gate robustness testing of SiC power MOSFETs[END_REF]. Concerning HTGS, DUT with α = 50% seems to have reached an asymptote at 1000 h of ageing.

It is common to extrapolate ageing data to see the critical parameters shift in a longer period of time. An average lifetime of 10 years is required for power converter applications. In this particular dynamic ageing case, extrapolation to 10 years indicates a shift of the threshold voltage of 20 % in the worst scenario (sample D18).

Gate oxide analysis

Characterizations of the Si02/SiC interface via C-V, charge pumping and gate leakage current add a deeper level of understanding.

Fig. 5 shows the gate current vs. the gate voltage (drain at 0 V) for three different temperatures: 25, 50 and 150 °C. Each sweep performed from 0 to 35 V is directly followed by another bias from 0 to -30 V. The gate leakage current starts to increase exponentially around 25 V and -15 V, and this increase is characteristic of the Fowler-Nordheim effect. It is interesting to note that the negative bias is not stable, as already showed in [START_REF] Fiorenza | Fowler-Nordheim tunneling at SiO2/4H-SiC interfaces in metaloxide-semiconductor field effect transistors[END_REF] on p-type MOS capacitors. For low negative VGS, the p region under a part of the gate is in accumulation mode and, therefore the gate current is due to injection of hole from SiC into the oxide. It cannot come from injection of holes from the n region because the n region is not in inversion but depletion mode. For high positive VGS, the FN regime is more stable (begins at ~28 V) and is due to the injection of electrons from accumulation layer in the n channel under the gate or from the N+ source region. Although such high values of VGS are not reached in ageing tests, these instabilities (present even at -8V or +20 V) may lead to small variations in effective charges trapped in the oxide, and to a small VTH shift.

Quantitative data are difficult to obtain due to a lack of knowledge on the exact geometry of the component. To extract value of barrier height, exact surface of channel or of p-type region under the gate are mandatory.

Fig. 6 shows C-V measurements on sample D36 (similar data are obtained on all samples). The setup of the Keysight B1506A is using Ciss measurement, with shorted Drain and Source pads.

For VGS < -5 V the n region under the gate is depleted, thus the measured capacitance is mainly the gate to source capacitance. No voltage shift is observed after ageing on this part of C-V. Fig. 5. C2M0045170D IG-VG characteristics vs. temperature Fig. 6. C-V characteristics measured at 100 kHz on sample D36 initial C-V @t = 0 h in red circle and final in blue square @t = 1030.1 h on DUT D36 at ambient temperature.

For VGS > -1 V, the n region under the gate is in accumulation mode, so the p region should be in inversion mode because VTH is the voltage to apply to pass the structure from weak inversion to deep inversion. Only in this part, a positive VTH shift (~230 mV) is observed, i.e. a negative charge has been injected in the oxide. It could be due to an injection of electrons in the oxide from the n region under the gate during the positive part of the ageing pulse. It is consistent with the fact that VTH shift increases with the duty cycle.

Charge pumping measurement is another technique that should give precise interface trap density in a SiC MOSFET [START_REF] Brugler | Charge pumping in MOS devices[END_REF]. As the power structure of a SiC MOSFET does not allow access to the substrate thus it is not possible to directly measure the current Icp, the method used is a modified three terminal, introduced by Passmore and al. in 2005 [START_REF] Passmore | Modified three terminal charge pumping technique applied to vertical transistor structures[END_REF]. Sending a trapezoidal signal of frequency f with a fixed amplitude on the gate, the current is measured on the Drain with a high resolution SMU. Drain and source are kept grounded. Then the base value Vb of this signal is increased. Fig. 7 shows the current measured for different frequencies and the pumped charge QCP vs. f which is linear. This last curve is typical from charge pumping technique. Fig. 8 shows different CP curves obtained with different gate signal amplitude, using the same low level of VGS. Maximum amplitude is obtained when the MOSFET is gone from inversion to accumulation mode. Shape of the curve is not usual for a MOSFET for microelectronics, but this could be explained by the complex structure of the power MOSFET, showing an oxide both on p-type and n-type regions. Shift of the curve towards negative voltage is consistent with the CP principle. The magnitude of the peak should be related to the interface trap density, and should be rather constant when V > VTH-VFB. Fig. 9 shows charge pumping characteristics measured before and after a voltage stress on the device. The peak magnitude does not increase despite the ageing thus the stress does not create additional interface traps. The shift of the curve towards positive voltage comes from negative charges being definitely trapped in the oxide, in the same manner that we have discussed about C-V. In this case again without data on the gate technology, it is not possible to draw quantitative conclusions and the interface trap density is barely achievable. If an area of 14.3×10 -6 m² is taken arbitrary (estimated roughly from the bare die datasheet and from SiC MOSFET available gate area), then the density of trapped charges near the interface state Dit can be approximated to 3.4 ×10 10 eV -1 .cm -2 . Fig. 7. Influence of frequency on charge pumping measurement on C2M0045170D at ambient temperature, for ∆V = 4 V. Qcp slope ≈ 2.46 ×10 -9 C.s Fig. 8. Influence of gate signal amplitude ∆V on charge pumping measurement on C2M0045170D at ambient temperature, for f = 10 kHz Fig. 9. Charge pumping measurement before and after a strong bias of 30 V for 12 h on a 1.7 kV gate.

Conclusion

This paper presents an ageing study on 1.7 kV 45 mΩ SiC-MOSFET. A lifespan of 10 years (~87600 h) is required by industrials on their application. In this scenario then for HTGB the shift is going up to 450 mV (12.7 %) of the initial threshold voltage value for D36. For HTGS the worst case is a 240 mV (14.2 %) shift for D16.

So the VTH evolution during both static and dynamic ageing is worrying.

We have pointed out a very important phenomenon related to the stability of the recorded VTH after ageing. Indeed, a relaxation phenomenon is observed and a measurement protocol was proposed to eliminate the component of VTH that is rapidly recovered. Also a deeper investigation was performed by means of gate leakage current analyses, C-V and three terminal charge pumping measurements. They show that the degradation observed on the 1.7 kV threshold voltage during ageing is real, and nonrecoverable and probably due to injection of electrons from the n-region into the oxide of the devices. Interface trap density does not seem to be modified after a high voltage stress (30 V for 12 h) on the gate.

In a near future, modeling of trapping/detrapping phenomenon will be performed. TCAD simulations with the complex structure of SiC MOSFET are also required for a better understanding of the semiconductor physics and charge pumping measurements.

Fig. 1 .

 1 Fig. 1. (a) Ageing setup on the hotplate. 18 MOSFET (9x2) can be aged simultaneously (b) Scheme of ageing test

Fig. 4 (Fig. 4 .

 44 Fig. 4. (a) Normalized threshold voltage shift during HTGS static ageing at ambient temperature. (b) HTGS2 VTH/ VTH (%) for three different duty cycle, α = 20%, 50% and 80%

TABLE I .

 I AGEING TESTS AND PARAMETERS

				Fig. 2. Normalized threshold voltage shift during
				HTGB+ static ageing at ambient temperature on 1.7kV
				SiC MOSFET
		Tests (1000 h 150 °C)
		HTGB+	HTGS
	VGS	20V	Low = -10V V GS V GS High = 25V
	VDS		Floating
	Ageing mode Static	Dynamic f = 20 kHz
	Devices	D29  D37 α20% :
			D10, 11, 12
			α50% :
			D13, 14, 15	Fig. 3. VTH shift measured directly after 20 min at 20
			α80% :	V at ambient temperature on a 1.7 kV SiC MOSFET
			D16, 17, 18
	Parameters	VTH, RDSON, IGSS

(open squares) VTH shift measured after an additional negative pulse on the gate VGS = -8V for 1 s, (open circles). Initial VTH is 2.65 V

Acknowledgement

This work was supported by a grant overseen by the French National Research Agency (ANR) as part of the "Investissement d'Avenir" Program (ANE-ITE-002-01).


Teng Zhang
email: ateng.zhang@insa-lyon.fr

Christophe Raynaud
email: bchristophe.raynaud@insa-lyon.fr

Dominique Planson
email: cdominique.planson@insa-lyon.fr

Multi-barrier height characterization and DLTS study on Ti/W 4H-SiC Schottky Diode

Keywords: SiC, Schottky, multi barrier, DLTS, annealing, I-V, C-V, stress

Schottky barrier height (SBH) has been characterized on 4H-SiC Schottky diodes with metal contact of Ti/W by Current-Voltage (I-V) and Capacitance-Voltage (C-V) measurements between 80 K and 400 K. Multi-barrier has been recognized and calculated according to different models. No clear difference has been found between single barrier diode and diode with multi-barrier from DLTS tests. Evolution on the I-V characteristics has been observed after high temperature annealing. The effect of annealing at room temperature (RT) and high temperature DLTS scan (stress under high temperature) have also been studied on both static characteristics and DLTS results.

Introduction

Inhomogeneous barrier, which manifests as abnormal high current under low forward bias namely double or multi barrier phenomenon, has attracted attention among Schottky barrier diode (SBD) characterization for a long history, and not restricted to SiC devices. Similar phenomenon has been reported especially at low temperature, and it was highlighted that these 'non-ideal' diodes occurred regardless of growth technique, pre-deposition cleaning method, or contact metal [1][2]. Schottky barrier height (SBH) has been characterized on Ti/W 4H-SiC Schottky diodes by I-V and C-V measurements under wide temperature range. The evolution on forward I-V characteristics has been observed and investigated by high temperature annealing and accelerated stress with the help of Deep Level Transient Spectroscopy (DLTS).

Experimental Setup

Three 4H-SiC SBDs with metal contact of Ti/W have been investigated and labeled diode #1, #2 and #3. All SBDs with a square surface of 2.48 mm 2 were provided by our collaborator with a process of etch after sputtering. I-V and C-V characteristic have been measured in the cryostat between 80 K and 400 K with a step of 20 K in the dark to rule out the influence of light, with a Keithley K2410 and a Keysight E4990A Impedance Analyser. Forward characteristics have been measured every 0.01 V with a limitation of 30 mA which is high enough for the study of exponential region of the diodes. Applied voltage varies from 0 V to -5 V with a frequency of 100 kHz to extract the linear performance on the plot of 1/C 2 vs. V. On top of that, the DLTS tests have been realized with FT-1230 HERA DLTS (High Energy Resolution Analysis Deep Level Transient Spectroscopy) system provided by PhysTech.

Results and discussion

SBH Identification on Diode #1. As shown in Fig. 1, the forward I-V curves of diode #1 present a second barrier for temperatures below 240 K. The ideality factor n and saturation current IS could be extracted from the 'linear' region in forward characteristic.

Modified Richardson plot.

With the help of shallow doping concentration ND and barrier height calculated from C-V measurements, flat-band SBH (ΦBF) and the modified Richardson constant A * can be obtained according to modified Richardson plot, as illustrated in Fig. 2 [3]. Two distinct ΦBF are extracted with a higher SBH of 1.20 eV over the whole temperature range studied and the other SBH of 0.72 eV shows up at low temperature only. If considering the constant A * over both paths and the effective surface of low SBH region should account for 10 -8 of the total area. However, its influence on forward I-V characteristics could be the major factor especially at low temperature.

80K 100K 120K 140K 160K 180K 200K 220K 240K 260K 280K 300K 320K 340K 360K 380K 400K Current (A) Voltage (V) 80 K
Fig. 1. Forward I-V characteristic on diode #1 between 80 K and 400 K. 1.20 eV

A*: 565 A•cm -2 •K -2 ln(I S /T 2 )+(1-1/n)ln(N C /N D ) (A*K -2 ) 1/nT (K -1 )  BF : 0.72 eV A*: 5.26×10 -6 A•cm -2 •K -2
Fig. 2. Modified Richardson plot based on ΦBF for diode #1. Gaussian Distribution Model. Assuming that SBH is normally distributed on Gaussian [4], its mean barrier height and standard deviation can be extracted with the zero-bias barrier height Φ0 calculated based on the A * obtained above, as shown in Fig. 3.

Potential Fluctuation Model. To explain the temperature dependence of the ideality factor, Werner et al. assumed the SBH to be normally distributed, but the mean and standard deviation vary linearly with voltage [5]:

02 BB V      . and 22 03 SS V     .
(1) Those parameters can be calculated with the plot illustrated in Fig. 4. Zero bias barrier height (eV) q/2kT (V -1 )

: 0.78 eV  2 s : 0.006 V 2 : 1.31 eV  2 s : 0.009 V 2
Fig. 3. Φ0 as a function of q/2kT on #1. q/2kT (V -1 )

Fig. 4. 1/n-1 as a function of q/2kT on #1. Discussion. Two different SBHs are recognized from various models. However, unlike Φ0 that is strongly depend on temperature as shown in Fig. 3, the recalculated ΦBF at each temperature based on its definition almost keeps constant and is closed to the SBH obtained by C-V (~1.19 eV). In addition, similar behavior between Gaussian distribution and potential fluctuation model is forecasted.

I-V Characteristics Evolution on Diode #2.

During DLTS study, several temperature scans have been applied on diode #2 as shown in Fig. 5 with that of diode #1 as a reference. Even with almost the same results on DLTS spectra which indicates that no clear difference on trap levels can be found among the temperature investigated, the I-V characteristics could be rather different, as illustrated in Fig. 6. Considering that samples are reserved in the vacuum dark chamber of DLTS system, the evolution of I-V characteristic could only result from the multi DLTS temperature scan applied. Annealing and Stress Investigation on Diode #3. In order to investigate the origin of the evolution on I-V characteristics, additional measurements have been adopted on diode #3 as listed in Table 1. Fig. 8. Evolution on DLTS on diode #3. Evolution on Forward I-V Characteristics. Fig. 7 shows the measured I-V at 60 K on diode #3 with labels illustrated in Table 1. It is clear that low temperature DLTS scans have no influence on I-V curve. However, multi-barrier shows up after high temperature annealing, while annealing at RT has no obvious effect on I-V characteristic. Furthermore, high temperature DLTS scans reshape the I-V curve, which indicates that possible pinch-off of low SBH takes place under both high temperature and bias stress, this could be annealed at RT to some degree.

Evolution on DLTS signals. Those DLTS signals (b1) are illustrated in Fig. 8. It is worth highlighting that the peak at around 190 K of the DLTS signal which can be annealed at high temperature will recover at RT. Meanwhile, this defect level is eliminated after high temperature DLTS scan, or in other words the accelerated bias stress due to high temperature, and no recovery is found after RT annealing.

Summary

Multi-barrier has been identified by different models on 4H-SiC Ti/W SBD. No trap level found can contribute to the evolution on multi-barrier formation. However, a second barrier shows up after annealing at high temperature, and cannot be annealed at RT. High temperature DLTS scan, which regarded as accelerated bias stress owing to high temperature, also influences the SBH by reforming the low SBH region, and can be recovered a little at RT. High temperature DLTS scan will also wipe out several defect levels, while annealing at high temperature has only temporary effects on them. Furthermore, the characterization of metal/semiconductor interface is preferred in the future work.

Fig. 5 .

 5 Fig. 5. DLTS signal (correlation b1) between 60 K and 550 K on diode #1 and #2 before and after double barrier shown up. The DLTS parameters selected for all the measurements are marked in the plot. No remarkable difference shows up on current DLTS between 20 K and 180 K either (not shown here).

Fig. 6 .

 6 Fig. 6. Forward I-V curves on diode #2 during (a): 1 st scan and (b): 3 rd scan.

Fig. 7 .

 7 Fig. 7. Evolution on I-V characteristic on #3.

Acknowledgement

Authors acknowledge the financial support of the Chinese Scholarship Council (CSC) n. 201506090154 as well as National Center of Microelectronics of Barcelona and Caly Technologies.


Traitement conservateur d'un second événement intra-mammaire homolatéral : intérêt de la classification d'irradiation partielle de GEC-ESTRO

Lucile Montagne

To cite this version:

Lucile Montagne. Traitement conservateur d'un second événement intra-mammaire homolatéral : intérêt de la classification d'irradiation partielle de GEC-ESTRO. Human health and pathology. 2019. dumas-02436910

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •


domain: stat math

text:
Didier Rémi Servien

Patrick J C Tardivel
email: tardivel@math.uni.wroc.pl

Didier Concordet

Simple expressions of the LASSO and SLOPE estimators in low-dimension

Keywords: Best linear unbiased estimator, LASSO, SLOPE β∈R

published or not. The documents may come

Simple expressions of the LASSO and SLOPE estimators in low-dimension

Introduction

Let us consider the following low-dimensional linear model

Y = Xβ * + ε, (1)
where X is a n × p fixed design matrix with ker(X) = 0 (i.e. n ≥ p), β * ∈ R p is an unknown parameter and ε is a centered random vector with an invertible and known covariance matrix Γ. The Least Absolute Shrinkage and Selection Operator (LASSO) estimator and the Sorted L-One Penalized Estimation (SLOPE) estimator respectively introduced by Tibshirani [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF] and Bogdan et al. [START_REF] Bogdan | Slope -adaptive variable selection via convex optimization[END_REF] are defined by βlasso := argmin

β∈R p 1 2 Y -Xβ 2 + λ β 1 and, (2)
βslope := argmin

In the second expression, the tuning parameters (λ i ) 1≤i≤p satisfy λ 1 ≥ λ 2 ≥ • • • ≥ λ p ≥ 0. The brackets [ . ] denote a permutation of {1, . . . , p} such that

|β [1] | ≥ • • • ≥ |β [p] |.
It is well known that when the design X is orthogonal (i.e X X = Id p ), the LASSO estimator leads to the following soft-thresholded Ordinary Least Squares (OLS) estimator [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF] βlasso = sign( βols

Popularized by the pioneer work of Tibshirani, the orthogonal design became a case study [START_REF] Chzhen | On lasso refitting strategies[END_REF][START_REF] Duan | Generalized lasso with under-determined regularization matrices[END_REF][START_REF] Sell | Sequential selection procedures and false discovery rate control[END_REF][START_REF] Lockhart | A significance test for the lasso[END_REF][START_REF] Tian | Selective inference with unknown variance via the squareroot lasso[END_REF][START_REF] Wen | On sparse vector recovery performance in structurally orthogonal matrices via lasso[END_REF]. Furthermore some properties such as the irrepresentable condition [START_REF] Bühlmann | Statistics for high-dimensional data: Methods, theory and applications[END_REF][START_REF] Meinshausen | High-dimensional graphs and variable selection with the lasso[END_REF][START_REF] Zhao | On model selection consistency of lasso[END_REF][START_REF] Zou | The adaptive lasso and its oracle properties[END_REF] hold when X is orthogonal. The orthogonal design is also a case study for the SLOPE estimator [START_REF] Bogdan | Slope -adaptive variable selection via convex optimization[END_REF][START_REF] Gossmann | Identification of significant genetic variants via slope, and its extension to group slope[END_REF][START_REF] Su | Slope is adaptive to unknown sparsity and asymptotically minimax[END_REF].

Whatever the considered estimator, the orthogonal design setting appears to be an ideal case. When we tried to generalize its properties to the non-orthogonal setting, we discovered a relevant orthogonalizing transformation U s := D(X Γ -1 X) -1 X Γ -1 , where s := (s 1 , . . . , s p ), s 1 > 0, . . . , s p > 0 and D := diag(1/ √ s 1 , . . . , 1/ √ s p ). Actually, if we consider the new model Ỹ = Xβ * + ε, where Ỹ = U s Y , X = U s X and ε = U s ε, the LASSO estimator βs after applying the transformation U s can simply be written as a function of the Best Linear Unbiased Estimator (BLUE) in the following way:

βs := argmin β∈R p 1 2 Ỹ -Xβ 2 + λ β 1 ⇔ βs = sign( βblue i )(| βblue i | -λs i ) + 1≤i≤p
.

(5) Similarly to the LASSO estimator, we also obtained a simple expression for the SLOPE estimator based on the BLUE. We can notice that, in low-dimension, the U s transformation giving [START_REF] Sell | Sequential selection procedures and false discovery rate control[END_REF] does not require X to be orthogonal or the components of ε to be independent; but ker(X) = 0 is necessary.

Notations

In this article, we denote J the SLOPE norm J : [START_REF] Bogdan | Slope -adaptive variable selection via convex optimization[END_REF] for the proof that J is a norm). The OLS and BLUE estimators of the model (1), denoted βols and βblue , are respectively equal to βols := (X X) -1 X Y and βblue :=

β ∈ R p → λ 1 |β [1] | + • • • + λ p |β [p] | where |β [1] | ≥ • • • ≥ |β [p] | and λ 1 ≥ • • • ≥ λ p (see for example
(X Γ -1 X) -1 X Γ -1 Y. ( 6
)
Whatever t ∈ R, we set (t) + = max{t, 0} and sign(t) = 1 t>0 -1 t<0 . Given a subset A ⊂ R p , conv(A) is the smallest convex set containing A. Finally, the notation Id p represents the p × p identity matrix.

Orthogonalization of the design: simple form of the LASSO and SLOPE

When the design is orthogonal, some algorithms provide the SLOPE estimation [START_REF] Bogdan | Slope -adaptive variable selection via convex optimization[END_REF] but the estimator writing is not explicit. To our knowledge, there does not exist currently any explicit formula for the SLOPE. In the following theorem, we provide the explicit expression of the SLOPE when X is orthogonal.

k s -k s-1 + , . . . , Ŝks -Ŝks-1 k s -k s-1 + ks -ks-1 components .
Let us notice that when βols has a continuous distribution over R p , the Cesàro sequence ( Ŝk /k) almost surely reaches its maximum at a unique point. In other terms, k 1 := argmax { Ŝk /k} is unique and the same argument applies for k 2 , . . . , k s . When As a consequence, when λ 1 = • • • = λ p = λ in the orthogonal setting, the formula of the SLOPE given in the theorem 2.1 coincides with the one of the LASSO given in (4).

λ 1 = • • • = λ p = λ,
Remark: When X is orthogonal, ε ∼ N (0, σ 2 Id n ) and λ i = σz(1 -iq/(2p)), i ∈ {1, . . . , p} (where q ∈ (0, 1) and z(η) is the η-quantile of the N (0, 1) distribution), the procedure rejecting the null hypothesis β * i = 0 when βslope i = 0 controls the FDR at level q [START_REF] Bogdan | Slope -adaptive variable selection via convex optimization[END_REF]. The explicit expression of the SLOPE shows that this procedure is close to the original Benjamini-Hochberg procedure [START_REF] Benjamini | Controlling the false discovery rate: a practical and powerful approach to multiple testing[END_REF] (actually, these procedures are equal when the sequence ( Ŝk /k) is decreasing) and this provides an intuitive explanation for the FDR control.

An illustration of the relationship between the OLS estimator and the SLOPE estimator when X is orthogonal is given by the figure 1 in the special case where p = 2, λ 1 = 2 and λ 2 = 1. This figure also illustrates the properties of the SLOPE: this estimator is sparse (i.e. some components are exactly equal to 0), and some components are equal in absolute value.

Remarks: Let us point out some relevant transformations allowing us to obtain simple expressions for the LASSO estimator and the SLOPE:

• Transformation which brings back to the orthogonal setting: When X is not orthogonal, applying the transformation U 1 := (X Γ -1 X) -1 X Γ -1 on each side of the model (1) brings back the orthogonal setting in which Ỹ := β * + ε, where Ỹ = U 1 Y and ε = U 1 ε. In this new model, the LASSO estimator has the expression described in [START_REF] Sell | Sequential selection procedures and false discovery rate control[END_REF] with s 1 = • • • = s p = 1 and the SLOPE has the expression described in the theorem 2.1 except that the OLS estimator is substituted by the more accurate BLUE estimator.

• Transformation which brings back the orthogonal columns setting:

When columns of X are not orthogonal (i.e. when X X is not diagonal), applying the transformation

U s := D(X Γ -1 X) -1 X Γ -1
, where s := (s 1 , . . . , s p ), s 1 > 0, . . . , s p > 0 and D := diag(1/ √ s 1 , . . . , 1/ √ s p ), on each side of the model (1) returns the orthogonal columns' setting. After applying this transformation, LASSO's expression is still explicit and its expression is described by formula (5).

To avoid confusion, we call soft-thresholded BLUE the LASSO-type estimator whose formula is given in [START_REF] Sell | Sequential selection procedures and false discovery rate control[END_REF] and LASSO the estimator solution of (2). The soft-thresholded BLUE is easier to compute than the LASSO estimator (because the LASSO estimator computation needs to solve numerically the optimization problem described in (2)) and the soft-thresholded BLUE estimator is also easier to interpret than the LASSO estimator (contrarily to the LASSO estimator, the relationship between BLUE and the soft-thresholded BLUE is explicit). Thus it is recommended that in low-dimension the soft-thresholded BLUE should be used instead of the LASSO estimator, except if there are particular reasons for using the latter. Finally, from procedures based on the LASSO estimator defined in (2) one can derive new procedures based on the softthresholded BLUE given in [START_REF] Sell | Sequential selection procedures and false discovery rate control[END_REF] as illustrated in the following remark. Remark: Janson and Su [START_REF] Janson | Familywise error rate control via knockoffs[END_REF] have developed a multiple testing procedure based on the knockoff-LASSO estimator defined as follows βkn-lasso := argmin

β∈R p 1 2 Y -X ko β 2 + λ β 1 .
Since the matrix X ko (defined in Barber and Candès [START_REF] Barber | Controlling the false discovery rate via knockoffs[END_REF]) satisfies ker(X ko ) = 0, up to a transformation, the knockoff-LASSO is just a soft-thresholded BLUE (where the BLUE is (X ko Γ -1 X ko ) -1 X ko Γ -1 Y , with Γ = var(Y )). Therefore, from the procedure described in [START_REF] Janson | Familywise error rate control via knockoffs[END_REF] one can derive a new procedure based on the soft-thresholded BLUE.

Comparison between the LASSO estimator and the soft-thresholded BLUE

When we want to recover the non-null components of β * , the soft-thresholded BLUE outperforms the LASSO estimator, solution of (2), as evidenced hereafter.

Support recovery

Let ε be a Gaussian vector of R 2 having a N (0, Id 2 ) distribution, let β * = (t, 0) ∈ R 2 with t = 0, let X be the design matrix described hereafter and let Y = Xβ * + ε.

X = 1/2 1/2 1/2 1/2 0 1 1 1 1 1 and thus βblue ∼ N t 0 , 5 -2 -2 1 .
When u ∈ R p , let us set supp(u) := {i ∈ {1, . . . , p} | u i = 0}. Now let us denote respectively βlasso (λ) and βs (λ) the LASSO estimator and the soft-thresholded BLUE to stress that these estimators depend on λ. Whatever λ > 0, and even if |t| is infinitely large, the following inequality [START_REF] Wainwright | Sharp thresholds for high-dimensional and noisy sparsity recovery using l1 constrained quadratic programming (lasso)[END_REF][START_REF] Tardivel | On the sign recovery given by the thresholded lasso and thresholded basis pursuit[END_REF] holds

P supp( βlasso (λ)) = supp(β * ) ≤ 1/2.
This inequality is illustrated in Figure 2 when λ ∈ {1/2, 1}.

On the other hand, if we take s 1 = √ 5, s 2 = 1 and λ as the 1 -α quantile of a N (0, 1) distribution in (5), we have P(supp( βs (λ)) = supp(β * )) ≈ 1 -α when |t| is large. Consequently, by using the soft-thresholded BLUE, the probability to recover supp(β * ) can be arbitrarily close to 1 when holds: |t| is large and α is small.

Estimation of β *

When the covariance matrix Γ is not a scalar matrix then, usually, the LASSO estimator is computed after whitening the noise as follows: βlasso,w (λ) := argmin

β∈R p 1 2 Γ -1/2 Y -Γ -1/2 Xβ 2 + λ β 1 . (7)
When λ = 0 the above estimator coincides with the BLUE and thus has a smaller L 2 risk than the LASSO estimator as computed in (2) which is equal to the OLS estimator. The estimator given in (8) also coincides with the BLUE when λ = 0. Let s := (s 1 , . . . , s p ) where s 1 , . . . , s p are standard errors of the BLUE (thus s 2 1 , . . . , s 2 p are the diagonal components of (X Γ -1 X) -1 ). The soft-thresholded BLUE is defined as Figure 2. These figures provide the relationship between βols and the set of non-null components of the LASSO estimator supp( βlasso (λ)) (as explained in [START_REF] Schneider | On the distribution and model selection properties of the lasso estimator in low and high dimensions[END_REF]) when λ = 0.5 (above) and λ = 1 (below). The xaxis and y-axis represent respectively the first and second component of the OLS estimator. One may notice that the estimator supp( βlasso (λ)) recovers the true set {1} when βols is in the red area. Because βols is in the red area, this implies that β ols follows βs (λ) := argmin

β∈R p 1 2 U s Y -U s Xβ 2 + λ β 1 ⇔ βs (λ) = sign( βblue i )(| βblue i | -λs i ) + 1≤i≤p . (8)
Hereafter we compare estimators described in ( 7) and ( 8) with respect to the L 2 risk.

Let Y = Xβ * + ε where X = Id p , β * ∈ R p and ε is a Gaussian vector in R p having a N (0, Γ) distribution. The performance of the LASSO as well as the soft-thresholded BLUE depend on the design X and on Γ. By taking X = Id p , we only focus on the whitening preliminary step for the LASSO. In Figure 3, we compare the following functions

λ ≥ 0 → E( βlasso,w (λ) -β * 2 ) and λ ≥ 0 → E( βs (λ) -β * 2 ).
in the particular settings described hereafter:

• Equicorrelated case: Let Γ be a 10 × 10 matrix whose diagonal and nondiagonal coefficients are respectively equal to 1 and c where c ∈ {0.3, 0.9} and let β * = (1, . . . , 1) ∈ R 10 so that β * 2 = E( Y -β * 2 ). • Correlation exponentially decreasing: Let Γ be a 10 × 10 matrix where Γ ij = c |i-j| for 1 ≤ i, j ≤ 10 and c ∈ {0.7, 0.9} and let β * = (1, . . . , 1) ∈ R 10 .

Figure 3 illustrates that, when the tuning parameter is appropriately selected for both βlasso,w and βs , the L 2 risk of the soft-thresholded BLUE is approximately equal or smaller than the L 2 risk of the LASSO estimator. Correlation exponenentially decreasing: c=0.9

Tuning parameter L2 risk L2 risk of LASSO after whitening L2 risk of soft-thresholded BLUE Figure 3. The x-axis and y-axis of these curves represent respectively the tuning paramater λ ≥ 0 and the L 2 risk for both LASSO and soft-thresholded BLUE estimators. To obtain these curves we simulated 10000 the random vector Y and we computed the soft thresholded BLUE and LASSO estimator for λ ∈ {0, 0.02, 0.04, . . . , 3}. Computation of the LASSO estimator have been done with glmnet package [START_REF] Friedman | Regularization paths for generalized linear models via coordinate descent[END_REF]. In every cases, when λ = 0 these two functions are equal to E( Y -β * 2 ) and when λ is very large these two functions are approximately equal to β * 2 2 . One can notice that the function λ ≥ 0 → E( βs (λ) -β * 2 ) does not change in these four settings since, in each case, components of the BLUE have a N (1, 1) marginal distribution. The minimum is 7.01 and is reached at λ = 0.74. Depending on the setting, the minimum of the function λ ≥ 0 → E( βlasso,w (λ) -β * 2 ) is 8.30, 6.71, 7.76 and 7.25 is reached at λ = 0.24, λ = 0.10, λ = 0.24 and λ = 0.10. Overall, after choosing properly the tuning parameter, one may notice that the L 2 risk is approximately equal or slightly smaller for the soft-thresholded BLUE than for the LASSO estimator.

Conclusion

Up to a transformation, the LASSO and the SLOPE have simple and explicit writing. In addition, our results point out that methods using the LASSO or the SLOPE in lowdimension can be derived as methods which only use the BLUE. Then, comparisons with the LASSO estimator showed the benefits of the soft-thresolded BLUE.

Proof of the proposition A.1

First, let us notice that when X is orthogonal the following equivalence holds βslope := argmin

β∈R p 1 2 Y -Xβ 2 + J(β) ⇔ βslope := argmin β∈R p 1 2 βols -β 2 + J(β).
Consequently, to prove theorem 2.1, one only needs to provide an explicit expression of the minimizer of the function φ defined hereafter

∀x ∈ R p , φ(x) = 1 2 y -x 2 + J(x)
, where y ∈ R p is a fixed vector.

Let us notice that φ is a coercive and strictly convex function, thus whatever y ∈ R p , φ has a unique minimizer. As suggested by assumption 2.1 in the article of [START_REF] Bogdan | Slope -adaptive variable selection via convex optimization[END_REF], one can restrict the study of the function φ to y 1 ≥ y 2 ≥ • • • ≥ y p ≥ 0. Actually finding the minimizer in this particular case makes it possible to recover easily the minimizer of φ when y is an arbitrary vector of R p as explained in [START_REF] Bogdan | Slope -adaptive variable selection via convex optimization[END_REF]. Let us remind some basic notions of sub-differentiability. Let > 0, let f : R p → R be a convex function, the sub-differential of f at the point x ∈ R p denoted ∂f (x) is the convex set described hereafter.

s ∈ ∂f (x) if ∀h ∈ B(0, ), f (x + h) -f (x) ≥ s, h ⇔ s ∈ ∂f (x) if ∀h ∈ R p , f (x + h) -f (x) ≥ s, h .
The sub-differentiability makes it possible to characterise the minimizer of φ (see e.g [START_REF] Hiriart-Urruty | Convex analysis and minimization algorithms i: Fundamentals[END_REF] page 177). The point x * is a minimizer of φ if and only if 0 ∈ ∂φ(x * ).

Proposition A.1. Let φ : x ∈ R p → 1 2 y -x 2 + J(x) with y 1 ≥ • • • ≥ y p ≥ 0, let (S k ) 1≤i≤p be a sequence such that S k = k i=1 y i -λ i and let 1 ≤ k 1 ≤ • • • ≤ k s = p be a partition of {1, . . . , p} such that k 1 = max argmax k∈{1,...,p} S k k and ∀i ∈ {2, . . . , s}, k i = max argmax k>ki-1 S k -S ki-1 k -k i-1 .
Let c 1 = S k1 /k 1 and for all i ∈ {2, . . . , s}, c i = (S ki -S ki-1 )/(k i -k i-1 ) and let x * ∈ R p be a vector such that ).

Then the unique minimizer of φ is x * .

Hereafter the SLOPE norm J is also denoted J λ1,...,λp , the set of permutations of {1, . . . , p} is denoted S p and given u ∈ R p , the permutation [ . ] ∈ S p is such that |u

[1] | ≥ • • • ≥ |u [p] |.
Lemma A.2. Properties i) and ii) deal with the sub-differential of J and property iii) deals with the sub-differential of φ.

i) If x 1 = • • • = x p > 0, then conv (λ r(1) , . . . , λ r(p) ) r∈Sp ⊂ ∂J(x). ∂φ s (x ks-1+1 , . . . , x ks ) then u ∈ ∂φ(x). Indeed φ(x + h) -φ(x) is equal to s i=0 φ i+1 (x ki+1 + h ki+1 , . . . , x ki+1 + h ki+1 ) -φ i+1 (x ki+1 , . . . , x ki+1 ) ≥ s i=0 u ki+1 h ki+1 × . . . , u ki+1 h ki+1 = u, h .
Consequently, u ∈ ∂φ(x), which ensures that iii) holds.

Lemma A.3. Let us assume that ∀i ∈ {1, . . . , p}, S i ≤ 0, then the unique minimizer of φ : x ∈ R p → 1 2 y -x 2 + J(x) is x * = (0, . . . , 0). Proof: To prove that x * = (0, . . . , 0) is a minimizer of φ, it suffices to show that 0 ∈ ∂φ(x * ). Let us give the following equivalences 0 ∈ ∂φ(x * ) ⇔ 0 ∈ -y + x * + ∂J(x * ) ⇔ y ∈ ∂J(0).

1 )(| βols 1 |

 11 -λ) + , . . . , sign( βols p )(| βols p | -λ) + .

  the Cesàro sequence is non-increasing and consequently the following equality holds ∀i ∈ {1, . . . , p}, βslope i = sign( βols i )(| βols i | -λ) + = βlasso i .

Figure 1 . 1 | = | βslope 2 | 1 || βslope 2 | 1 | 2 | > 0 and | βslope 1 |

 11212121 Figure 1. This figure illustrates the relationship between the OLS estimator and the SLOPE, the x-axis and y-axis represent respectively the first and second component of the OLS estimator. Let Ŝ1 , Ŝ2 be defined as in the theorem 2.1. When Ŝ1 ≤ 0 and Ŝ2 ≤ 0 then βols is on the black area and βslope = 0. When Ŝ1 ≤ Ŝ2 /2 and Ŝ2 /2 > 0 then βols is on the red area and | βslope 1 | = | βslope 2 | > 0. When Ŝ1 > Ŝ2 /2 with Ŝ1 > 0 and Ŝ2 -Ŝ1 < 0 then βols is on the grey area, βslope = 0 and | βslope 1 || βslope 2 | = 0. Otherwise βols is on the white area then | βslope 1 | > 0, | βslope 2 | > 0 and | βslope 1 | = | βslope 2 |.

1 β ols 2 ≤ 0 , and because P( βols 1 βols 2 ≤ 0 )

 20120 Figure2. These figures provide the relationship between βols and the set of non-null components of the LASSO estimator supp( βlasso (λ)) (as explained in[START_REF] Schneider | On the distribution and model selection properties of the lasso estimator in low and high dimensions[END_REF]) when λ = 0.5 (above) and λ = 1 (below). The xaxis and y-axis represent respectively the first and second component of the OLS estimator. One may notice that the estimator supp( βlasso (λ)) recovers the true set {1} when βols is in the red area. Because βols is in the red area, this implies that β ols 1 β ols 2 ≤ 0, and because P( βols 1 βols 2 ≤ 0) ≤ 1/2, one may deduce that

x * = ((c 1 )

 1 + , . . . , (c 1 ) + k1 components , (c 2 ) + , . . . , (c 2 ) + k2-k1 components , . . . , (c s ) + , . . . , (c s ) + ks-ks-1 components

By lemma 2 ,

 2 the sub-differential of φ at 0 contains the set C given hereafterC := conv   r∈Sp [-λ r(1) , λ r(1) ] × • • • × [-λ r(p) , λ r(p) ]   ⊂ ∂J(0).Let us remind that a closed convex set is the intersection of all closed half-spaces containing it. Let a 1 x 1 + • • • + a p x p ≥ b be an arbitrary closed half-space containing C. To prove that y ∈ C, we are going to show thata 1 y 1 + • • • + a p y p ≥ b. Let (.)be a permutation of {1, . . . , p} such that |a (1) | ≤ • • • ≤ |a (p) | and let us denote u i = |a (i+1) |-|a (i) | with i ∈ {1, . . . , p -1}. Because v := (-λ p sign(a (1) ), . . . , -λ 1 sign(a (p) )) ∈ C and because whatever r ∈ S p , (v r(1) , . . . , v r(p) ) ∈ C, one may deduce that a (1)v 1 + • • • + a (p) v p = -λ p |a (1) | -• • • -λ 1 |a (p) | ≥ b. The following implications show that a 1 y 1 + • • • + a p y p ≥ -λ p |a (1) | -• • • -λ 1 |a (p) | ≥ b.We deduce from this last inequality thata 1 y 1 + • • • + a p y p ≥ -λ p |a (1) | -• • • -λ 1 |a (p) |, ⇔ a (1) y (1) + λ p |a (1) | + • • • + a (p) y (p) + λ 1 |a (p) | ≥ 0, ⇔ |a (1) | sign(a (1) )y (1) + λ p + • • • + |a (p) | sign(a (p) )y (p) + λ 1 ≥ 0, comes from the identity |a (1) |b 1 + • • • + |a (p) |b p = |a (1) |(b 1 + • • • + b p ) + u 1 (b 2 + • • • + b p ) + • • • + u p-1 b p . Finally, because y 1 ≥ • • • ≥ y p ≥ 0 one may deduce

Acknowledgements

Patrick Tardivel thanks Professor Bogdan to allow him to continue his research at the Mathematics Institute of Wroc law University. Authors thank Artur Bogdan for his careful reading.

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

This work is part of the project GMO90+ supported by the grant CHORUS 2101240982 from the Ministry of Ecology, Sustainable Development and Energy in the national research program RiskOGM. Patrick Tardivel is partly supported by a PhD fellowship from GMO90+. We also received a grant for the project from the IDEX of Toulouse 'Transversalité 2014'.

Appendix A. Proof of the theorem 2.1

Sketch of proof

Theorem 2.1 is a straightforward consequence of proposition A.1. Our goal is thus to prove this proposition, which provides an explicit expression for the minimizer of x ∈ R p → 1 2 y -x 2 + J(x) (where y ∈ R p is a fixed vector such that y 1 ≥ y 2 ≥ • • • ≥ y p ≥ 0).

Looking at the output of the algorithm described in [START_REF] Bogdan | Slope -adaptive variable selection via convex optimization[END_REF] allowed us to conjecture the following two minor results:

1) The null vector is the unique minimizer of x ∈ R p → 1 2 y -x 2 + J(x) when (S k ) 1≤k≤p is negative.

2) The vector (S p /p, . . . , S p /p) is the unique minimizer of x ∈ R p → 1 2 y-x 2 +J(x) when the Cesàro sequence (S k /k) 1≤k≤p reaches its maximum at k = p and when S p > 0.

Statements 1) and 2) are proved respectively in lemma A.3 and in lemma A.4. These two lemmas are the keystones of the proof of proposition A.1.

In these lemmas we need to prove that an element u belongs to a closed convex set C. To prove this belonging, we use the fact that C is the intersection of all halfspaces that contain it (see [START_REF] Boyd | Convex optimization[END_REF] page 49). Consequently, if

|, and let r be an arbitrary permutation of {1, . . . , p}. One of the key inequalities in lemma A.2 is the following one

p be a partition of {1, . . . , p} such that

Let j ∈ {0, . . . , s} and let us define functions φ 1 , . . . , φ s+1 as follows

Then the sub-differential of φ satisfies

Proof: First, let us prove i). Because whatever r ∈ S p , the two following expressions hold

Consequently, whatever r ∈ S p we have (λ r(1) , . . . , λ r(p) ) ∈ ∂J(x). Furthermore, because ∂J(x) is a convex set, one may deduce result i). Now, let us prove ii), whatever s 1 ∈ [-1, 1], . . . , s p ∈ [-1, 1] whatever r ∈ S p , the following inequality holds

Finally, let us show iii). Let h ∈ R p be small enough so that whatever i ∈ {1, . . . , s}, the inequality x ki -h ∞ > x ki+1 + h ∞ occurs (such a small h ensures that the k th 1 largest components of x + h are x 1 + h 1 , . . . , x k1 + h k1 and so on). As a consequence, the SLOPE norm satisfies the following equality

When h is small enough, one may deduce that whatever u ∈ ∂φ 1 (x 1 , . . . , x k1 ) × • • • × the inequality given hereafter which ensures that a 1 y 1 + • • • + a p y p ≥ b. In other terms,

Consequently, y ∈ C and so x * = (0, . . . , 0) is the unique minimizer of φ.

Lemma A.4. Let us assume that ∀i ∈ {1, . . . , p}, S i /i ≤ S p /p and S p > 0, then the unique minimizer of φ : x ∈ R p → 1 2 y -x 2 + J(x) is x * = (S p /p, . . . , S p /p) . Proof: To prove that x * is a minimizer of φ, it suffices to show that 0 ∈ ∂φ(x * ). Let us gives the following equivalences

By the lemma A.2, conv (λ r(1) , . . . , λ r(p) ) r∈Sp ⊂ ∂J(x * ). Hereafter we are going to show -y + x * ∈ conv (λ r(1) , . . . , λ r(p) ) r∈Sp . Let us remind that a closed convex set is the intersection of all closed half-spaces containing it. Let a 1 x 1 + • • • + a p x p ≥ b be an arbitrary closed half-space containing conv (λ r(1) , . . . , λ r(p) ) r∈Sp . To prove that y -x * ∈ conv (λ r(1) , . . . , λ r(p) ) r∈Sp , it suffices to prove that a 1 (y 1 -x * 1 ) + • • • + a p (y p -x * p ) ≥ b. Let (.) be a permutation of {1, . . . , p} such that a (1) ≤ • • • ≤ a (p) and let us denote u i = a (i+1) -a (i) with i ∈ {1, . . . , p -1}. By definition of the half-space a 1 x 1 + • • • + a p x p ≥ b, an appropriate permutation r ∈ S p ensures that a [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF]

The last expression comes from the identity a

0 and because whatever i ∈ {1, . . . , p}, S i /i ≤ S p /p one may deduce the inequalities given hereafter which ensures that a 1 (-

Consequently, -y + x * ∈ conv (λ r(1) , . . . , λ r(p) ) r∈Sp thus x * = (S p /p, . . . , S p /p) is the unique minimizer of φ.

Proof of the proposition A.1: First, let us show that

thus let us show that whatever i ∈ {1, . . . , s -1}, the inequality c i+1 = c i cannot occur. Indeed, the following equality always holds

(by setting k 0 = 0 and S k0 = 0). ).

Lemma A.4 ensures that whatever i ∈ {1, . . . , s}, we have 0 ∈ ∂φ i (c i , . . . , c i ). Thus 0 ∈ ∂φ(x * ), which ensures that x * is a minimizer of φ. Now, if 0 ≥ c 1 > • • • > c s then the sequence (S i ) 1≤i≤p is negative, thus lemma A.3 ensures that x * = (0, . . . , 0) is a minimizer of φ. )∂ × φ i0+1 (0), with φ i0+1 as in lemma A.2. Lemma A.4 ensures that whatever i ∈ {1, . . . , i 0 }, we have 0 ∈ ∂φ i (c i , . . . , c i ). Furthermore, because ∀i > k i0 , (S i -S ki 0 ) ≤ 0, lemma A.3 ensures that 0 ∈ ∂φ i0+1 (0). Thus 0 ∈ ∂φ(x * ), which ensures that x * is a minimizer of φ.