Luck Darnière

Marcus Tressl

Defining integer valued functions in rings of continuous definable functions over a topological field

Let K be an expansion of either an ordered field (K, ), or a valued field (K, v). Given a definable set X ⊆ K m let C(X) be the ring of continuous definable functions from X to K. Under very mild assumptions on the geometry of X and on the structure K, in particular when K is o-minimal or P -minimal, or an expansion of a local field, we prove that the ring of integers Z is interpretable in C(X). If K is o-minimal and X is definably connected of pure dimension 2, then C(X) defines the subring Z. If K is P -minimal and X has no isolated points, then there is a discrete ring Z contained in K and naturally isomorphic to Z, such that the ring of functions f ∈ C(X) which take values in Z is definable in C(X).

Introduction

We give a first order definition of the ring of integers within rings of continuous functions that are first order definable in expansions of ordered and valued fields. Before describing a more technical outline of the contents, we explain the context of the article and the results in a colloquial way.

Rings of continuous functions on topological spaces are central objects in functional analysis, topology and geometry. To name an example: They are rings of sections for the sheaf of continuous (say, real valued) functions on a topological space and as such play the algebraic role in the study of topological (Hausdorff) spaces.

By a ring of definable continuous functions we mean the following. Let K be an expansion of an ordered field (K, ) (e.g. the real field) or a valued field (K, v) (e.g. the p-adics). In both cases K carries a topology that turns K into a topological field. Let X ⊆ K n be definable (with parameters) in K and let C(X) be the set of all functions X -→ K that are continuous and definable in K. If K is the real or the p-adic field, definable is the same as semi-algebraic. Then C(X) is a ring and similar to the classical case mentioned above, C(X) carries the algebraic information of the definable homeomorphism type of X. This is amply illustrated in the case when K is a real closed field: Let S be the category of semi-algebraic subsets X ⊆ K n , n ∈ N, with continuous semi-algebraic maps as morphisms. Let C be the category of all the rings C(X) of continuous K-definable functions and K-algebra homomorphisms as morphisms. Then the functor C : S -→ C that sends X to C(X) and a morphism f : X -→ Y to the Kalgebra homomorphism C(Y ) -→ C(X); g → g • f , is an anti-equivalence of categories. This can also be seen in analogy with algebraic geometry, where varieties defined over a field K are anti-equivalent to affine K-algebras, leading to the modern language of schemes. For semi-algebraic sets the machinery is developed in this vain in terms of so called semi-algebraic spaces, see [Sch87]. The same connection exists between definable sets X in the p-adic context and the rings C(X) for that context. In each case, one may thus think of rings of continuous definable functions as "coordinate rings" when studying topological properties of definable sets.

Model theoretic studies of rings of continuous (definable) functions may be found in [START_REF] Prestel | Model theory of real closed rings[END_REF], [START_REF] Tressl | Super real closed rings[END_REF] for the case of real closed fields and in [START_REF] Bélair | Anneaux p-adiquement clos et anneaux de fonctions définissables[END_REF],[Bél95], [GT08] in the p-adic case. Model theory of rings of continuous functions on topological spaces have been studied in [Che80], which serves as a main source for inspiration for us.

In this article we study model theoretic properties of rings of continuous definable functions under fairly mild assumption on the base structure. The principal goal is to show that, in most cases, these rings interpret the ring of integers in a uniform way; in particular these rings are not decidable. This is established in Theorem 5.5. Now undecidability was already known in some cases. For example, one can interpret the lattice of closed subsets of R n in the ring C(R n ) of continuous semi-algebraic functions R n -→ R. When n 2, this lattice itself is undecidable by [START_REF] Grzegorczyk | Undecidability of some topological theories[END_REF] and indeed interprets the ring of integers, see [Tre17]. However, in this lattice, one cannot interpret C(R n ) in a uniform way as we will see in Remark 5.8.

In the p-adic case, the first author has shown in [START_REF] Darnière | Model-completion of scaled lattices[END_REF] that the lattice of closed subsets of Q n p is even decidable and it was unknown whether C(Q n p ) has a decidable theory at all.

In section 6 we show that in many cases the rings C(X) actually define the subset of constant functions with values in a natural isomorphic copy of Z, see Theorem 6.7 for the precise formulation. When K is an o-minimal expansion of a field and X is definably connected of local dimension 2 everywhere, then indeed C(X) defines the ring of constant functions with values in Z. As a consequence, when K expands the real field we obtain that the real field, seen as constant functions R n -→ R is definable in C(X). This implies that the projective hierarchy is definable in these rings.

Detailed description of the main results and set up.

We consider an expansion K of a topological field (K, O) where O is either the unit interval [-1, 1] of a total order on K, or the ring of a non-trivial valuation v on K. We endow K m with the corresponding topology, for every integer m 0. We will make almost everywhere the following assumption on K.

(BFin)

Every definable subset of K that is closed, bounded and discrete, is finite.

As is well known, every discrete definable subset of Given any two definable sets X ⊆ K m and Y ⊆ K n we let C(X, Y ) denote the set of continuous functions X → Y that are definable with parameters in K. If Y is a subring of K, e.g. when Y = Z, then C(X, Y ) is considered as a ring, where addition and multiplication is given point-wise. When Y = K, we just write C(X). Let

K is finite if K is o-minimal [vdD98
τ Z = τ k : k ∈ Z for some non-zero τ ∈ O with 1/τ / ∈ O.
We also furnish τ Z with a ring structure so that the bijection Z -→ τ Z , k → τ k is an isomorphism. Again, pointwise addition and multiplication turns C(X, τ Z ) into a ring.

Our main result is that, under very general conditions on K and X, the ring C(X, Z) (resp. C(X, τ Z )) is definable (resp. interpretable) in the ring structure of C(X) expanded by the set

B = {s ∈ C(X) | ∀x ∈ X : s(x) ∈ O}.
In many cases, in particular when K is o-minimal case or an expansion of a p-adically closed field, we will see that B is already definable in C(X), see Remark 4.4. On the other hand O will not be definable in the ring structure when K is algebraically closed.

A crucial input and starting point of the paper is the definability of the Nullstellensatz in a weak, but surprisingly general form in Theorem 3.4. This says that for almost any ring A of functions from some given set S to a given field K, the n + 1-ary relation

{f 1 = 0} ∩ . . . ∩ {f n = 0} ⊆ {g = 0} of A is the Jacobson radical relation of A.
For example it suffices to ask that K is not algebraically closed, A contains the constant functions with value in K and that all functions in A without zero are invertible. This is explained in section 3. In the case of o-minimal or P -minimal structures it implies that the lattice of closed definable subsets of X is interpretable in C(X).

The technical heart of this paper is Section 5 where we prove our first main result (Theorem 5.5 and Corollary 5.6).

Theorem A Assume that K satisfies (BFin). Let τ be a non-zero and non-invertible element of O. Let X ⊆ K m be a definable set which has arbitrarily many germs1 at some point p 0 .

• If O = [-1, 1] then the ring of functions f ∈ C(X) such that f (p 0 ) ∈ Z is definable in (C(X), B).
• In any case the set of functions f ∈ C(X) such that f (p 0 ) ∈ τ Z is definable in (C(X), B), and its natural ring structure is interpretable in (C(X), B).

As a consequence the ring of integers Z is interpretable in (C(X), B).

If K = (K , . . . ) is an elementary extension of K, and X is the subset of K m defined by the same formula as X, there is a natural embedding of (C(X), B) into (C(X ), B ). It follows from the above result that, surprisingly enough, this is not an elementary embedding in general (Corollary 5.7).

Note that the above theorem is fairly general: it only assumes that K satisfies (BFin) and X has arbitrarily many germs at p 0 . In Section 6 we improve it by assuming, in addition to (BFin), that K has a good dimension theory for definable sets (see the axioms list (Dim) in Section 2). This holds true if K is o-minimal, P -minimal or C-minimal, and more generally if it is dp-minimal (see Remark 2.4). We can then prove our second main result.

Theorem B Assume that K satisfies (Dim) and (BFin). Let τ be a non-zero and noninvertible element of O. Let X ⊆ K m be a definable set of pure dimension d 2.

(1)

If O = [-1, 1] then C(X, Z) is definable in (C(X), B).
(2) In any case C(X, τ Z ) is definable in (C(X), B). Theorem 6.7 actually gives a more precise and more general statement. Let us also mention that in the P -minimal case (among others, see condition (Z) in Section 6) the condition on the pure dimension of X can be relaxed: the result holds true whenever X has no isolated point.

Model theoretic and topological set up

Let K be a field and let O be either the unit interval [-1, 1] of a total order of K, or the ring of a non-trivial valuation v of K.

In both cases we let O × denote the set of non-zero a ∈ O with a -1 ∈ O. This is a multiplicative subgroup of

K × = K \ {0}. We let | | : K × → K × /O × be
\ {0} = K × /O × .
The latter is a totally ordered abelian (multiplicative) group. We denote it by |K × |, or v(K × ) when additive notation is more appropriate.

For every x ∈ K m we set x = max(|x 1 |, . . . , |x m |), and for all X ⊆ K m we write X = { x : x ∈ X}; if m = 1 we simply write |X| (or v(X) in additive notation). Open and closed balls in K m with center c ∈ K m and radius r ∈ K × are defined as usually; both are clopen in the valued case. We endow K m with the topology defined by the open balls, and |K| with the image of this topology (which induces the discrete topology on |K × | in the valued case). For any set S, we write S for the topological closure of S and ∂S = S \ S for the frontier of S.

In this paper, definable means "first-order definable with parameters". Let L og = {e, * , } be the language of (additive or multiplicative) ordered groups, let L ring = {0, 1, +, -, ×} be the language of rings and let L be an extension of L ring containing a unary predicate symbol O. We will be working with expansions K of an ordered or valued field K to L, where the symbol O is interpreted as explained above. Definability refers to L for subsets of K m and to L og for subsets of |K × |. Now let C(X) be the ring of continuous definable functions as explained in the introduction. For all f, g ∈ C(X) the sets {f = g}, {f = g}, {|f | |g|} and so on, are defined as the subsets of X on which the corresponding relation holds true. For example {f = 0} = {x ∈ X : f (x) = 0} is the zero-set of f . On C(X) we work with the relation

g f ⇐⇒ ∀x ∈ X g(x) = 0 ⇒ f (x) = 0 .
We prove in Theorem 3.4 that this relation is definable in the ring C(X). Definition 2.1 Let X ⊆ K m be a definable set, p 0 ∈ X and let U be a definable neighborhood of p 0 in X. We say that a function s ∈ C(U ) vanishes on a germ at p 0 if s has a non-isolated zero at p 0 . We say that s 1 , . . . , s k ∈ C(U ) vanish on separated germs at p 0 if there are δ 1 , . . . , δ k ∈ C(U \ {p 0 }) with the following properties.

(S1) The sets S i = {s i = 0} \ {p 0 } are pairwise disjoint.

(S2) Each s i vanishes on a germ at p 0 .

(S3) For i = j, the function δ i is constantly 1 on S i and constantly 0 on S j .

(S4) Each δ i is bounded on U \ {p 0 }.
We call functions δ 1 , . . . , δ k with these properties separating functions for s 1 , . . . , s k . Finally we say that X has arbitrarily many germs at p 0 if for every positive integer k there is a definable neighborhood U of p 0 in X and k functions in C(U ) that vanish on separated germs at p 0 .

Examples 2.2 Intuitively this means that p 0 can be approached in X through arbitrarily many disjoint ways.

(1) For every r 2, X = K r has arbitrarily many germs at the origin (see Proposition 6.4).

(2) If K is o-minimal then X = K does not have arbitrarily many germs at 0, because any definable set S ⊆ K \ {0} whose closure contains 0 will necessarily meet one of the two intervals (-∞, 0) or (0, +∞).

(3) If K is any expansion of the real field R including the sin function then X = K has arbitrarily many germs at 0: take s i (x) = x sin((1 -ix)/(kx)) for 1 i k.

(4) If K is any expansion of a valued field whose value group is a Z-group then K has arbitrarily many germs at 0: take s i (x) = xχ i (x) where χ i is the characteristic function of the (clopen) set of elements of O \ {0} whose valuation is congruent to i modulo k.

A coordinate projection π : K m -→ K r is a map of the form π(x) = (x i ) i∈I for some I ⊆ {1, . . . , m} of size r 0; we write π I when necessary. The dimension of a non-empty set X is the maximal r m such that π(X) has non-empty interior for some coordinate projection π :

K m → K r . This is extended to dim(∅) = -∞. The local dimension of X at a point x ∈ K m is dim(X, x) = min dim B ∩ X : B is an open ball centered at x . Note that dim(X, x) = -∞ if and only if x /
∈ X, and that dim(X, x) = 0 if and only if x is an isolated point of X.

Definition 2.3 For every integer d 0 we write

∆ d (X) = x ∈ X : dim(X, x) = d .
Further we write W d (X) for the set of all x ∈ X for which there is an open ball B centered at x and a coordinate projection π : K m → K d that induces by restriction a homeomorphism between B ∩ X and an open subset of

K d .
Since the open balls are uniformly definable in L ring ∪{O} the sets ∆ d (X) and W d (X) are definable in L.

We say that K satisfies (Dim) if for every definable set X ⊆ K m and every definable map f : X → K r the following properties hold true.

(Dim1) dim(f (X)) dim(X). (Dim2) dim(X) = dim(X) and if X = ∅, then dim(∂X) < dim(X). (Dim3) If dim(X) = d 0 then dim(X \ W d (X)) < d. 2
Remark 2.4 These properties hold true in every dp-minimal expansion of a field which is not strongly minimal ([SW18]). This implies and generalises known results on o-minimal, C-minimal and P -minimal fields (see [vdD98], [START_REF] Haskell | Cell decompositions of C-minimal structures[END_REF], [START_REF] Haskell | A version of o-minimality for the p-adics[END_REF] and [CKDL17]).

Note that the sets ∆ d (X) are pairwise disjoint and that l

d ∆ l (X) is closed in X for each d, while W d (X) is open in X.
Property 2.5 Property (Dim1) implies that the dimension is preserved by definable bijections and that W d (X) ⊆ ∆ d (X) for every d 0. In particular the sets W d (X) are pairwise disjoint.

Proof: The first assertion is obvious, we prove the second one. If x ∈ W d (X), then there is a definable neighborhood U of x in X, a coordinate projection π :

K m → K d and an open subset V of K d such that π| U is a homeomorphism onto V . In particular dim U = d by the first assertion, hence dim W d (X) d. For every sufficiently small open ball B centered at x we have B ∩ X ⊆ U , hence π| B∩X is a homeomorphism onto a non-empty open subset of K d and so dim(B ∩ X) = d. This proves dim(X, x) = d hence W d (X) ⊆ ∆ d (X).
Since the sets ∆ d (X) are pairwise disjoint, so are the sets W d (X).

Definability of the poset of zero sets

Let X be a set and let A be a ring of functions X -→ K for some field K. We show in Theorem 3.4 that for a huge class of examples, the (n + 1)-ary relation {f 1 = 0} ∩ . . . ∩ {f n = 0} ⊆ {g = 0} of A is equivalent to g being in the Jacobson radical of the ideal (f 1 , . . . , f n ). In particular, theses relations are 0-definable in the ring A. The crucial ingredients are contained in Proposition 3.2 and Proposition 3.3.

Let I be an ideal of a ring A. The Jacobson radical Jac(I) of I is defined as the intersection of the maximal ideals of A containing I (cf. [Mat89, p. 3]). The Jacobson radical of the ring A is defined as Jac(0). Proposition 3.2 Let K be a field.

(1) If K is not algebraically closed, then there are polynomials u(x, y), v(x, y) in two variables over K such that the unique zero of xu(x, y) + yv(x, y) in K 2 is (0, 0).

(2) Assume that R is a real closed field and K = R[i] is its algebraic closure. We write a * for the complex conjugation of a ∈ K with respect to R. Then for the functions u(x, y) = x * and v(x, y) = y * defined on K 2 , the unique zero of xu(x, y)

+ yv(x, y) in K 2 is (0, 0). (3) Assume that K is a topological field (see [War93]) of characteristic = 2
, where a basis of neighborhoods of 0 ∈ K is given by the non-zero ideals of a ring O with fraction field K and non-zero Jacobson radical3 .

Then there are τ -continuous functions u, v : K 2 -→ K that are definable in the expansion (K, O) of K by the set O, such that the unique zero of xu(x, y)+yv(x, y) in K 2 is (0, 0).

Proof: (1) Since K is not algebraically closed there is a polynomial p

(x) = x d + a d-1 x d-1 +. . .+a 0 ∈ K[t] without zeroes in K such that d 1. Then its homogenization q(x, y) = x d + a d-1 x d-1 y + . . . + a 0 y d has a unique zero in K 2 , namely (0, 0): If b = 0, then q(a, b) = 0 just if a = 0. If b = 0, then q(a, b) = b d • p( a b ) = 0. We choose u(x, y) = x d-1 and v(x, y) = a d-1 x d-1 +. . .+a 0 y d-1
and see that xu(x, y)+ yv(x, y) = q(x, y) has the required properties.

(2) is clear. Claim. For every topological space X, each continuous function f : X -→ K and all

x 0 ∈ X with f (x 0 ) = 0 there is an open neighborhood U of x 0 in X such that the restriction of |f | : X -→ |K| to U is constant. Proof. Replacing f by 1 f (x 0 ) • f if necessary, we may assume that f (x 0 ) = 1. By the assumption in (iii) there is a non-zero element ε in the Jacobson radical of O such that 1 + εO is an open neighborhood of 1 in K. By continuity of f at x 0 there is a neighborhood U of x 0 in X with f (U ) ⊆ 1 + εO. Since εO ⊆ Jac(0), no element in 1 + εO can be in any maximal ideal of O. Hence f (U ) ⊆ 1 + εO ⊆ A × , which shows that |f (x)| = |1| for all x ∈ U .
Returning to the proof of (3), we define

U = |x -y| < |x + y| , V = 0 = |x -y| ≮ |x + y| , Z = x -y = x + y = 0 .
This is a partition of K 2 in (K, O)-definable sets. Note that x + y = 0 in U and x -y = 0 in V . Using the claim with X = K 2 it is easy to see that U and V are open in K 2 . Further, the functions u, v :

K 2 -→ K defined by u(x, y) = x + y if (x, y) ∈ U, x -y if (x, y) ∈ K 2 \ U and v(x, y) = x + y if (x, y) ∈ U, y -x if (x, y) ∈ K 2 \ U,
are continuous on U and on V . Moreover, since x -y = x + y = 0 on Z, both u and v tend to 0 at every point of Z. Thus u and v are continuous on X. By construction they are also definable in (K, O). On U , xu(x, y) + yv(x, y) = (x + y) 2 has no zero. On V , xu(x, y) + yv(x, y) = (x -y) 2 has no zeroes. Since K has characteristic = 2, the set Z is {(0, 0)}, which establishes the assertion.

Proposition 3.3 Let K be a field and let u, v : K 2 -→ K be functions such that the unique zero of xu(x, y) + yv(x, y) in K 2 is (0, 0). Let X be a set and let A be a ring of functions X -→ K such that A is closed under composition with u and v, i.e., u • (f, g), v • (f, g) ∈ A for all f, g ∈ A.

(1) For all f 1 , . . . , f n ∈ A there are g 1 , . . . , g n ∈ A with

{f 1 = 0} ∩ . . . ∩ {f n = 0} = {g 1 f 1 + . . . + g n f n = 0}. (2) If every f ∈ A without zeroes in X is a unit in A then for all f 1 , . . . , f n , g ∈ A we have {f 1 = 0} ∩ . . . ∩ {f n = 0} ⊆ {g = 0} =⇒ g ∈ Jac(f 1 , . . . , f n ).
(

) If A contains all constant functions X -→ K, then for all f 1 , . . . , f n , g ∈ A we have g ∈ Jac(f 1 , . . . , f n ) =⇒ {f 1 = 0} ∩ . . . ∩ {f n = 0} ⊆ {g = 0}. 3
Proof: (1). By induction on n, where n = 1 is trivial. The induction step readily reduces to the claim in the case n = 2. By assumption on A we know that h :=

f 1 • (u • (f 1 , f 2 )) + f 2 • (v • (f 1 , f 2 )) ∈ A.
By assumption on u, v we see that the zero set of h in X is {f 1 = 0} ∩ {f 2 = 0}. Hence we may take

g 1 = u • (f 1 , f 2 ) and g 2 = v • (f 1 , f 2 ).
(2). Assume n i=1 {f i = 0} ⊆ {g = 0} and let m be a maximal ideal of A containing f 1 , . . . , f n . We need to show that g ∈ m. By (1), there is some h in the ideal (f 1 , . . . , f n ) generated by f 1 , . . . , f n in A with zero set (3). Let x ∈ n i=1 {f i = 0} and let e : A -→ K be the evaluation map at x. Since A contains the constant functions we know that e is surjective, hence m = ker(e) is a maximal ideal of A. Then f 1 , . . . , f n ∈ m and so by assumption g ∈ m, i.e. g(x) = 0. Theorem 3.4 Let X be a set and let A be a ring of functions X -→ K containing the constant functions such that every f ∈ A without zeroes in X is a unit in A. Suppose one of the following conditions hold:

(a) K is not algebraically closed, or, (b) K is the algebraic closure of a real closed field R and A is closed under conjugation of K = R[i], or, (c) K is a topological field of characteristic = 2
, where a basis of neighborhoods of 0 ∈ K is given by the non-zero ideals of a ring O with fraction field K and nonzero Jacobson radical. Further assume that w•(f, g) ∈ A for every (K, O)-definable continuous function w : K 2 -→ K.

The following conditions are equivalent for all f 1 , . . . , f n , g ∈ A.

(1)

{f 1 = 0} ∩ . . . ∩ {f n = 0} ⊆ {g = 0} ⇐⇒ g ∈ Jac(f 1 , . . . , f n ).
(2) g ∈ Jac(f 1 , . . . , f n ).

(3) For all h ∈ A, the element 1 + hg is a unit modulo the ideal (f 1 , . . . , f n ).

Consequently, the (n + 1)-ary relation

{f 1 = 0} ∩ . . . ∩ {f n = 0} ⊆ {g = 0} of A is definable in the ring A by the L ring -formula ∀x ∃y 1 , . . . , y n , z (1 + x•g)•z = 1 + y 1 f 1 + . . . + y n f n .
Of particular interest for us is the case n = 1. Hence the binary relation f g defined as {f = 0} ⊆ {g = 0}, is 0-definable in A.

Proof: The assumptions in Proposition 3.3 hold by Proposition 3.2. Hence Proposition 3.3(2),(3) imply the equivalence of (1) and (2). The equivalence of (2) and (3) holds by Remark 3.1.

Examples 3.5 Let X be a topological space. Then Theorem 3.4 applies to the following rings A of functions X -→ K.

(a) K is an ordered field or a p-valued field and

• A is the ring of continuous functions X -→ K, or,

• X ⊆ K n is definable in K and A is the ring of definable continuous functions X -→ K, or,

• X ⊆ K n is open (definable) and A is the ring of (definable) k-times differentiable functions X -→ K, or,

• X ⊆ K n is a variety and A is the ring of rational functions X -→ K without zeroes on X (sometimes referred to as regular functions in the literature).

In each case, condition (a) of Theorem 3.4 applies.

(b) A is the ring of continuous functions X -→ C, or, X ⊆ C n and A is the ring of continuous semi-algebraic functions X -→ C. In both cases, condition (b) of Theorem 3.4 applies.

(c) K is a valued field of characteristic = 2, furnished with the valuation topology and A is the ring of continuous functions X -→ K, or, X ⊆ K n is definable and A is the ring of definable continuous functions X -→ K. In both cases, condition (c) of Theorem 3.4 applies, where O is the valuation ring of K.

Lemma 4.1 (1)

There is ν m ∈ C(K m ) such that |ν m (x)| = x for every x ∈ K m . Now let X ⊆ K m be a definable set, a ∈ X and let B be a closed ball (resp. B 0 an open ball) with radius r ∈ K × (resp. r 0 ∈ K × ) and center a.

(2) Each of the sets {a}, B and B c 0 = K m \ B 0 are zero sets of functions from C(X). We pick such functions and denote them by δ a , δ B and δ B c 0 respectively. Proof: (1). For every

x = (x 1 , . . . , x m ) let ν m (x) = x 1 if x = |x 1 |, ν m (x) = x 2 if x = |x 2 | > |x 1 |
, and so on. It is obviously definable, and easily seen to be continuous on K m .

(2) and (3). We may take δ a (x) = ν m (x -a) restricted to X, using (1). For the other functions, in the valued case we may take the indicator function of X \ B for δ B , and the indicator function of B 0 for δ B c 0 as well as for δ B,B 0 (they are continuous because B and B 0 are clopen). In the ordered case we have 0 < r 0 < r and |K| identifies with the set of non-negative elements of K so we may take δ B (x) = max{0, x -a -|r|}, δ B c 0 (x) = max{0, r -x -a } and δ B,B 0 (x) = max{0, min{1, u(x)}} where u(x) = (r -x -a )/(r -r 0 ). Lemma 4.2 There are formulas Point(f ) and Inter(f, g, h) in L ring , and a formula Isol(s, p) in L ring ∪ {B} such that for every definable set X ⊆ K m :

(1) C(X) |= Point(f ) ⇐⇒ f has a single zero in X.

(2) C(X) |= Inter(f, g, h) ⇐⇒ {f = 0} ∩ {g = 0} = {h = 0}.

(3) (C(X), B) |= Isol(s, p) ⇐⇒ {p = 0} is an isolated point of {s = 0}.

Proof: Let V(X) = {s = 0} : s ∈ C(X) ordered by inclusion. It is a bounded distributive lattice. Indeed for every f, g ∈ C(X) we have

{f = 0} ∪ {g = 0} = {f g = 0} {f = 0} ∩ {g = 0} = {ν 2 (f, g) = 0},
where ν 2 ∈ C(K 2 ) is the function given by Lemma 4.1 (1). For every a ∈ K m , {a} = {δ a = 0} where δ a is given by Lemma 4.1 (2), so the atoms of (V(X), ⊆) are exactly the singletons. The two first points then follow from the fact that (V(X), ⊆) is uniformly interpretable in C(X) as the quotient ordered set of the preorder , see Theorem 3.4.

For item (3), assume that {p = 0} = {p 0 }. If p 0 is an isolated point of {s = 0}, let B 0 be an open ball with center p 0 which is disjoint from S = {s = 0} \ {p 0 }. By Lemma 4.1 (2) there is a function

δ B c 0 ∈ C(X) such that δ B c 0 (x) = 0 if and only if x / ∈ B 0 . Finally let u = ν 2 (s, δ B c 0 )
where ν 2 is given by Lemma 4.1 (1). Then {u = 0} = {s = 0} ∩ B c 0 = S, hence s pu and s u. Conversely if there is u ∈ C(X) such that s pu and s u then {u = 0} is a neighborhood of p 0 disjoint from S hence p 0 is an isolated point {s = 0}. So this property is axiomatized by the formula Isol(s, p) ≡ Point(p) ∧ p s ∧ ∃u s pu ∧ s u .

Proposition 4.3 For every parameter-free formula ϕ(y) in L ring (resp. L ring ∪ {O}) in k free variables, there is a parameter-free formula [ϕ] in L ring (resp. L ring ∪ {B}) in k + 1 variables such that for every definable set X ⊆ K m , every h ∈ C(X) k and every s ∈ C(X) we have

(C(X), B) |= [ϕ](h, s) ⇐⇒ ∀x ∈ {s = 0}, K |= ϕ(h(x)).
Proof: If ϕ(y) is a polynomial equation P (y) = 0 (y a single variable), we may take [ϕ](y, s) as s P (y). If ϕ(y) is the formula P (y) ∈ O, we may take [ϕ](y, s) as

∀p Point(p) ∧ p s -→ ∃f f ∈ B ∧ p P (y) -f ,
expressing the fact that a function P (y) has values in O on the zero set of s just if for all x ∈ {s = 0}, P (y) agrees with a function f ∈ B at x. This proves the result for atomic formulas. For arbitrary formulas, [ϕ] is defined by induction as follows.

• [ϕ ∧ ψ] is [ϕ] ∧ [ψ]. • [¬ϕ](y, s) is ∀p (Point(p) ∧ p s) → ¬[ϕ](y, p) . • [∃w ϕ](y, s) is ∀p (Point(p) ∧ p s) → ∃w [ϕ](w, y, p) .
We only show the right to left implication of the claimed equivalence in the case of an existential quantifier, all other implications are straightforward. Take h ∈ C(X) k , s ∈ C(X) and assume that K |= ∃w ϕ(w, h(x)) for every x ∈ {s = 0}. We need to show that (C(X), B) |= ∀p (Point(p) ∧ p s) → ∃w [ϕ](w, y, p) . So take p ∈ C(X) whose unique zero p 0 satisfies s(p 0 ) = 0. By assumption there is some c ∈ K with K |= ϕ(c, h(p 0 )). Let h 0 ∈ C(X) be the constant function with value c. Now, for every z ∈ X and every q ∈ C(X) that has a unique zero z, if p(z) = 0 then K |= ϕ(h 0 (z), h(z)). By induction, this means (C(X), B) |= [ϕ](h 0 , h, p), as required.

Finally, by choice of the formulas [ϕ] we see that [ϕ] is an

L ring -formula, if ϕ is an L ring -formula. Remark 4.4 If O is definable in K by a formula ϕ(x, a) in L ring ,
where a is an n-tuple of parameters from K, then Proposition 4.3 applied to ϕ(x, y) implies that B is definable in C(X) by the L ring -formula [ϕ](x, a * , 0) with parameters a * , the constant function of C(X) n with value a. There are plenty of such fields with definable orders or valuations (including all the non-algebraically closed local fields):

• real-closed and p-adically closed fields;

• the field of rational numbers (by Lagrange's Four Squares Theorem);

• one-variable functions fields over number fields [START_REF] Miller | On existential definitions of C.E. subsets of rings of functions of characteristic 0[END_REF];

• dp-minimal valued fields that are not algebraically closed [START_REF] Johnson | On dp-minimal fields[END_REF];

• the field of Laurent series F ((t)) with the natural valuation [START_REF] Ax | On the undecidability of power series fields[END_REF]; and numerous others: see [FJ17] for a summary on Henselian valuation rings definable in their fraction field.

Having arbitrarily many germs at a point is a local property, but it can be made a bit more global as follows.

Property 4.5 Let X ⊆ K m be a definable set and let p 0 ∈ X. Then X has arbitrarily many germs at p 0 (see Definition 2.1) if and only if for every positive integer k there exist k functions in C(X) that vanish on separated germs at p 0 with separating functions in C(X \ {p 0 }).

Proof: One implication is obvious. For the converse, assume that a definable neighborhood U of p 0 in X is given together with v 1 , . . . , v k ∈ C(U ) that vanish on separated germs at p 0 , and with corresponding separating functions d 1 , . . . , d k ∈ C(U \ {p 0 }). Restricting U if necessary, by continuity at p 0 , we may assume that each v i is bounded on U and that U = B 0 ∩ X for some open ball B 0 with center p 0 . Let B be a closed ball with center p 0 contained in B 0 . By Lemma 4.1 (3) we have a bounded function

h = δ B,B 0 ∈ C(X) with h(x) = 0 on X \ B 0 and h(x) = 1 on B ∩ X.
For each i k let u i (x) = 0 on X \ B 0 and u i (x) = h(x)v i (x) on B 0 ∩ X. Similarly let δ i (x) = 0 on X \ B 0 and δ i (x) = h(x)d i (x) on B 0 ∩ X \ {p 0 }. Each u i (resp. δ i ) is continuous on B 0 ∩ X and tends to 0 at every point in ∂(B 0 ∩ X) (because v i and δ i are bounded) hence u i ∈ C(X) and δ i ∈ C(X \ {p 0 }). Now let s i = ν 2 (δ B , u i ), where ν 2 , δ B ∈ C(K 2 ) are given by Lemma 4.1 (1),(3). By construction δ i is bounded on X \ {p 0 }, δ i (x) = d i (x) on B ∩ X \ {p 0 } and {s i = 0} equals {v i = 0} ∩ B. So the functions s i , δ i inherit from v i , d i all the properties (S1)-(S4) of functions vanishing on separated germs, cf. Definition 2.1.

Constructing integers using limit values and chunks

The present section is devoted to our first main results of interpretability and definability. Our construction is based on the following subset of K. Let X ⊆ K m be a definable set. Let s, p ∈ C(X) be such that the zero-set of p is a single point p 0 and s vanishes on a germ at p 0 . Then for all f, g ∈ C(X) for which g has no zeroes in the set S = {s = 0} \ {0}, we consider the continuous function f /g on S and write Γ for its graph. We define

L s,p (f /g) = l ∈ K : (p 0 , l) ∈ Γ .
Informally, L s,p is the set of "limit values" at p 0 of f /g restricted to S. Note that this is always a closed definable subset of K. We will also consider the following relations on C(X). Note that they are definable in L ring ∪ {B} using Proposition 4.3.

|f | s |g| ⇐⇒ {s = 0} ⊆ {|f | |g|} |f | < s |g| ⇐⇒ {s = 0} ⊆ {|f | < |g|}.
Lemma 5.1 There is a parameter-free formula Limit(f, g, h, s, p) in L ring such that for any definable set X ⊆ K m , we have C(X) |= Limit(f, g, h, s, p) if and only if

• the zero-set of p is a single point p 0 ,
• s vanishes on a germ at p 0 ,

• g has no zeroes in the set S = {s = 0} \ {p 0 } and

• h(p 0 ) ∈ L s,p (f /g).
Proof: The first three properties are defined by the conjunction χ(g, s, p) of the formulas in Lemma 4.2. Hence we define χ(g, s, p) as Point(p) ∧ ¬Isol(s, p) ∧ Inter(g, s, p) ∨ Inter(g, s, 1) .

Let f, g, h, s, p ∈ C(X) be such that C(X) |= χ(g, s, p) and let p 0 be the zero of p. Pick τ ∈ K × with |τ | < 1. For every ε, v ∈ C(X) with ε(p 0 ) = 0 and v(p 0 ) = 0, by continuity, there is an open ball B 0 centered at p 0 such that B 0 ∩ X ⊆ {v = 0} and for every

x ∈ B 0 ∩ X, |ε(x)| > |τ ε(p 0 )| and |h(x) -h(p 0 )| < |τ ε(p 0 )|. So if h(p 0 ) ∈ L s,p (f /g), then
there is a point q 0 ∈ B 0 ∩ X \ {p 0 } with s(q 0 ) = 0 and f (q 0 ) g(q 0 ) -h(q 0 ) ε(q 0 ) .

Note that q 0 ∈ B 0 \ {p 0 } implies that q pv, and that s(q 0 ) = 0 implies g(q 0 ) = 0, so the second condition above is equivalent to |f (q 0 ) -g(q 0 )h(q 0 )| |g(q 0 )ε(q 0 )|. Conversely, if there are such points q 0 for any ε, v ∈ C(X) with ε(p 0 ) = 0 and v(p 0 ) = 0, then setting v = δ B c 0 for any open ball B 0 centered at p 0 (with δ B c 0 given by Lemma 4.1(2)), we get that h(p 0 ) ∈ L s,p (f /g). Thus we can take for Limit(f, g, h, s, p) the conjunction of χ(g, s, p) and the formula ∀ε, v p vε → ∃q Point(q) ∧ q pv ∧ q s ∧ |f -gh| q |gε| . Lemma 5.2 Let l 1 , . . . , l k ∈ K and let X ⊆ K m be a definable set. Let p ∈ X be such that p has a single zero p 0 at which X has arbitrarily many germs. Let s 1 , . . . , s k ∈ C(X) be vanishing on separated germs at p 0 and let S i = {s i = 0} \ {p 0 }. Then there are f, g ∈ C(X) with g(x) = 0 on S = S 1 ∪ • • • ∪ S k such that the restriction of f /g to each S i has constant value l i . In particular:

In other words, Theorem 5.5 says essentially that, given any τ ∈ K × with |τ | < 1 and any definable set X ⊆ K m with arbitrarily many germs at some point p 0 , then the set of all h ∈ C(X) taking values in τ Z at p 0 , is definable in (C(X), B); the definition is independent of X, K and m and is in the two definable parameters p 0 , τ . Also notice that by Remark 4.4, the set B is definable in L ring in many cases, hence Theorem 5.5 applies to the pure ring C(X) in all these cases.

Proof: In order to ease the notation, in this proof we write C(X) |= χ(. . . ) instead of (C(X), B) |= χ(. . . ), for every formula in L ring ∪ {B}.

We first construct a formula which axiomatizes (uniformly) the property that τ = τ * (p 0 ) ∈ O \ {0} and L s,p (f /g) is a bounded τ Z -chunk. We use the formula Limit(f, g, h, s, p) stated in Lemma 5.1. It will be convenient to abbreviate it as ϕ σ (h) where σ = (f, g, s, p). Thus for every f, g, v, s, p ∈ C(X) we have

v(p 0 ) ∈ L s,p (f /g) ⇐⇒ C(X) |= ϕ σ (v).
Let ψ σ (α, α , β, γ, p, τ * ) be the conjunction of the following formulas.

1. p αα -1 → ϕ σ (α ). The formula LimBCh × (f, g, s, p, τ * ) is then defined as LimCh × (f, g, s, p, τ * ) ∧ ∃δ∀α ϕ σ (α) → |α| p |δ| .

Clearly it holds true if and only if τ = τ * (p 0 ) ∈ I and L s,p (f /g) is a bounded τ Z -chunk. Finally, let Int × (h, p, τ * ) be the formula ∃f, g, s, Limit(f, g, h, s, p) ∧ LimBCh × (f, g, s, p, τ * ).

If C(X) |= Int × (h, p, τ * ) then p has a single zero p 0 , s vanishes on a germ at p 0 and h(p 0 ) ∈ L s,p (f /g); further, by the above, τ = τ * (p 0 ) ∈ I, and L s,p (f /g) is a bounded τ Z -chunk (in particular it is a bounded discrete subset of K). Since L s,p (f /g) is is a closed definable subset of K it is then finite by (BFin). Thus by Remark 5.4 (in multiplicative notation) there is a non-negative integer n such that L s,p (f /g) = {τ i : -n i n}. We obtain L s,p (f /g) ⊆ τ Z and so h(p 0 ) ∈ τ Z .

Conversely, assume that p has a single zero p 0 and for this zero we have τ = τ * (p 0 ) ∈ I and h(p 0 ) = τ k for some k ∈ Z. Let T h = {τ i : -|k| i |k|}. Since X has arbitrarily many germs at p 0 , we may invoke Property 4.5 to obtain functions s i ∈ C(X) that vanish at separated germs at p 0 , for -|k| i |k| 5 . Let s be the product of the s i 's and S = {s = 0}. By Lemma 5.2 there are f, g ∈ C(X) such that g(x) = 0 on S and

T h = L s,p (f /g) = l(p 0 ) : l ∈ C(X) and p 0 ∈ S ∩ {f = gl} .
Thus C(X) |= Limit(f, g, h, s, p) ∧ LimBCh × (f, g, s, p, τ * ), hence C(X) |= Int × (h, p, τ * ). This finishes the proof of the first equivalence of the theorem.

If K has a total order that is definable in L ring ∪ {O} and compatible with +, then by Proposition 4.3 the following relations are definable in L ring ∪ {O}.

g s f ⇐⇒ {s = 0} ⊆ {g f } g ≺ s f ⇐⇒ {s = 0} ⊆ {g ≺ f }
Let Φ σ (α, β, γ, p, τ * ) be the conjunction of the following formulas.

1. ϕ σ (-α ).

2. -γ p α + β p γ → ϕ σ (α + β).

∀u -

γ p u p γ → ∃ξ ϕ σ (ξ) ∧ ξ p u ≺ p ξ + τ * . 4. ∀u, ξ, ξ B(u, ξ, p, τ * ) ∧ B(u, ξ , p, τ * ) → p ξ -ξ ,
where B(u, ξ, p, τ * ) stands for ϕ σ (ξ) ∧ ξ p u ≺ p ξ + τ * .

We then let LimCh + (f, g, s, p, τ * ) be the conjunction of 0 ≺ p τ * with ϕ σ (τ * ) and

∀α, β, γ ϕ σ (α) ∧ ϕ σ (β) ∧ ϕ σ (γ) → Φ σ (α, β, γ, p, τ * ) ,
and let LimBCh + (f, g, s, p, τ * ) be LimCh + (f, g, s, p, τ * ) ∧ ∃δ∀α ϕ σ (α) → α p δ .

Finally we take Int + (h, p, τ * ) as ∃f, g, s Limit(f, g, h, s, p)∧LimBCh × (f, g, s, p, τ * ) . The proof that it satisfies the second equivalence in the Theorem is analogous to the multiplicative case and left to the reader.

Corollary 5.6 Assume that K satisfies (BFin) and take τ ∈ K × with |τ | < |1|. Let X ⊆ K m be a definable set having arbitrarily many germs at a given point p 0 . Let p, τ * ∈ C(X) be such that p 0 is the unique zero of p and τ * (p 0 ) = τ . Then the ring of integers (Z, +, ×) is interpretable in (C(X), B) with parameters (p, τ * ).

Proof: For every k, l ∈ Z let l|k denote the divisibility relation. It is well known (see for example [Ric89]) that multiplication is 0-definable in (Z, +, |, ), hence it suffices to interpret the latter structure in (C(X), B). Let Z = {f ∈ C(X) : f (p 0 ) ∈ τ Z }, and for every f ∈ Z let σ(f ) be the unique k ∈ Z such that f (p 0 ) = τ k . Then

f (p 0 ) ∈ τ Z ⇐⇒ (C(X), B) |= Int × (f, p, τ * ),
by Theorem 5.5 and therefore Z is definable in (C(X), B). The equivalence relation σ(f ) = σ(g) on Z is also definable, because

σ(f ) = σ(g) ⇐⇒ f (p 0 ) = g(p 0 ) ⇐⇒ (C(X), B) |= p f -g.
For every f, g ∈ Z we have obviously σ(f ) + σ(g) = σ(f g) and

σ(g) σ(f ) ⇐⇒ |g(p 0 )| |f (p 0 )| ⇐⇒ (C(X), B) |= |g| p |f |.
This gives an interpretation of (Z, +, ) in (C(X), B). Moreover σ(g) divides σ(f ) if and only if f (p 0 ) ∈ (τ σ(g) ) Z = (g(p 0 )) Z . We obtain

σ(f )|σ(g) ⇐⇒ (C(X), B) |= Int × (f, p, g)
in the terminology of Theorem 5.5. Consequently (Z, +, |, ) is interpretable in (C(X), B), from which the result follows.

We say that

|K × | is τ -archimedean for some τ ∈ K if τ Z is coinitial in |K × |, that is for every x ∈ K × there is an integer k such that |τ k | |x|.
In the ordered case, K is 2-archimedean if and only if it is archimedean in the sense that Z is cofinal in K.

Corollary 5.7 Let K = (K , . . . ) be any elementary extension of K and assume that K and K satisfy (BFin). Let X ⊆ K m be a definable set having arbitrarily many germs at a given point p 0 . Let X ⊆ K m be defined by a formula defining X. On the other hand, since K is not τ -archimedean, there exists an element a ∈ K such that |a | |τ k | for every k ∈ Z. Let f be the constant function on X with value a . For every

If |K × | is τ -archimedean for some τ ∈ K and |K × | is not τ -archimedean, then the natural L ring ∪ {B}-embedding of C(X) into C(X )
g ∈ C(X ) such that g (p 0 ) ∈ τ Z , |f (p 0 )| |g (p 0 )|, hence the formula in ( * ) is not satisfied in (C(X ), , B).
Remark 5.8 Let R ⊆ S be real closed fields. Let L(R n ) be the lattice of closed and semi-algebraic subsets of R n . In [START_REF] Astier | Elementary equivalence of lattices of open sets definable in o-minimal expansions of real closed fields[END_REF] it is shown that the natural embedding L(R n ) → L(S n ) is an elementary map. However, if n 2 and R = R S, then we know from 5.7 that the natural embedding C(R n ) → C(S n ) is not elementary. Consequently for real closed fields R, there is no interpretation of the ring C(R n ) in the lattice L(R n ) that is independent of R.

Defining integers using local dimension

In this section, assuming (Dim), we show that the set of functions in C(X) that take values in τ Z or τ Z for some τ ∈ K × at some point p 0 as in Theorem 5.5, is definable in (C(X), B), provided p 0 is not of small local dimension. In order to do so we first prove that the points at which X has arbitrarily many germs are dense among those at which X has local dimension 2.

Recall from Definition 2.3 that ∆ k (X) denotes the set of x ∈ X of local dimension k and W k (X) denotes the set of x ∈ X such that there is an open ball B centered at x and a coordinate projection π : K m → K k which induces by restriction a homeomorphism between B ∩ X and an open subset of K k . Lemma 6.1 Assume that K satisfies (Dim). For every definable set X ⊆ K m and every integer k 0, W k (X) is a dense subset of ∆ k (X). If non-empty, both of them have dimension k.

Proof: We already now that W k (X) ⊆ ∆ k (X) by Property 2.5. Pick x ∈ ∆ k (X) and an open ball B ⊆ K m centered at x. By shrinking B if necessary we may assume that

dim(B ∩ X) = k. From (Dim3) we know W k (B ∩ X) = ∅. On the other hand, W k (B∩X) ⊆ W k (X) because B∩X is open in X. Consequently W k (B∩X) ⊆ B∩W k (X) and so B ∩ W k (X) = ∅. This proves density. By (Dim2) it only remains to prove that W k (X) has dimension k, provided it is not empty. Clearly dim W k (X) k. If dim W k (X) = l > k then by (Dim3) W l (W k (X)) is non-empty. But W k (X) is open in X, hence W l (W k (X)) is contained in W l (X). So W l (W k (X)
) is contained both in W l (X) and in W k (X), a contradiction since W l (X) and W k (X) are disjoint by Property 2.5.

For any two subsets A, B of a topological space X, the strong order is defined by

B A ⇐⇒ B ⊆ A \ B.
If X ⊆ K m is definable and f, g ∈ C(X) we define g f ⇐⇒ {g = 0} {f = 0}. Lemma 6.2 For every definable set X ⊆ K m and every f, g ∈ C(X) g f ⇐⇒ C(X) |= ∀h (f gh → g h).

In particular is definable in L ring .

Proof: For every definable set U ⊆ X that is open in X and each p 0 ∈ U there is a function h ∈ C(X) with h(p 0 ) = 0 and h(x) = 0 on X \ U (using Lemma 4.1 it suffices to take h = δ B c 0 for any open ball B 0 with center p 0 such that B ∩ X is contained in U ), hence p 0 ∈ {h = 0} ⊆ U . Hence the sets {h = 0} with h ∈ C(X) form a basis of Proof: For each k 1 we need to define s i and δ i verifying conditions (S1)-(S4) of Definition 2.1. It suffices to do the case of K r (and then take restrictions to U of the functions s i and δ i built for K r ). Let V k = K r , take a hyperplane H ⊂ K r containing p 0 and k distinct lines L 1 , . . . , L k passing through p 0 and not contained in H. Let σ i : K r → L i be the projection onto L i along H. Now let s i (x) = ν r (x -σ i (x)) where ν r is given in Lemma 4.1 (1). Conditions (S1) and (S2) are fulfilled since {s i = 0} = L i for each i. Finally let δ i = j =j δ i,j where δ i,j (x) = s j (x)/s j (σ i (x)). Clearly δ i,j (x) ∈ C(K r \ {p 0 }) and we have δ i,j (x) = 1 on L i \ {p 0 } and δ i,j (x) = 0 on L j \ {p 0 }, hence δ i has properties (S3) and (S4).

The intuition coming from real geometry suggests that having local dimension 2 should be necessary (and sufficient) for a definable set to have arbitrarily many germs at a point. In contrast, the next result shows that local dimension 1 (which is obviously necessary) is sufficient at least if K has the following property, which holds true for example in P -minimal structures like the p-adics.

(Z) v(K × ) is elementarily equivalent to (Z, +, ) (i.e., it is a Z-group) and for every definable set X ⊆ K × , v(X) is definable in v(K × ). and let S i = v -1 (A i ) ∩ U . The sets S i are pairwise disjoint, clopen in U \ {p 0 }, and p 0 belongs to the closure of each of them. So the functions defined by s i (x) = 0 if x ∈ S i and s i (x) = δ p 0 (x) otherwise (where {δ p 0 = 0} = {p 0 }, see Lemma 4.1) are continuous on U . Clearly each of them vanishes on a germ at p 0 since {s i = 0} = S i ∪ {p 0 }. Remark 6.6 For all definable sets X ⊆ K m , Y ⊆ K n and every point p 0 ∈ X, if there is a definable homeomorphism ϕ : U → V such that U (resp. V ) is a definable neighborhood of p 0 in X (resp. of ϕ(p 0 ) in Y ) then X has arbitrarily many germs at p 0 if and only if Y has arbitrarily many germs at ϕ(p 0 ), because this property is local. It then follows from Proposition 6.4 that for every definable set X ⊆ K m and every integer k 2, X has arbitrarily many germs at every point of W k (X) for every k 2 (and even for k = 1 if K satisfies (Z) by Proposition 6.5).

For every integer k 0, every definable set X ⊆ K m and every Y ⊆ K let C k (X, Y ) (resp. C k (X, Y )) be the set of functions f ∈ C(X) such that f (x) ∈ Y for every x ∈ ∆ k (X) (resp. x ∈ l k ∆ l (X)), and f (x) = 0 otherwise.

Remark 3. 1

 1 One checks easily that Jac(0) = {a ∈ A | ∀x ∈ A : 1 + ax ∈ A × }, where A × denotes the units of A. Translating this description for A/I back to A shows that Jac(I) = {a ∈ A | ∀x ∈ A ∃y ∈ A, z ∈ I : (1 + ax)y = 1 + z}.

( 3 )

 3 We write |K| = {aO | a ∈ K} for the set of principal fractional ideals of O and |a| = aO for a ∈ K. Further we write |a| |b| instead of |a| ⊆ |b|.

  n i=1 {f i = 0}. In particular h ∈ m and {h = 0} ⊆ {g = 0}. Suppose g / ∈ m. Then there is some a ∈ A and some m ∈ m with ag + m = 1, in particular {g = 0} ∩ {m = 0} = ∅. Since {h = 0} ⊆ {g = 0} we get {h = 0} ∩ {m = 0} = ∅. By (1) again, there are b 1 , b 2 ∈ A such that b 1 h + b 2 m has no zeroes. But then by assumption on A, b 1 h + b 2 m is a unit of A, contradicting h, m ∈ m.

( 3 )

 3 If B ⊆ B 0 , then there is a function δ B,B 0 ∈ C(X) with values in O such that δ B,B 0 vanishes on B c 0 and {δ B,B 0 = 1} = B.

  2. |1|p |αβγ| p |γ 2 |. 3. ∀u |1| p |uγ| p |γ 2 | → ∃ξ ϕ σ (ξ) ∧ |ξτ * | < p |u| p |ξ| . 4. ∀u, ξ, ξ A(u, ξ, p, τ * ) ∧ A(u, ξ , p, τ * ) → p ξ -ξ , where A(u, ξ, p, τ * ) stands for ϕ σ (ξ) ∧ |ξτ * | < p |u| p |ξ|.Clearly τ = τ * (p 0 ) ∈ I and L s,p (f /g) is a τ Z -chunk if and only if C(X) satisfies the conjunction, which we will denote LimCh × (f, g, s, p, τ * ), of |0| < p |τ * | < p |1|, of ϕ σ (τ * ) and of ∀α, α , β, γ ϕ σ (α) ∧ ϕ σ (β) ∧ ϕ σ (γ) → ψ σ (α, α , β, γ, p, τ * ) .

  is not an elementary embedding. Proof: Let p ∈ C(X) be such that {p = 0} = {p 0 }, and τ * be the constant function on X with value τ . The assumption that |K × | is τ -archimedean implies that |0| < |τ | < |1|. For every f ∈ C(X), there exists k ∈ Z such that |f (p 0 )| |τ k |, so there exists g ∈ C(X) such that |f (p 0 )| |g(p 0 )| and g(p 0 ) ∈ τ Z (it suffices to take g = (τ * ) k ). Thus by Theorem 5.5 ( * ) (C(X), B) |= ∀f ∃g, |f | p |g| ∧ Int × (g, p, τ * ).

Proposition 6. 5

 5 If K satisfies (Z), then every definable open subset U of K has arbitrarily many germs at every point of U .Proof: For any p 0 ∈ U , the set {v(u -p 0 ) : u ∈ U \ {p 0 }} is a Presburger set by (Z), and it is not bounded above in v(K × ). Hence it contains some setA = ξ ∈ v(K × ) : α ξ and ξ ≡ a [N ]for some α ∈ v(K × ) and some integers a and N 1 (where ξ ≡ a [N ] denotes the usual congruence relation ξ -a ∈ N v(K × )). Given an integer k 1, for 1 i k let A i = ξ ∈ A : α ξ and ξ ≡ a + iN[kN ]

  ], P -minimal[START_REF] Haskell | A version of o-minimality for the p-adics[END_REF] or C-minimal[START_REF] Haskell | Cell decompositions of C-minimal structures[END_REF], and more generally if it is a dp-minimal ordered or valued field[START_REF] Simon | On dp-minimal ordered structures[END_REF],[START_REF] Jahnke | Dp-minimal valued fields[END_REF]. The same holds true if K is any Henselian

(non-trivially) valued field of characteristic 0, or any algebraically bounded expansion of such [vdD89]. But property (BFin) is also obviously satisfied if O is compact, hence if (K, O) is any expansion of R, the field of real numbers, or any valued local field (that is a finite extension of the field Q p of p-adic numbers, or the field of Laurent series F ((t)) over a finite field F ).

  the residue map and extend it by |0| = 0. In the ordered case this is just the usual absolute value, in the valued case |x| is a multiplicative notation for the valuation. The set |K| = {|x| : x ∈ K} is totally ordered by the relation |y| |x| if and only if y ∈ x•O. The multiplication defined on |K| by |x| • |y| = |xy| extends the multiplication of |K|

	Note that in the valued case, |y|	|x| if and only if v(y)	v(x).

Roughly speaking, X has arbitrarily many germs at p0 if there exists arbitrarily many disjoint definable subsets of S of X \ {p0} such that p0 ∈ S. See Section

for a precise definition.

In the classical cases of o-minimal, C-minimal and P -minimal structures, the set ∆ d (X) is usually considered instead of W d (X) in (Dim3). However it is this slightly stronger statement with W d (X) which we need in Section 6. It appears in Proposition 4.6 of [SW18].

Our principal example here is a proper valuation ring O of K.

Basic properties of zero setsWe collect a few basic facts, which will be used in the rest of the paper. We continue to work with the set up of the introduction and section 2.

This obviously happens in the ordered case, where K is an ordered field and O is the interval [-1, 1]. But Theorem

5.5 does not restrict to this case, as it does not assume any relation at all between O and the order .

Here |k| denotes the ordinary absolute value in Z, not the image of k in |K|.

(1) {l 1 , . . . , l k } = L s,p (f /g), with s = s 1 s 2 • • • s k .

(2) {l 1 , . . . , l k } = l(p 0 ) : l ∈ C(X) and p 0 ∈ S ∩ {f = gl} .

Proof: Let δ 1 , . . . , δ s ∈ C(X \ {p 0 }) be separating functions for s 1 , . . . , s k . Clearly f = i k l i δ i p (extended by 0 at p 0 ) and g = p have the required properties, from which items (1) and (2) follow immediately. For the second item, note that p 0 ∈ S ∩ {f = gl} simply means that the function l takes the same values as f /g on a subset of S having points arbitrarily close to p 0 , hence l(p 0 ) ∈ L s,p (f /g) by continuity of l.

Definition 5.3 Let (G, +, ) be totally pre-ordered abelian group, hence is a total preorder satisfying x y ⇒ x+z y +z. For τ ∈ G with τ > 0 we write τ Z = {τ k : k ∈ Z}.

We call a subset T of G a τ Z-chunk of G if τ ∈ T and for all α, β, γ ∈ T :

(1) -α ∈ T ;

(2

This should be seen as a finitary, and hence definable, version of integer parts as studied by [START_REF] Mourgues | Every real closed field has an integer part[END_REF] in the case of real closed fields.

Remark 5.4 For every finite τ Z-chunk of G one checks easily that there is some integer n with G = {-nτ, . . . , -τ, 0, τ, 2τ, . . . , nτ }.

In the ordered case (and more generally when there is a total pre-order on K, definable in L ring ∪ {O} and compatible with the group structure of (K, +)), we can consider chunks of (K, +, ). In the valued case however we have to lift chunks from |K × | to K × . So we will consider chunks of the totally pre-ordered multiplicative group (K × , ×, O ) where O is the inverse image of the order of |K × |, i.e., x O y is just |x| |y|. In this case we adapt the terminology to multiplicative language. For example, for τ ∈ K × with |τ | < 1 we write τ Z = {τ k : k ∈ Z} instead of τ Z.

Theorem 5.5 Assume that K satisfies (BFin). Let I = {x ∈ K : |0| < |x| < |1|}. There is a parameter-free formula Int × (p, h, τ * ) in L ring ∪ {B} such that for every definable set X ⊆ K m , if p has a unique zero p 0 ∈ X, if X has arbitrarily many germs at p 0 and if the value

If K has a total order compatible with + and definable 4 in L ring ∪ {O}, then there is similarly a formula Int + (h, p, τ * ) such that, with the same assumptions on p,

closed sets of the topology of X. Since g f just if every closed definable set containing {f = 0} \ {g = 0} also contains {g = 0}, we see that g f if and only if for all h ∈ C(X)

Proposition 6.3 Assume that K satisfies (Dim). For every integer k 0 and every p 0 ∈ X the following are equivalent.

(1) dim(X, p 0 ) k.

Furthermore, there is a parameter-free formula χ k (p) in L ring such that C(X) |= χ k (p) if and only if p has a single zero p 0 ∈ X and dim(X, p 0 ) = k.

Proof: (1)⇒(2). Assume that dim(X, p 0 ) = l k and take v ∈ C(X) with v(p 0 ) = 0. Then {v = 0} is a neighborhood of p 0 in X hence by Lemma 6.1 there is a point q 0 ∈ {v = 0} ∩ W l (X). Let V be a definable neighborhood of q 0 in X, contained in {v = 0} ∩ W l (X) and homeomorphic to an open subset W of K l via the restriction of a coordinate projection π :

Let B be a closed ball in K m centered at q 0 and contained in V . Take δ B ∈ C(K m ) with {δ B = 0} = B as given by Lemma 4.1 (2). Finally let f i be the restriction of ν 2 (g i •π, δ B ) to X, where ν 2 is defined in Lemma 4.1 (1). Clearly f i ∈ C(X), f 0 (q 0 ) = 0 hence f 0 / ∈ C(X) × , {f k = 0} ⊆ B is disjoint from {v = 0}, and f 0 • • • f k because π maps the zero-set of each f i homeomorphically to {g i = 0} ∩ π(B).

(2)⇒(1). Let B 0 be an open ball in K m centered at p 0 . We have to prove that dim(B 0 ∩ X) k. Let v ∈ C(X) be such that v(x) = 0 on X \ B 0 and v(p 0 ) = 0 (e.g. one can choose v = δ B c 0 as in Lemma 4.1 (2)). By assumption there are f 0

by (Dim2). Item (1) now follows by induction on i.

Using Lemma 6.2 and Lemma 4.2, the equivalence of (1) and (2) implies the existence of a parameter-free formula χ k (p) in L ring , such that for every p ∈ C(X) we have C(X) |= χ k (p) just if p has a single zero p 0 ∈ X and dim(X, p 0 ) k. We then take χ k as χ k ∧ ¬χ k+1 . Proposition 6.4 Every definable open subset U of K r with r 2 has arbitrarily many germs at every point p 0 ∈ U . Theorem 6.7 Assume that K satisfies (Dim) and (BFin). Let X ⊆ K m be a definable set and let τ be a non-zero element of K.

(1)

hence so is C 2 (X, τ Z ). If moreover K satisfies (Z) the same holds true for k 1.

(2) If K has a total order compatible with + and definable in L ring ∪ {O} then the set C k (X, τ Z) is definable in L ring ∪ {B} for every integer k 2, hence so is C 2 (X, τ Z).

By Remark 4.4, Theorem 6.7 leads to definability in L ring in many cases (in particular if K is an expansion of a local field or if the theory of K is dp-minimal) provided K is not algebraically closed.

Remark 6.8 The formulas given by the proof of Theorem 6.7 involve only one parameter: the function with constant value τ . Thus if τ ∈ Z these definitions of C k (X, τ Z ) and C k (X, τ Z) are parameter-free.

Proof: Replacing τ by 1/τ if necessary we may assume that |τ | < |1|. Let χ k (p) and Int × (h, p, τ * ) be the formulas given by Proposition 6.3 and Theorem 5.5 respectively. Consider the formula Z k (h, τ * ) defined as

and any open ball B 0 ⊆ K m centered at p 0 let p = δ p 0 and v = δ B c 0 be the functions from Lemma 4.1. Then C(X) satisfies χ k (p) and p v; hence Z k (h, τ * ) gives q ∈ C(X) having a single zero q 0 with q 0 ∈ B and h(q 0 ) ∈ τ Z . In other words p 0 belongs to the closure of D h . By continuity we obtain h(p 0 ) ∈ τ Z , hence h ∈ C k (X, τ Z ).

Conversely assume that h ∈ C k (X, τ Z ). Take p ∈ C(X) with C(X) |= χ k (p). Then p has a single zero p 0 and p 0 ∈ ∆ k (X). For any v ∈ C(X) with C(X) |= p v, by continuity, the set V = {v = 0} is a neighborhood of p 0 in X. Hence by Lemma 6.1 there is a point q 0 ∈ V ∩W k (X). In particular q 0 ∈ ∆ k (X) by Property 2.5 and therefore h(q 0 ) ∈ τ Z . By Remark 6.6 the assumptions on K ensure that X has arbitrarily many germs at p 0 . Thus C(X) |= Int × (h, q, τ * ) with q = δ q 0 defined in Lemma 4.1 (2), which shows that C(X) |= Z k (h, τ * ).

The proof of the second statement is similar. Replacing τ by -τ if necessary we may assume that τ 0. Then let τ * ∈ C(X) be the function with constant value τ and replace Int × (h, q, τ * ) by Int + (h, q, τ * ) in Z k (h, τ * ).


Romain Derollepot

Emmanuel Vinot

Sizing of a combined series-parallel hybrid architecture for river ship application using genetic algorithm and optimal energy management

Keywords: Hybrid architecture, Optimal control, Sizing process, River ship

Sizing of a combined series-parallel hybrid architecture for river ship application using genetic algorithm and optimal energy management

Romain Derollepot, Emmanuel Vinot

Introduction

In a context of reduction of carbon dioxide emissions, Inland Waterways Transport (IWT) may be a good alternative to road transportation. Commercial river ships that can carry from 2000 to 5000 tons still present a better ratio of fuel consumption per tons carried than trucks [START_REF] Pillot | Exhaust emissions from in-service inland waterways vessels[END_REF], [START_REF] Minami | A newly developed plug-in hybrid electric boat (pheb)[END_REF]. However, few effort has been made trying to reduce the fuel consumption of ships compared to road transportation.

A solution to reduce the fuel consumption of ships is to replace the existing powertrain by a hybrid architecture. Electrical mode and engine downsizing can then be used in an efficient manner to reduce fuel consumption and pollutant emissions. Many authors focus on serial-architecture [START_REF] Bellache | Energy management in hybrid electric boat based on frequency distribution approach-using diesel, lithium-battery and supercapacitors[END_REF] for submarine [START_REF] Skinner | Multi-objective design optimisation of submarine electric drive systems[END_REF], small passenger ship [START_REF] Lhomme | Diesel-electric excursion ships enhancement using energetic macroscopic representation[END_REF], or boat used in protected marine area [START_REF] Barabino | A new energy storage and conversion system for boat propulsion in protected marine areas[END_REF]. However this architecture should be not optimal due to cascading efficiency in electrical machines and converters. More complex architecture should then be more efficient as described in [START_REF] Hofman | System-level design optimization of a hybrid tug[END_REF] for a hybrid tug.

This paper focuses on a commercial river ship of pusher type fig. 1. This ship is designed to push barges tied to its front. On the figure the pusher is only the small ship at the end of the convoy and is mainly composed of an engine room and a wheelhouse. With such a type of ship, only the barges are docked during loaded phases when the ship pushes other barges. Thus the ship is often in use when the barges are tied and untied. The fuel consumption of such ships can then reach 400 000 liters of fuel per year.

Three hybrid architectures have been sized and compared in this study. The key-points are then the sizing and the management of the system to ensure the best fuel economy. This paper focuses on these two points in order to find the best sizing and energy management for a given use of the boat.

To solve the combined optimization problem of the sizing and the management of the system a global approach has been proven to be the most efficient for road hybrid vehicle with one engine [START_REF] Vinot | Comparison of different power-split architectures using a global optimisation design method[END_REF], [START_REF] Silvas | Review of optimization strategies for system-level design in hybrid electric vehicles[END_REF], [START_REF] Nuesch | Cost and fuel-optimal selection of hev topologies using particle swarm optimization and dynamic programming[END_REF]. The energy management can then be calculated in an optimal manner [START_REF] Reinbold | Optimal sizing of an electrical machine using a magnetic circuit model: application to a hybrid electrical vehicle[END_REF], [START_REF] Bertram | Multi-objective optimization of a parallel hybrid electric drive train[END_REF], [START_REF] Song | Component sizing optimization of plug-in hybrid electric vehicles with the hybrid energy storage system[END_REF] or with methods close to optimality [START_REF] Zhao | A fully-analytical fuel consumption estimation for the optimal design of light-and heavy-duty series hybrid electric powertrains[END_REF], [START_REF] Hofman | System-level design optimization of a hybrid tug[END_REF].

In this paper an optimal energy management using the dynamic programming (DP) method has been developed [START_REF] Bellman | Dynamic Programming[END_REF], and is inserted in a main optimization loop managed by a genetic algorithm. The power-train architecture is much more complex than that of a road vehicle especially due to the number of engines. Thus the energy management is more complex to develop due to more possible operating modes and more degrees of freedom. The DP method has then been developed to a complex hybrid architecture with three engines which can be operated in four modes. It allows to evaluate the potential of fuel economy according to the size of the components and especially the size of the main engines and battery.

First, this paper presents the proposed hybrid architectures. Then, the sizing process and its application to a complex hybrid architecture are described in part 3. Section 4 and 5 present the modeling of the components of the powertrain, of the uses of the ships and the energy management method developed to assess the fuel consumption. Finally, the application of the developed sizing process and the obtained Pareto front are presented sec. 6.

Hybrid architectures

As many commercial river ships, the powertrain of the pusher (fig. 2) possesses two independent propulsion lines. Each line is composed of an internal combustion engine (ICE) of high power, multi-disk clutch, final-gear and propeller. One or more electrical generator (ICE + electrical machine (EM)) provide the power to the servitudes of the ship (front propeller, onboard electricity, hydraulic-pressure, winch, wheelhouse...). Different possibilities of hybridization can be studied and compared. In the scope of this paper, only the possibilities of series, parallel or "combined" series-parallel hybridization have been studied and compared to the conventional (fig. 2) architecture. Power split architectures [START_REF] Vinot | Model simulation, validation and case study of the 2004 ths of toyota prius[END_REF], [START_REF] Kamiya | Development of traction drive motors for the toyota hybrid system[END_REF], as those used in automobile, have been judged too complex for river ships due to the presence of a planetary gear and two electrical machines for one propulsion line.

Three architectures have been sized and compared:

• a parallel architecture (adding one electrical machine mechanically connected to each ICE shaft, fig. 3),

• a series architecture with no direct mechanical link between engines and propellers (fig. 4),

• a combined series-parallel architecture (fig. 5). Electrical machines are mechanically connected to the ICE shaft directly or by mean of a coupling device. One electrical generator (ICE + electrical machine) is connected to the electric network. The power of the generator can be high enough to provide a part of the required power to the propulsion and can allow pure series operation. The operating point of the system is chosen to obtain the minimum fuel consumption along a uses cycle (part. 5). It is noted that more than one electrical generator can be used, thus allowing a downsizing of the engine.

In the conventional architecture used in river ship, one or more auxiliary engines are used to provide the on-board electricity (fig. 2) and the power to the front propeller (by mechanic or hydraulic coupling). In the proposed hybrid architectures, the electrical generator is removed and the front propeller is connected to an electrical machine linked to the on-board electric network. The next section proposes a sizing process of this complex hybrid architecture combining optimal control management and genetic algorithm.

Sizing process

The sizing of a hybrid architecture is highly dependent on the management of the system. To give a suitable answer to this issue, the energy management can be determined using an optimal control method inserted in a main optimization process (fig. 6). Coupling dynamic programming [START_REF] Bellman | Dynamic Programming[END_REF], [START_REF] Vinot | Time reduction of the dynamic programming computation in the case of hybrid vehicle[END_REF] for the energy management and Meta-heuristic algorithm has been proven an efficient solution to size hybrid architectures dedicated to road transportation [START_REF] Silvas | Comparison of bilevel optimization frameworks for sizing and control of a hybrid electric vehicle[END_REF], [START_REF] Bertram | Multi-objective optimization of a parallel hybrid electric drive train[END_REF], [START_REF] Nuesch | Cost and fuel-optimal selection of hev topologies using particle swarm optimization and dynamic programming[END_REF]. This process is applied in this study for the sizing of more complex architectures dedicated to river ship (fig. 6). A Genetic Algorithm [START_REF] Deb | A fast and elitist multiobjective genetic algorithm: Nsga-ii[END_REF] is used to find the optimal components size that minimizes different objectives such as fuel consumption and size or cost of the components (Main optimization process, fig. 6). The fuel consumption is calculated by means of the dynamic programming (DP) method. For each sizing DP finds the optimal energy management of the system on a uses cycles (lower level control optimization, fig. 6). For one operating point of the propeller, the system can be operated in different manners (electric, series, combined mode) leading to a sub-level of optimization included in the DP process (system operation optimization, fig. 6). In the design and sizing process of hybrid architectures [START_REF] Huang | Multi-objective genetic algorithm for hybrid electric vehicle parameter optimization[END_REF], [START_REF] Reinbold | Optimal sizing of an electrical machine using a magnetic circuit model: application to a hybrid electrical vehicle[END_REF], we commonly need models of the components and uses cycles in order to simulate the powertrain operations and to calculate the fuel consumption which is one of the objectives to be reduced. Next part focuses on those two points.

Modeling of uses and components

To simulate the fuel consumption of hybrid architectures, models of the architectures and thus of the components must be developed and validated (sec. 4.2). In addition, uses cycles have to be developed to represent the operations of the boat (main propeller speed, front propeller power and onboard servitude power).

Uses cycles

In order to simulate the operations of the ship in a representative manner, uses cycles describing the main propeller speed, the front propeller electrical power and the electrical power consumption of the servitudes have been generated using measurements. The ship has been monitored in a period of 6 months, then the data have been compacted in a representative cycle of 2 hours which can be used to simulate the operations of the ship. To generate this cycle, the main propeller speed measurements (fig. 7) have been sorted in classes of speed. Then the time passed in each class during the six months of measurement has been determined. The cycle has been generated with constant speed phases with a duration proportional to those measured. Figure 7 shows the engine speed on a period of 7 hours extracted from the 6 months measurements. The time between two phases where the engine is at idle speed (63 rd/s) corresponds to the time between two locks1 . The mean period between two locks is around 2 hours on the trip. In fact, on the monitoring journey, the boat stop only when it reaches locks and final destination. A maneuver phase is then obserced (engine speed around 80 rd/s) as wall as a stop of the boat on the lock where the main engine is at idle speed (63 rd/s). This duration of two hours has thus been chosen to generate a representative cycle (fig. 8). The other speeds of the engine (128 rd/s and 158 rd/s) correspond to cruising phases. The higher engine speeds are linked to a narrow pass on the river where the flow is high, a high propulsion power (1900 kW) is then required to cross this pass in complete security.

The same approach has been applied for the front propeller, taking into account that it is only used in phases of maneuver where the main propeller is shut off (engine on idle).

Finally, the electrical consumption power has also been sorted. The maximum observed in the power of the servitudes corresponds to the activation of the wheelhouse, which goes down before the passage of bridges and then goes up again. The mean duration between two peaks has been determined to be around 12 minutes which is the periodicity retained for the on onboard electrical consumption.

It has been verified that these cycles are energetically representative of the operations of the conventional ships. Table 1 present the fuel consumption of the engines, the mean on-board electrical consumption and the mean front propeller power calculated using the 6 months of measurement. The simulation results have been obtained with the model of the conventional ships using the components model described in section 4.2. The comparison between measurement and simulation shows less than 2 % of differences. It is noted that the speed evolution is only increasing in the generated cycle. In fact the sequences of speed do not influence the fuel consumption in our case. Using the DP the model of the battery is independent of SoC. Moreover, the dynamic of the shaft of the propeller is neglected. The torque demand on the propeller can be modeled in different manners. A model of the hull of the ship can be performed and used to provide the drag force as it is usually done in modeling ground vehicles. Nevertheless, such an approach is difficult to perform in our case as the real speed of the ship on the water flow is difficult to evaluate. Moreover, the drag force in confined environment, such as rivers, is strongly dependent on the width and depth of the river (also difficult to measure). A choice has thus been made to model the drag force regarding only the propeller torque versus propeller speed characteristic.

The torque versus speed characteristics of the propellers has been deduced from steady state measurements. Fig. 9 presents the measurement (torque and speed) performed on the left and right propeller for chosen points in steady state operations. The red points correspond to the right propeller and the blue points to the left propeller.

The propeller is an object in motion in the water. It is therefore only subject to fluid resistance. Thus it has been verified that its drag torque is a square function of the rotation speed eq. 1. Going upstream to the others components, the final gear is represented with a gear ratio and a constant efficiency. The engines are modeled by classical fuel consumption maps which have been deduced using measurements on steady-state operations. The electrical machine and the inverter are represented with efficiency maps. The battery pack is modeled by classical serial resistive scheme (open circuit voltage in series with a resistance depending on the direction of the current (charge or discharge). The battery is composed of Lithium-Phosphate cells (9.12 Ah, 30 Wh) arranged in series and parallel.

Γ = α × Ω 2 + β (1)
Considering the model of each components, the VEHLIB library [START_REF] Vinot | Model simulation, validation and case study of the 2004 ths of toyota prius[END_REF] is used to simulate the global system and solve the DP problem. The components are connected in a backward manner going upstream from propellers to engines and fuel consumption. For each time of the cycle previously described the fuel consumption can then be calculated for each possible mode in order to choose the more efficient (sec. 5).

Energy management

5.1. Dynamic Programming applied to combined series-parallel hybrid architecture The fuel consumption of a hybrid system is strongly dependent on the energy management of the system. In a sizing process it is often solved in an optimal manner [START_REF] Gao | Hybrid vehicle design using global optimisation algorithms[END_REF], [START_REF] Vinot | Global optimized design of an electric variable transmission for hevs[END_REF]. The problem is then to find the optimal control of the system in order to minimize the fuel consumption (eq.2) along a known in advance driving cycle under a constraint of battery state of charge (SoC) (eq.3):

J = t f t 0 ṁf (t))dt (2)
∆ SOC = t f t 0 δ SOC (t)dt = K (3)
In the case of the proposed series-parallel architecture (fig. 5), for a given speed of the main propeller several operating modes of the system are possible:

• pure electrical mode (C1 closed and C2 open, main engine and EG engine switched off),

• pure parallel hybrid mode (C1 and C2 closed, Engine of EG switched off),

• pure series hybrid mode (C1 closed and C2 open main engine switched off),

• combined hybrid mode (C1 and C2 closed and engine of EG on).

Moreover, when the EG is in operation, for a given electrical power P EG a choice of the speed (torque) of the engine of the EG can be performed to minimize the fuel consumption.

Dynamic programming (DP), [START_REF] Bellman | Dynamic Programming[END_REF] is a well known method to solve in an optimal manner the problem of energy management in hybrid vehicles. It has been proven an efficient method to solve optimal control for vehicles with one storage source [START_REF] Vinot | Time reduction of the dynamic programming computation in the case of hybrid vehicle[END_REF], [START_REF] Sundstrom | On implementation of dynamic programming for optimal control problems with final state constraints[END_REF] as a battery. The SoC is considered as the state variable and depends only on the battery current.

The DP problem can be represented in a SoC sample area (fig. 10) where the states of a column are linked to the states of the following column by a spread of edges. Each edge corresponds to a battery current (SoC variation), and is associated with a fuel consumption. The maximum battery current -maximum discharge-is that of the pure electrical mode. The SoC can be sampled in order to ensure that, at each step of time, the pure electric mode (which is the maximum battery current) corresponds to one edge of the graph (fig. 10), [START_REF] Vinot | Time reduction of the dynamic programming computation in the case of hybrid vehicle[END_REF]. In this case P EG is equal to zero. Using the Bellman principle [START_REF] Bellman | Dynamic Programming[END_REF] the optimal battery current is then determined and thus the electrical/hybrid mode choice is performed. The other states are by definition hybrid modes. The choice between the different hybrid modes is performed by minimizing the fuel consumption versus EG power for a known battery current i.e. on each edge of the graph.

Looking at the equations of the system eq.4 to eq.9 we obtain 6 equations and 7 unknown variables (Ω ice , Ω em , T ice , T em , P em , P batt , P EG ). It is then clear that choosing P EG for example will fix the operation of the system. For that P EG is sampled from 0 to the power of the electric generator in serial mode (P EG-HS ) in each edge. The value corresponding to the minimum of fuel consumption is chosen. A value of zero corresponds to the pure parallel hybrid mode. P EG-HS is the maximum possible value of P EG . It corresponds to the electrical power which has to be provided by the electrical generator if the main engines are switched-off (pure series mode) or to the maximum value of EG power. Intermediate values correspond to a combined series-parallel mode.

The optimization of the operations of the Electrical Generator -choice of torque and speed for a given electrical power-is performed once for all, generating an optimal power curve based on the characteristics of the engine and the electrical machine.

Ω ice = Ω prop k f g (4
)

Ω em = Ω prop k cpl k f g (5)
(

T ice + T em k cpl η cpl )η f g k f g = T prop (6)
P em = T em Ω em + P t em (T em , Ω em ) (7)
P batt + P EG = P aux + P f rontprop + 2 • P em (8)

P batt = I batt • E batt -R batt • I batt 2 (9)

Results on Management

Fig. 11 presents the operations of the system along a uses cycle. The following representative variables are presented:

• mechanical power of one of the main propellers,

• electrical power of the on-board servitudes and of the front propeller,

• SoC of the battery,

• electrical power of the battery and of the electrical generator and mechanical power of one of the main engines. The DP is applied for a sizing with three engines (two main engines and one engine for the electrical generator) of 375 kW and a battery of 300 kWh, (sec. 6). In this study only operations in charge sustaining mode are considered. No external battery recharge is taken into account and the final battery SoC is thus equal to the initial one (K=0, eq.3).

It is noted that the SoC variation on the representative cycle is only on a window of +10%, -18 %. The battery can be smaller with no significant effect on the fuel consumption. In fact its size is imposed by a constraint on battery energies to respect a minimum on-board available power (sec.3).

The parameters of the DP process are presented in Table 3. It takes around 52s on an Intel Xeon E5-2637 @ 3.5 gHz to solve the DP problem with a time sample of 1s. Around 60 % of the time is dedicated to the calculus of the costs of the edges (choice of optimal operations of the system for a given battery current). In fig. 11, looking at the power of the battery, of the main engine and of the EG, the different operations of the system can be observed:

• pure electrical mode from t=400s to 1500s for example. It corresponds to maneuver phases with low required power on the main propeller or on the front propeller, the battery provides all the energy required by the main propeller and the on board servitude,

• from 100s to 123s the front propeller is driven by the electrical generator in a serial manner. The power of the front propeller is too high to operate the system in pure electrical mode in an efficient manner considering charge sustaining,

• parallel hybrid mode can be observed from 1500s to 1950s for example. It corresponds to low speed propulsion phases. The battery is charging during these phases, the engines provide the power required for the propulsion, the auxiliary and to recharge the battery,

• pure mechanical phases can be observed (from 2150s to 2800 s) when the energy is quite only provided by the main engine with direct mechanical connection to the propeller,

• finally the system is operated in a combined series-parallel mode from 3665s to the end of the cycle, which corresponds to high speed propulsion phases. The power of the main engines is lower than the power required by the propellers. In these phases the engines are at their full-power. When the required power is not too high the battery is recharged and the stored energy will be used to provide the required power at the end of the cycle,

• to reach the final required SOC, the combined hybrid mode is also used to recharge the battery for example from instant 2790 to 2870 when it alternates with pure parallel hybrid mode. The EG is then on when the on-board servitude power is high.

Results and discussion

This part present the sizing of a combined series-parallel architecture for a river-ship of pusher type. The previously proposed method has been applied. Multi-objective algorithm (NSGA-II, [START_REF] Deb | A fast and elitist multiobjective genetic algorithm: Nsga-ii[END_REF]) is used to solve the sizing problem with an embedded optimal energy management previously presented, to asses the fuel consumption.

The following three objectives to minimize are chosen:

• the fuel consumption calculated using optimal energy management (sec. 5).

• the power of the main engine which is representative of its cost and can be an important parameter of choice (see discussion),

• the battery energy storage capacity which is also representative of its cost and size,

The power of the electrical machine in parallel (EM) and of the electrical generator (engine plus electrical machine) can also be some objectives to minimize but are in fact fixed by the constraint of the systems (see the following).

It is quite impossible to evaluate the cost of the overall architecture. In fact it strongly depends on components which are not developed in high series. The cost of the main engines is thus not linked to its power. And the cost to prototype the electrical part is quite impossible to asses. Thus, to answer this issue we chose the previous three objectives. After the sizing process, the power of each components is known, a choice can then be performed integrating different considerations like cost, or volume of the systems.

Four variables of optimization are used in the NSGA process:

• the main engine power (P ICE ),

• the power the electrical machines in parallel (P EM ),

• the power of the electrical generator (P EG ),

• the battery energy storage capacity (E BAT T ). This will vary considering the number of cells of the battery pack.

The following constraints are applied to the systems:

• the global propulsion power needs to be superior or equal to those of the original architecture , i.e. 950 kW for each shaft:

P em + P ice >= 950kW (10)
• the maximum power of the conventional architecture (1900 kW) has to be available for 10 minutes. As the global power provided by the engines is lower than 2*950 kW, this will fix the battery energy taking into account that the battery will be discharged by only 50 % in 10 minutes.

E sbatt >= ( 1900 -2 • P ice η em -P EG ) • T d • 100 ∆SoC (11)
E sbatt >= 1 3 ( 1900 -2 • P ice η em -P EG ) (12)
Where E sbatt is the energy stored in the battery fully charged (expressed, in Wh), η em is the mean efficiency of the electrical machine and converter, T d is the time of discharge (1/6 hours in ours case) and ∆SoC is the discharge of the battery during T d (50 %).

Table 4 presents the main parameters used to run the NSGA algorithm. The sizing process take around 3 days on an Intel Xeon E5-2637 @ 3.5 gHz. The NSGA optimization is parallelized on height core. The parameters used for DP are those presented tab. 3 for DP. Figure 12 shows the Pareto front, fuel consumption versus engine power and battery energy storage capacity, obtained with the previously proposed sizing process for a series-parallel hybrid architecture (fig. 5). The red point represent the constraint on the battery energy storage capacity E BAT T (eq. 10) for the maximum power of the EG (750 kW). The sizing with smaller engine and /or battery are not in the studied domain.

The two inequality constraints on the global propulsion power (eq. 10) and the battery energy storage (eq.12) behave in fact as equality constraint. The global energy power is quite equal to 950 kW and the battery energy storage capacity also corresponds to the minimum allowed by the constraint. Figure 13 represents the difference between the global power on each shaft minus 950kW, in percent of the minimum power allowed (950kW) by the constraint:

(P ICE + P EM ) -950 950 * 100 (13)
It is clear that the great majority of the individuals respect the constraint with less than 5 % of exceed. The same results are observed considering the battery energy storage capacity. This validates the choice not to consider EM and EG power as objectives.

The Pareto front is quite a plane. This means that for a given engine power the influence of the size of the battery does not act on the fuel consumption. In fact to respect the constraints, the EG size grows up when the battery size becomes smaller. But no significant difference can be observed in fuel consumption if energy is provided by battery or EG. Some interesting sizings can be extracted from the Pareto front (fig. 14) as they present other interests than only a potential gain in fuel consumption. Table 5 presents this sizing with the corresponding power of the main engine (ICE) of the parallel Electrical Machine (EM), of the electrical generator (EM) and of the weight of the battery (BATT) considering a mass energy of 100Wh/kg.

The minimum of fuel consumption (architecture III, tab.5) is expected with an engine of 500 kW (green line on fig. 14). A gain of 12 % is then obtained compared to the conventional architecture. Different couples EG/battery can match the constraint (eq. 10 and eq.12), from pure parallel architecture to architectures with one or two electrical generators of 375 kW. An architecture with one EG of 375 kW is presented in the table.

In this architecture, the main engines are downsized (500 kW compared to 900 kW in the conventional powertrain) but not enough to use engines dedicated to trucks which have a maximum available power of around 375 kW. River ships which are currently propelled by conventional powertrain are often not equipped with depollution devices (catalytic converter, partic-ulate filter ...). The pollution of the ships per tons/km is thus at a level between euro II and euro V ground vehicle norms [START_REF] Pillot | Exhaust emissions from in-service inland waterways vessels[END_REF]. This will change in 2020 as the euro VI norm is planed to be applied to IWT. Adapting road engines dedicated to trucks can be a good solution to meet this issue and to reduce the global cost of hybridization while using already mass production engines. Architecture IV (tab.5) presents a solution using 3 engines of 375 kW dedicated to trucks (red point on fig. 14 ). Two engines are on the main shaft and one is used for EG. The fuel consumption is then a bit higher than the minimum possible (140.8 l/h versus 137.5 l/h). However, 10 % of fuel economy can be expected knowing that a pusher uses 400 000 l of fuel per year. The size of the battery (3 tonnes) remains reasonable (for a boat of 2000 tonnes). Moreover, compared to a pure parallel architecture, the presence of the electrical generator can provide driving comfort in maneuver phases if not possible using only the battery.

Two other sizings (II and V) have been treated separately from the sizing process. Solution II presents the case of a minimalist hybridization based on the existing architecture and adding two small EMs and a small battery. It shows a good potential of fuel economy (7%) at small cost. Finally, solution V presents the case of pure series architecture. It shows that for such an application this solution is of no interest regarding fuel consumption. In fact, most of the energy is used for the main propellers. The decoupling between the propeller speed and the engine speed which allows to operate the engines in good specific consumption area cannot compensate the cascading efficiencies of the EM machines and converters.

Conclusion

This paper presents a study of the potential of combined series-parallel hybridization of river ships. An optimal sizing process embedding an optimal control has been developed and applied on a complex series-parallel architecture. A Pareto front representing fuel consumption versus the size of the components is then obtained and can be used to make a final choice of the architecture. Fuel economy between 12 and 7% can be performed depending on the size of the different components.

This study shows that the use of engines dedicated to trucks is possible. It can reduce the pollutant emission drastically using standard depollution devices. With three engines -two connected in parallel and one in seriesand 3 tons of battery, it is shown that a fuel economy of 10% can be expected. Taking into account that the fuel consumption of a ship of pusher type is around 400 000 l/year, 40 000 liters of fuel can be saved each year.

Figure 1 :

 1 Figure 1: Pusher with barge

Figure 2 :

 2 Figure 2: Conventional architecture

Figure 3 :

 3 Figure 3: Parallel architecture.

Figure 4 :

 4 Figure 4: Serial architecture.

Figure 5 :

 5 Figure 5: Combined series-parallel architecture.

6

 6

Figure 6 :

 6 Figure 6: Sizing process

Figure 7 :

 7 Figure 7: Main propeller speed: extraction on a small period of the six month of measurement

Figure 8 :

 8 Figure 8: Uses cycle

Figure 9 :

 9 Figure 9: Propeller model

Figure 10 :

 10 Figure 10: Dynamic programming optimal trajectory and limits

Figure 11 :

 11 Figure 11: Energy Management Strategy

Figure 12 Figure 13 :

 1213 Figure 12: Pareto front

Figure 14 :

 14 Figure 14: Pareto front

Table 1 :

 1 Fuel consumption and electrical power An important step to model existing conventional architectures and then hybrid architectures is to model the components of the power train.

		measurement simulation on uses cycles
	Fuel consumption l/h	142	144.8
	Mean electrical power	15.4	15.6
	Mean front propeller power	1.2	1.2
	4.2. Components model

Table 2 :

 2 Symbols defintion

	Symbol Quantity
	Ω ice	Main engines speed
	T ice	Main engines torque
	Ω em	Electrical Machines speed
	T em	Electrical Machines torque
	P em	Electrical Machines electrical
		power
	P t em	Electrical Machines losses
	P EG	Electrical generator electrical
		power
	P aux	Onboard Servitude electrical
		power
	P f rontprop Front	propeller	electrical
		power
	P batt	Battery electrical power
	E batt	Battery open circuit voltage
	I batt	Battery current
	R batt	Battery internal resistance
	η cpl	coupling device on electrical
		machine efficiency
	k cpl	coupling device on electrical
		machine ratio
	η f g	final gear efficiency
	k f g	final gear ratio

Table 3

 3

	: graph parameter
	step of time	1 s
	SoC Sample	0.02 %
	number of edges	5 • 10 8

Table 4 :

 4 NSGA parameter

	number of individuals	60
	number of generations	500
	crossover rate	0.8
	P ICE min/max	[50 950] kW
	P EM min/max	[50 950] kW
	P EG min/max	[50 750] kW
	E BAT T min/max	[30 600] kW.h

Table 5 :

 5 selected architectures

locks : gate on the river or canal to go up or down

Acknowledgements

This work was performed during the Project PROMOVAN (http://www.promovan.tla.net/) regrouping French government agencies and industrial actors of the IWT domain. This Project was supported by European Regional Development Fund (ERDF) and the CPIER fund involving the French Rhône-Alpes Région.