MAR ÍA ISABEL Alicia Dickenstein

Bernard Mourrain
email: bernard.mourrain@inria.fr

CURVE VALUATIONS AND MIXED VOLUMES IN THE IMPLICITIZATION OF RATIONAL VARIETIES

Keywords: Mathematics Subject Classification. 14M25, 14T05, 68W30 implicitization, Newton polytope, tropical geometry, generalized Puiseux series

We address the description of the tropicalization of families of rational varieties under parametrizations with prescribed support, via curve valuations. We recover and extend results by Sturmfels, Tevelev and Yu for generic coefficients, considering rational parametrizations with non-trivial denominator. The advantage of our point of view is that it can be generalized to deal with non-generic parametrizations. We provide a detailed analysis of the degree of the closed image, based on combinatorial conditions on the relative positions of the supports of the polynomials defining the parametrization. We obtain a new formula and finer bounds on the degree, when the supports of the polynomials are different. We also present a new formula and bounds for the order at the origin in case the closed image is a hypersurface.

Introduction

A classical question that has thrived research in Computational Algebraic Geometry is the problem of implicitization. The aim is to describe the prime ideal of polynomial relations among the coordinates of a rational map f . We concentrate on sparse elimination and we consider a family of rational maps of the following form: the input is a field K of characteristic zero, n + 1 finite sets A 0 , . . . , A n of lattice points in Z d and n + 1 non-zero Laurent polynomials in d variables f 0 , . . . , f n ∈ K[x ±1 1 , . . . , x ±1 d ] supported on these sets. More precisely, (1.1)

f i = a∈Ai c i,a
x a , x = (x 1 , . . . , x d ), c i,a ∈ K, for all i ∈ {0, . . . , n}, and our parametrization is of the form

(1.2) f : (K * ) d (K * ) n f = f 1 f 0 , . . . , f n f 0 .
As the ideal of relations among the coordinates of f is defined over any subfield containing the coefficients, we assume in what follows without loss of generality that K is algebraically closed. We can think of the implicitization problem as a moduli question, as the answer differs depending on the genericity of the coefficients (c i,a ) n i=0 . We consider K with the trivial valuation. In [START_REF] Sturmfels | The Newton polytope of the implicit equation[END_REF], Sturmfels, Tevelev and Yu approached this question with tools of tropical geometry, thus exploiting the combinatorial nature of the problem. Instead of finding the ideal of relations, we can compute its tropicalization. This is a rational weighted balanced polyhedral fan in R n that captures the combinatorics of this ideal. Tropical implicitization techniques are well suited to study the generic points of this space via the theory of Geometric Tropicalization developed by Hacking, Keel and Tevelev [START_REF] Cueto | Implicitization of Surfaces via Geometric Tropicalization[END_REF][START_REF] Hacking | Stable pair, tropical, and log canonical compactifications of moduli spaces of del Pezzo surfaces[END_REF]. These techniques are hard to address for special choices of coefficients, since they depend on the process of resolution of singularities.

We approach this question from a different perspective. Namely, we go back to the valuative approach to tropical geometry, manifested in the Fundamental Theorem of Tropical Geometry [START_REF] Draisma | A tropical approach to secant dimensions[END_REF][START_REF] Einsiedler | Non-Archimedean amoebas and tropical varieties[END_REF], and we reinterpret tropical implicitization in this elementary language. The main advantage of this approach is that we recover in Theorem 2.8 a straightforward generalization of the description of [START_REF] Sturmfels | The Newton polytope of the implicit equation[END_REF]Theorem 2.1] for generic parametrizations, while developing tools that could be further refined to study the non-generic case. In fact, Theorem 2.8 can be deduced from their result, as we discuss in Section 2. [START_REF] Draisma | A tropical approach to secant dimensions[END_REF].

We also study in detail the degree of the closed image of f (that is, the closure of the image of f ) which we denote by S, in case it has dimension d. When S is a hypersurface (that is, when d = n -1), one can get the direction and the length of the edges of the Newton polytope N (H) of a defining equation H for S from the description of the cones in the tropicalization of the image of f and their multiplicities. Then, N (H) can be algorithmically reconstructed, see for instance [START_REF] Jensen | Computing Tropical Resultants[END_REF]. We could then from this information extract the degree deg(H) = deg(S). In the case of lattice polygons, there is a clear description of N (H) in [START_REF] Sombra | The Newton polygon of a rational plane curve[END_REF]. If f 0 = 1 and all supports are equal, deg(S) is computed in [START_REF] Sturmfels | Minimal polynomials and sparse resultants[END_REF] by means of resultants. In case f 0 is a monomial, the description of the Newton polytope of H was also studied in general by Esterov and Khovanskii in [START_REF] Esterov | Elimination theory and Newton polytopes[END_REF], where they develop an elimination theory of polytopes via mixed fiber polytopes in the sense of McMullen. This gives the Newton polytope N (H) only for generic coefficients, from which deg(S) can be read in this case. We give a precise description in Theorem 3.3.

There is a straightforward upper bound for deg(S), given by the lattice volume of the convex hull (conv(∪ n j=0 A j )) of the union of the supports (see for instance [START_REF] Emiris | Implicit polynomial support optimized for sparseness[END_REF] in case d = n -1). The divergence from this upper bound for generic coefficients is related to the following question. Given A 0 , . . . , A n and Laurent polynomials f 0 , . . . , f n in d variables with these supports respectively and generic coefficients, when d generic linear combinations 1,0 f 0 + • • • + 1,n f n , . . . , d,0 f 0 + • • • + d,n f n with generic coefficients i,j ∈ K, have generic coefficients with respect to the support ∪ n j=0 A j ? In Theorem 3.3 we give conditions on the relative positions of A 0 , . . . , A n under which this upper bound and some refined ones are attained. The conditions are similar to the conditions of monotonicity of the mixed volume studied in [START_REF] Bihan | Criteria for Strict Monotonicity of the Mixed Volume of Convex Polytopes[END_REF] for the case of n sparse polynomials in n -1 variables, but they are in fact different as Example 3.9 shows.

When S = (H = 0) is a hypersurface, we also compute by means of curve valuations the order at the origin ord 0 (S) = ord 0 (H). Again, if the Newton polytope N (H) is known, this information can be extracted from it. Under our hypotheses, the origin is never in the image of f , but it could be in its closure. We compute ord 0 (S) in Theorem 4.2 under a condition on the family of supports that holds in many cases. This order can be greater than one (that is, the origin is a singular point of the closure of S in K n ) even for generic coefficients, depending again on the relative positions of the supports A 0 , . . . , A n . We also give along the paper examples not satisfying the hypotheses of our results, but where we can still compute degree and order adapting the arguments in our proofs.

In Section 2 we describe the tropicalization of S. We prove the basic Theorem 2.4 and the main Theorem 2.8. We also show in § 2.5 how this result and its proof can be used in generic and non-generic cases. Section 3 deals with the degree computation for any d. Our main result there is Theorem 3.3 and in § 3.1 we give different conditions for equality in the inequalities in its statement, in case d = n -1. Section 4 deals with the computation of the order of S at the origin for d = n -1. We present our main result Theorem 4.2 and we discuss examples that show how its proof can be used to study cases in which the hypotheses are not satisfied.

Acknowledgements: We are very grateful to M. Angélica Cueto, who worked with us at an earlier stage of this project. In particular, she suggested to us the example we present in Section 2.5 and provided the first picture in Example 2.14. We thank the two referees for their constructive comments to improve our article and Paco Santos for his immediate help with a polytope question.

The tropicalization of the image of f

We fix a family of finite integer sets A 0 , . . . , A n ⊂ R d and we take n + 1 Laurent polynomials f 0 , . . . , f n as in (1.1) with these respective supports and coefficients in K. The goal of this section is to describe the tropicalization of the closure im(f ) of the image of f in (1.2) in case the coefficients of f 0 , . . . , f n are generic. Since the parametrization is invariant under multiplication of all the f i 's by the same monomial, that is under a common translation of the supports, we assume all along, without loss of generality, that our supports A 0 , . . . , A n lie in the positive orthant

(Z ≥0 ) d , that is, that f 0 , . . . , f n ∈ K[x 1 , . . . , x d ].
Our main result in this section is Theorem 2.8, which describes the tropicalization of the closure of the image of f as a set. It extends [START_REF] Sturmfels | The Newton polytope of the implicit equation[END_REF]Theorem 2.1] to the case of rational functions. Our proof is built from Theorem 2.4 and Lemma 2.11 below using curve valuations. The main advantage of this approach is that it only involves elementary techniques and it hints on how to proceed in case the coefficients fail to be generic. We develop this idea in § 2.5. In Section 2.6 we recover the associated multiplicities from the results in [START_REF] Sturmfels | Elimination theory for tropical varieties[END_REF].

2.1. Tropical varieties. In this section, we present some basic definitions and notation. In particular, we give an introduction to useful statements on tropical geometry from the valuative perspective and recall the connection between initial ideals, tropicalizations and power series solutions to Laurent polynomial equations. Our main result here is Theorem 2.4.

We start by recalling the definition of the tropical variety of an ideal from the point of view of Gröbner theory. We consider an algebraically closed field K of characteristic zero with trivial valuation. As usual, we set

K * = K \ {0} and we call (K * ) d the d-torus. Definition 2.1. For any nonzero Laurent polynomial g = g α x α ∈ K[x ±1 1 , . . . , x ±1 d ]
and w ∈ R n , we denote by in w (g) the subsum of those terms g α x α in g for which g α = 0 and w, α is minimum.

We denote by in w (I) the ideal generated by all the initials in w (h) with respect to w of all nonzero polynomials h ∈ I. Given an ideal

I ⊂ K[x ±1 1 , . . . , x ±1 d ],
the support of the associated tropical variety or tropicalization T (I) of I is the set:

(2.1)

T (I) = {w ∈ R d | in w (I) does not contain a monomial},
that is, in w (h) contains at least two nonzero terms for any h ∈ I \ {0}.

Given an algebraic variety V ⊆ (K * ) d , we set T (V ) = T (I(V )), where I(V ) denotes the ideal of Laurent polynomials vanishing on V . By Hilbert's Nullstellensatz, T (I) records all points w such that the associated variety V K * (in w (I)) is nonempty. When the ideal I is principal, generated by a Laurent polynomial h, the set T (I) consists of all directions ω for which the initial form in ω (h) is not a monomial. Thus, T (I) consists precisely of the codimension one cones in the inner normal fan of the Newton polytope N (h) of h, that is, of the convex hull of the exponents of the monomials occuring in h with nonzero coefficient.

Tropical geometry can be approached from the perspective of valuations, when we think of them as Bieri-Groves' sets [START_REF] Bieri | The geometry of the set of characters induced by valuations[END_REF]. To state explicitly the link between equality (2.1) in Definition2.1 and the characterization of the tropicalization of T (I) via curve valuations, we introduce the algebraically closed field of power series with real exponents and wellordered supports L = K{{ε R }}, introduced by Hahn in 1907. These series form the ring of generalized Puiseux series, also known as the Mal'cev-Neumann ring (see e.g. Example 2.1.7 in [START_REF] Maclagan | Introduction to tropical geometry[END_REF] and [START_REF] Poonen | Maximally complete fields[END_REF]). Definition 2.2. An element σ ∈ L * has the form σ = b 0 ε α0 + h.o.t.(ε), where b 0 ∈ K * , α 0 ∈ R and h.o.t.(ε) is a sum of terms of the form b i ε αi with b i ∈ K and α i+1 > α i for all i ≥ 0. We call in(σ) := b 0 ε α0 its initial term and val(σ) := α 0 its order. We extend these notions to tuples in L d coordinatewise.

Note that in particular val(c) = 0 for all c ∈ K * .

The Fundamental Theorem of Tropical Geometry [START_REF] Draisma | A tropical approach to secant dimensions[END_REF][START_REF] Einsiedler | Non-Archimedean amoebas and tropical varieties[END_REF][START_REF] Maclagan | Introduction to tropical geometry[END_REF] asserts that given an ideal I in the Laurent polynomial ring K[x ±1 1 , . . . , x ±1 d ], the tropical variety T (I) is the set of all valuations of nonzero generalized Puiseux series solutions to I:

(2.2) T (I) = {val(ζ) | ζ ∈ V L * (I)}.
Note that by choosing the field of generalized Puiseux series L which is algebraically closed and complete [START_REF] Poonen | Maximally complete fields[END_REF] with value group R, it is not necessary to take closure on the right side [START_REF] Markwig | A field of generalized Puiseux series for tropical Geometry[END_REF]. The tropicalization T (I) is a rational polyhedral set that can be given a (non-unique rational polyhedral) fan structure and any such fan can be refined to a tropical fan structure.

Definition 2.3. The multiplicity of a maximal cone σ in a fan structure of T (I) is defined as the sum over all minimal associated primes P of the initial ideal in w (I), of the multiplicities:

mult(σ) = mult(P, in w (I)),
where w is any point in the relative interior of σ.

These multiplicities satisfy a balancing condition. We refer the reader to [START_REF] Sturmfels | Elimination theory for tropical varieties[END_REF] for details. Consider now nonzero Laurent polynomials f 0 , . . . , f n and the map f from (1.2). Denote by

f L : (L * ) d L n the extension of f to L. If I f ⊂ K[y ±1 1 , . . . , y ±1 n ]
is the ideal that defines the variety im f and J f ⊂ L[y ±1 1 , . . . , y ±1 n ], the ideal that defines the variety im f L , then J f = I f ⊗ K L. That is, any system of generators of I f over K is also a system of generators of J f over L via the inclusion K → L. Given a set X, denote by cl(X) its closure in the Euclidean topology. The next result, which is an easy consequence of (2.2), shows the connection between the tropicalization of the variety parameterized by the rational map f and the valuations of points in its image.

Theorem 2.4. Let f 0 , . . . , f n be nonzero Laurent polynomials in d variables with coefficients in K and let f : K d (K * ) n be the rational map they define as in (1.2). Then,

(2.3) T (I f ) = cl{val(f L (σ)) | σ ∈ (L * ) d , f L (σ) ∈ (L * ) n }. Proof. Let V = {val(f L (σ)) | σ ∈ (L * ) d , f L (σ) ∈ (L * ) n }.
The Fundamental Theorem of Tropical Geometry implies that val(f L (σ)) ∈ T (I f ) for every σ ∈ (L * ) d in the preimage of the torus (L * ) n belonging to the domain of f L . As tropical varieties are closed in the Euclidean topology, we have the inclusion cl(V ) ⊂ T (I f ) in (2.3). Moreover, since V is dense in T (I f ), we have equality in (2.3) by Theorem A in [START_REF] Bieri | The geometry of the set of characters induced by valuations[END_REF]. Indeed, Bieri-Groves Theorem A asserts that all maximal cones in T (I f ) have dimension equal to dim(im f L ). Let Y ⊂ im f L be an algebraic variety of codimension 1 such that im f L Y ⊂ im f L ⊂ (L * ) n . Then, all maximal cones in T (I(Y )) have dimension equal to dim(Y ) < dim(im f L ). Consequently, T (I f ) \ T (I(Y )) is dense in T (I f ), as well as V .

2.2. Defining the cones and the statement of Theorem 2.8. We present in Definition 2.7 the cones that occur in the tropicalization of the image of f . We first need some notation concerning supports and polytopes. Let B be a finite set in Z d and h ∈ L[x 1 , . . . , x d ] a polynomial h = q∈B c q x q with support B with c q = 0 for all q ∈ B. For any α ∈ R d , we define

(2.4) m α (h) = min q∈B {val(c q ) + α, q } and in α (h) = mα(h)=val(cq)+ α,q in(c q )x q .
In particular, if h ∈ K[x 1 , . . . , x d ], m α (h) does not depend on its coefficients and we have

(2.5) m α (B) = m α (h) = min q∈B { α, q } and in α (h) = mα(B)= α,q c q x q ,
as at the beginning of § 2.1. Let Q ⊂ R d be a lattice polytope and α ∈ R d . We call face α (Q) the face of Q which has α as an interior normal vector, that is, were the inner product with α is minimized:

(2.6) face α (Q) = {q ∈ Q | α, q = m α (Q)}.
If B ⊂ Z d is a finite set, we denote by face α (B) the intersection of the face of the convex hull of B defined by α with B (and we say that it is a face of B). Thus, for any nonzero Laurent polynomial g, the Newton polytope of in α (g) is face α (N (g)).

Given a lattice polytope Q ⊂ R d , we use the notation vol(Q) for its normalized volume with respect to the lattice Z d (so vol(Q) equals d! times its Euclidean volume vol e (Q)). Given a family of polytopes Q Definition 2.5. Given B = (B 1 , . . . , B r ) a family of finite sets in R d (or convex polytopes) and F j a face of B j for all j ∈ [r], we say that

1 , . . . , Q d ∈ Z d ,
F = (F 1 , . . . , F r ) is a coherent collection of faces of B if there exist α ∈ R d such that F j = face α (B j ) for all j ∈ [r]. Definition 2.6. A family of finite sets B 1 , . . . , B r in Z d is said to be essential if for any subset J ⊂ [r] of size at most d, it holds that (2.8) dim( i∈J conv(B i )) ≥ |J|.
With the previous notation, given a set J ⊂ [r], we say that a coherent collection

F = (F 1 , . . . .F r ) of faces of B is adapted to J if |J| ≤ d and the family {F j } j∈J is essential.
Consider again the family of supports A 0 , . . . , A n ⊂ Z d ≥0 . Given any α ∈ R d , we define (2.9)

ψ i (α) = m α (A i ) -m α (A 0 ), i ∈ [n],
and we consider the map ψ : R d → R n defined by

(2.10) ψ(α) = (ψ 1 (α), . . . , ψ n (α)).
We next define the cones on which the description of T (im f ) is based. We use the notation 1 = (1, . . . , 1) ∈ R n . Given a subset J ⊂ [n] 0 , we use the standard notation (R ≥0 ) J to denote the real vectors y ∈ R n satisfying y j ≥ 0 for all j ∈ J and y j = 0 for j ∈ J. Definition 2.7. Given a set J ⊂ [n] 0 and a coherent collection F = (F 0 , . . . , F n ) of faces of (A 0 , . . . , A n ) adapted to J, we define the following cones in R n : * if 0 / ∈ J,

(2.11) C F J := {ψ(α) + u | F i ⊆ face α (A i ) for all i ∈ [n] 0 , and u ∈ (R ≥0 ) J }, * if 0 ∈ J, set J = J -{0} and (2.12) C F J := {ψ(α) + u -λ 1 | F i ⊆ face α (A i ) for all i ∈ [n] 0 , u ∈ (R ≥0 ) J and λ ∈ R ≥0 }. Note that coherent collections (F 0 , . . . , F n ) of faces of A 0 , . . . , A n correspond to faces n j=0 F j of n j=0
A j , which are in bijection with cones of the inner normal fan of conv( n j=0 A j ). We now present Theorem 2.8. Our statement is a direct generalization of [21, Theorem 2.1], which applies to the case when f 0 is a monomial (A 0 consists of a single point). Our proofs are different, based on curve valuations, and give a hint on how to describe with this tool the tropicalization of the image when the coefficients are not generic.

Theorem 2.8. Consider the rational map f as in (1.2) where f 0 , . . . , f n ∈ K[x 1 , . . . , x d ] have supports A 0 , . . . , A n and generic coefficients. Then, the following subsets of R n coincide:

(i) the tropical variety T (im f ), (ii) the union of the cones C F J , for all J ⊆ [n] 0 such that |J| ≤ d and all coherent collections F of faces of (A 0 , . . . , A n ) adapted to J.

The proof is given in § 2.4 below. We give a simple example of the statement of Theorem 2.8

Example 2.9. Denote by P i the convex hull of A i for any i ∈ [n] 0 . For instance, in the unmixed case when all P i are dilations of a fixed polytope P ∈ R d , that is, when there exists positive integers d 0 , . . . , d n , such that

P i = d i P for all i ∈ [n] 0 [21, Section 3.2], we have that ψ(α) = m α (P ) • v d ,
where v d denotes the integer vector (d 1 -d 0 , . . . , d n -d 0 ). The image of ψ varies as in [START_REF] Sturmfels | The Newton polytope of the implicit equation[END_REF] according to whether P contains the origin or not.

When P contains the origin, the image of ψ is the half-line R ≤0 v d and the tropical variety

T (im f ) is then the union of the cones R ≤0 v d + (R ≥0 ) J over all |J| = dim(P ) -1 such that 0 ∈ J, R J ≥0 if 0 / ∈ J and |J| = dim P , R ≤0 v d + (R ≥0 ) J\{0} + R ≥0 (-1, . . . , -1) over all |J| = dim(P ) -1 such that 0 ∈ J and R J\{0} ≥0
+ R ≥0 (-1, . . . , -1) if 0 ∈ J and |J| = dim P .

When P does not contain the origin, the image of ψ is the line

R v d and T (im f ) is the union of the cones R v d +(R ≥0 ) J over all |J| = dim(P ) -1 such that 0 ∈ J, and R v d + (R ≥0 ) J\{0} + R ≥0 (-1, . . . , -1) over all |J| = dim(P ) -1 such that 0 ∈ J, R J ≥0 if 0 / ∈ J and |J| = dim P , and R J\{0} ≥0
+ R ≥0 (-1, . . . , -1) if 0 ∈ J and |J| = dim P . Note that the cones in our description overlap and a cone in the tropicalization can be equal to the union of some of them, as in Example 2.13.

We could use Theorem 2.8 to compute the dimension of the variety parameterized by f , which equals the maximal dimension of a cone C F J , because the tropical variety T (im f ) and im f have the same dimension by [START_REF] Bieri | The geometry of the set of characters induced by valuations[END_REF]. For instance, let d = n -1. If there exists a subset

J 0 ⊂ [n] 0 with |J 0 | = n -1 and C J0 = ∪ F C F J0 = ∅, then f parameterizes a hypersurface.
Note that the reciprocal is not true. For instance, consider the rational map f :

(C * ) 2 (C * ) 3 defined by f (x 1 , x 2 ) = (c 1 x 1 , c 2 x 1 , x 2 ),with c 1 , c 2 = 0.
Then, f parameterizes a hypersurface but C J = ∅ for all J = ∅, in particular those with cardinal 2. This is not a contradiction because in this case dim(C ∅ ) = 2.

We next show that Theorem 2.8 can be seen as a consequence of [START_REF] Sturmfels | The Newton polytope of the implicit equation[END_REF]Theorem 2.1]. Consider again finite sets A 0 , . . . , A n ⊂ Z d and polynomials f 0 , . . . , f n with these respective supports such that the closure of the image of the associated rational map f has dimension d.

Let ρ be the homomorphism of tori with its corresponding linear map A of lattices:

(2.13)

ρ : (K * ) n+1 (K * ) n (z 0 , . . . , z n ) → z1 z0 , . . . , zn z0 A : Z n+1 → Z n (w 0 , . . . , w n ) → (w 1 -w 0 , . . . , w n -w 0 )
.

Denote by F the rational map:

(2.14) [START_REF] Sturmfels | Elimination theory for tropical varieties[END_REF]). The cones of the tropical variety T (im f ) can be deduced from the description of the tropical variety T (im F).

F : (K * ) d (K * ) n+1 F = (f 0 , f 1 , . . . , f n ). Then, f = ρ • F and thus T (im f ) = A(T (im F)) (see Remark 1.2 in
As we show in Section 2.5, our direct and simple approach via curve valuations can be extended to deal with non-generic parametrizations. Also, curve valuations are used in Section 4 to give a formula for the order at the origin of S. The following lemma for polynomials with nonnegative exponents is straightforward.

∈ (L * ) d with in(σ) = b ε α = (b 1 ε α1 , . . . , b d ε α d ), b i ∈ L * , α i ∈ R, we have (2.15) h(σ) = in α (h)(b) ε mα(h) + h.o.t.(ε).
Notice that, a priori, we could have in α (h)(b) = 0 so Lemma 2.10 implies that val(h(σ)) ≥ m α (h), with equality if and only if in α (h)(b) = 0. Lemma 2.11 below gives conditions which ensure that a vector w can be realized as the (coordinatewise) valuation of a point in im(f L ).

Lemma 2.11. Let g 1 , . . . , g r be non-zero polynomials in K[x 1 , . . . , x d ] with r ≤ d and call B 1 , . . . B r in Z d ≥0 their respective supports. Given α ∈ R d , assume the initial polynomials in α (g 1 ), . . . , in α (g r ) have a common zero b ∈ (K * ) d such that their Jacobian matrix J inα(g) (b) has maximal rank r. Then, (i) for each ω ∈ R r verifying that ω j > m α (B j ) for all j ∈ [r], there exist σ ∈ (L * ) d with in(σ) = bε α such that val(g j (σ)) = ω j for all j ∈ [r]. (ii) if g 1 , . . . , g r have generic coefficients, for each ω ∈ R r verifying that ω j ≥ m α (B j ) for all j ∈ [r], there exist σ ∈ (L * ) d with val(σ) = α such that val(g j (σ)) = ω j for all j ∈ [r].

Proof. Given α, write ε α x = (ε α1 x 1 , . . . , ε α d x d ). For any g i = a∈Bi c i,a x a , i = 1, . . . , n, we denote g i (x) = ε -mα(Bi) g i (ε α x). These polynomials now lie in O[x 1 , . . . , x d ]. By Lemma 2.10, we know that in 0 (g i ) = in α (g i ) for all i ∈ [r] and J in0(g) = J inα(g) . Therefore, we can assume that α = 0 and m α (g i ) = 0, with

g i ∈ O[x 1 , . . . , x d ] for all i ∈ [r].
To prove item (i), under these assumptions on α and the polynomials g 1 , . . . , g r , consider the polynomials h i = g i -ε wi for every i ∈ [d] and w ∈ (R >0 ) r . By Lemma 2.10, val(h i (b)) > 0 for all i ∈ [r]. Since in 0 (h i ) = in 0 (g i ) and J in0(g) (b) has rank r, for any d -r generic affine linear forms 1 , . . . , d-r ∈ K[x 1 , . . . , x d ] that vanish on b, the Jacobian matrix J in0(h),l of the family of polynomials in 0 (h 1 ), . . . , in 0 (h r ), 1 , . . . , d-r is invertible when evaluated at b. Consider now the Jacobian matrix J h,l of the family of polynomials h 1 , . . . , h r , 1 , . . . , d-r . Since det(J h,l (b)) = det(J in0(h),l (b)) + h.o.t.(ε) and det(J in0(h),l (b)) ∈ K * , val(det(J h,l (b))) = 0. Then, by Hensel's Lemma there exist σ ∈ O n such that in(σ) = b and h i (σ) = 0 for all i ∈ [r], and so g i (σ) = ε ωi .

To prove item (ii), note that for a generic point c ∈ (K * ) d we have val(g i (c)) = 0 for all i ∈ [r], thus ω = 0 ∈ R r satisfies the statement. Consider now any ω ∈ (R ≥0 ) r . We need to show that there exists a point σ ∈ (L * ) d with val(σ) = 0 and such that val

(g i (σ)) = ω i for all i ∈ [r]. Consider the set I = {i ∈ [r] | ω i > 0}.
Since in 0 (g 1 ), . . . , in 0 (g r ) are generic polynomials and have a common zero b ∈ (K * ) d , the solution set in (K * ) d is an equidimensional variety of dimension d -r. If we denote by (B 1 ) 0 , . . . , (B r ) 0 the supports of these initial polynomials, ∆ d = {0, e 1 , . . . , e d } the set with e i the i-th vector of the canonical basis of R d and ∆

(d-r) d when it is repeated d -r times, then the mixed volume M V ((B 1 ) 0 , . . . , (B r ) 0 , ∆ (d-r) d
) is a positive integer. By the monotony of the mixed volume,

M V ({(B i ) 0 } i∈I , {(B i ) 0 ∪ {0}} i ∈I , ∆ (d-r) d
) is also positive. Given 1 , . . . , d-r generic affine linear forms in K[x 1 , . . . , x d ] and {λ i } i ∈I ⊂ K * generic numbers, denote by g • i the following polynomials

g • i (x) =      g i (x) for every i ∈ I g i (x) + λ i for every i ∈ I, i ≤ r i-r (x) for every r < i ≤ d. Then, the initial polynomials in 0 (g • 1 ), . . . , in 0 (g • d ) have an isolated simple common zero c ∈ (K * ) d . Consider now the polynomials h 1 , . . . , h d ∈ O[x 1 , . . . , x d ] defined as h i (x) = g • i (x) -ε ωi if i ∈ I and h i (x) = g • i (x) otherwise. Since c ∈ (K * ) d ⊂ O d satisfies val(h i (c)) > 0 for all i ∈ [d] and J in0(h) (c) = J in0(g),l (
c), then val(J h (c)) = 0. Then, by Hensel's Lemma there exists σ such that in(σ) = c and h i (σ) = 0 for all i ∈ [r], or equivalently, g i (σ) = ε wi for all ω i > 0 and g i (σ) = -λ i ∈ K * for all ω i = 0.

2.4. The proof of Theorem 2.8. We first show that T (I) is contained on the union of the cones C F J . We consider for any J ⊆ [n] 0 the cone C J = ∪ F C F J , where the union is over all coherent collections F of faces of (A 0 , . . . , A n ) that are adapted to J. 1 Then, * if 0 / ∈ J, C J := {ψ(α) + u, α ∈ τ J , and u ∈ (R ≥0 ) J }, * if 0 ∈ J, set J = J -{0} and of those indices for which in α (f i )(b) = 0. Since the polynomials f i are generic, in α (f i )(b) = 0 for i ∈ J implies that α ∈ τ J . We deduce from Lemma 2.10 that val(f i (σ)) = m i (α) for all i ∈ [n] 0 \J and val(f i (σ)) > m i (α) for all i ∈ J, so val(f L (σ)) = ψ(α) + u ∈ C J where α ∈ τ J and u ∈ R J ≥0 . This shows that val(f L (σ)) ∈ C J . On the other hand, if in α (f 0 )(b) = 0, this means that val(f 0 (σ)) = m 0 (α) + λ > m 0 (α). Take J be the set of indices j ≥ 1 satisfying in α (f j )(b) = 0 and J = J ∪ {0}, it follows by the same argument that val(f L (σ)) ∈ C J .

C J := {ψ(α) + u -λ 1, α ∈ τ J , u ∈ (R ≥0 ) J ,
This shows the inclusion T (im f ) ⊂ ∪ J C J = ∪ F adapted to J C F J . Our next task is to prove that the cones C J lie in T (I) if the polynomials f i have generic coefficients. Let α ∈ τ J . Since the polynomials f i are generic, we may assume that there exists b ∈ (K * ) d with in α (f j )(b) = 0 if and only if j ∈ J, with the additional property that the Jacobian matrix Jac (inα(fj )) j∈J (b) associated to those initial polynomials has maximal rank. In particular, in α (f i (b)) = 0 for every i ∈ [n] 0 \J and, by Lemma 2.10, val(f i (σ)) = m α (P i ) for every σ with in(σ) = b ε α .

First, assume 0 / ∈ J and consider ψ(α) + u ∈ C J where u ∈ (R >0 ) J and ω ∈ R n defined as ω = ψ(α) + u + m α (P 0 )(1, . . . , 1). Applying Lemma 2.11 to the polynomials (and coordinates of ω) indexed by J, there exists σ ∈ (L * ) d with in(σ) = bε α and val(f i (σ)) = ω i for every i ∈ J. Hence, val(f L (σ)) = ω -m α (P 0 )(1, . . . , 1) = ψ(α) + u. Since T (I) is a closed set, we deduce that C J ⊂ T (I). In case 0 ∈ J, consider ψ(α) + u -λ(1, . . . , 1) ∈ C J where u ∈ (R >0 ) J and λ > 0 and ω = (ω 0 , . . . , ω n ) = (m α (P 0 ) + λ, m α (P 1 ) + u 1 , . . . , m α (P n ) + u n ) ∈ R n+1 . Applying Lemma 2.11 to the polynomials {f i } i∈J (and the corresponding coordinates of ω), there exists σ ∈ (L * ) d with in(σ) = bε α and val(f i (σ)) = ω i for all i ∈ J. Then val(f L (σ)) = ψ(α) + u -λ(1, . . . , 1). Remark 2.12. For any choice of coefficients, the Newton polytope of an equation of im f lies inside the Newton polytope corresponding to the case of generic polynomials. However, this need not induce an inclusion in any direction between the cones in the corresponding tropicalizations, as in Example 2.14 below. We show in this example how to use the arguments in our proof of Theorem 2.8 to extend them beyond the case of generic coefficients. 1 Unlike the cones C F J , the cones C J are not necessarily convex.

Our combinatorial definitions in Def. 2.6 and Def. 2.7 are simplified by appealing to the generic behaviour of the coefficients to make them independent of the particular polynomials. We can modify the definition of the cones C F J depending not only on the supports but also on the given f 0 , . . . , f n to get one containment as follows. Given (F 0 , . . . , F n ) a collection of faces of (A 0 , . . . , A n ), we call τ F,f J the cone of those α ∈ R d such that F j = face α (A j ) for all j, and there exists a simple common zero b ∈ (K * ) d of in α (f j ) for all j ∈ J, such that in α (f i )(b) = 0 for any i / ∈ J. Simple means that the associated Jacobian matrix has maximal rank. So, τ F,f J is empty for every J unless there exists α ∈ R d such that F j = face α (A j ) for all j. For such directions we are requiring that the initials of f 0 , . . . , f n with respect to α behave as in the generic case. We then define the cones C F,f J as follows: In both cases, we consider polynomials f 0 , . . . , f 3 with the following respective supports: A 0 = {(0, 0)}, A 1 = {(2, 0), (3, 0), (0, 2)}, A 2 = {(0, 2), (0, 3), (2, 0)}, A 3 = {(1, 1), (0, 3), (1, 2), (2, 1), (3, 0)}. We depict the corresponding Newton polytopes

* if 0 / ∈ J, C F,f J := {ψ(α) + u, α ∈ τ F,f J , and u ∈ (R ≥0 ) J }, * if 0 ∈ J, set J = J -{0} and C F,f J := {ψ(α) + u -λ 1, α ∈ τ F,f J , u ∈ (R ≥0 ) J , and λ ∈ R ≥0 }. Then, C F,f J ⊂ T (im f ). 2.
P i = conv (A i ) for i = 1, 2, 3: - - - | | |
f 0 = 1 f 1 = a 1 x 2 1 + b 1 x 2 2 + x 3 1 f 2 = a 2 x 2 2 + b 2 x 2 1 + x 3 2 f 3 = x 1 x 2 + a 3 x 3 1 + b 3 x 2 1 x 2 + c 3 x 1 x 2 2 + d 3 x 3 2 ,
where a 1 , b 1 , a 2 , b 2 , a 3 , b 3 , c 3 , d 3 are generic nonzero elements of K. It is straightforward to check that in this case the cones in the statement of Theorem 2.8 are as follows:

• For J ⊂ [3] 0 with at least 3 elements, there is no possible coherent collection of faces adapted to J.

• For #J = 2 the only coherent collection of faces adapted to J is (A 0 , A 1 , A 2 , A 3 ) and the associated cones are:

C 1 = R ≥0 r 1 +R ≥0 r 2 , C 2 = R ≥0 r 1 +R ≥0 r 3 , C 3 = R ≥0 r 2 +R ≥0 r 3
, defined by the rays r 1 = (1, 0, 0), r 2 = (0, 1, 0) and r 3 = (0, 0, 1). • For #J = 1 we obtain the cones

C 4 = R ≥0 r 3 + R ≥0 r 4 , C 5 = R ≥0 r 1 + R ≥0 r 5 , C 6 = R ≥0 r 2 + R ≥0 r 5
, where we define the rays r 4 = (-1, -1, -1) and r 5 = (1, 1, 1), and

C a,1 = R ≥0 r 1 + R ≥0 r 6 , C a,2 = R ≥0 r 2 + R ≥0 r 7 , C a,3 = R ≥0 r 3 + R ≥0 r 8
, where r 6 = (-2, -3, -3), r 7 = (-3, -2, -3) and r 8 = (2, 2, 3). • Finally, for J = ∅, the image of ψ give us the cones

C b,1 = R ≥0 r 4 + R ≥0 r 6 , C b,2 = R ≥0 r 4 + R ≥0 r 7 , C b,3 = R ≥0 r 5 + R ≥0 r 8 .
Here, the union of cones C a,i ∪ C b,i = C i+6 for i = 1, 2, 3 yields the cones

C 7 = R ≥0 r 1 + R ≥0 r 4 , C 8 = R ≥0 r 2 + R ≥0 r 4 , C 9 = R ≥0 r 3 + R ≥0 r 5 .
In fact, the Newton polytope of a (reduced) polynomial H that defines the closure of the image of f is a truncated simplex with vertices {(4, 0, 0), (0, 4, 0), (0, 0, 4), (9, 0, 0), (0, 9, 0), (0, 0, 9)}. The cones C 1 to C 9 correspond to the 9 edges of N (H). We explain how to compute a priori the multiplicities (the lengths of the edges) in § 2.6.

ψ(0, 1) = (0, 0, 0) ψ(1, 2) = (2, 2, 3) ψ(1, 1) = (2, 2, 2) ψ(2, 1) = (2, 2, 3) ψ(1, 0) = (0, 0, 0) ψ(-2, -3) = (-6, -9, -9) ψ(-1, -1) = (-3, -3, -3) ψ(-3, -2) = (-9, -6, -9)
Figure 1. The image by ψ of the cones spanned by {(-3, 2), (0, 1)}, {(0, 1), (1, 2)}, {(2, 1), (1, 0)}, {(1, 0), (-2, -3)} is one dimensional. The black ray corresponding to ψ(1, 1) is both red and green.

Example 2.14 (Non-generic coefficients). Consider now the rational map f : (C * ) 2 (C * ) 3 associated to the polynomials

f 0 = 1 f 1 = x 2 1 -x 2 2 -x 3 1 f 2 = x 2 2 -x 2 1 -x 3 2 f 3 = 4x 1 x 2 -(x 1 + x 2 ) 3 .
The union of the zeros of these f i 's (that is, the base locus of f ) has a highly singular point at the origin:

Let H be a (reduced) polynomial that defines the hypersurface parameterized by f . Its Newton polytope N (H) is the pyramid in (Z ≥0 ) 3 with equation X +Y +Z ≤ 9 cut by the two half-spaces 3 X +3 Y +2 Z ≥ 12 and 2 X +2 Y + Z ≥ 7. Note that it is contained in the convex hull P of the support in the generic case in Example 2.13. We depict N (H) on the right and P on the left:

The tropicalization T (H) is not equal to the union of cones of Theorem 2.8. The cones corresponding to the edges on the top face coincide but looking at the facets of the polytopes P and N (H) visible from the origin, it is straightforward to check that there is no inclusion of T (H) and the cones occurring in the case of generic coefficients in any of the two directions. However, we can describe some of the cones in T (H) by considering the approach in the proof of Theorem 2.8 and using it as a recipe to find the tropicalization.

In Theorem 2.8, to prove that a cone C J lies in T (I), the genericity condition is used to ensure that there is a common zero b ∈ (C * ) n-1 of in α (f j ) if and only if j ∈ J and that it satisfies all hypotheses from Lemma 2.11. But this Lemma does not require that the polynomials are generic. Using this idea, we consider for all possible J the common zeros of {in α (f j )} j∈J that are not zeros of any polynomial in {in α (f j )} j ∈J . If either 0 ∈ J or #J ≥ 3, then there is no possible α such that {in α (f j )} j∈J have common zeros. If J = ∅, for all α ∈ R 2 there are points b ∈ (C * ) 2 that are not zeros of any in α (f j ). Then, as in the generic case (see Example 2.13), (2.16)

ψ(R 2 ) = C b,1 ∪ C b,2 ∪ C b,3
and this set is included in T (I). If J∈{{1, 3}, {2, 3}} the only vector α ∈ R 2 for which there are common zeros of {in α (f j )} j∈J that are not zeros of any in α (f j ) for all j ∈ J or of det(J {inα(fj )} j∈J ) is α = (0, 0). For J = {1, 2}, a possible vector is α = (0, 0) (although is not the only one). Then, C 1 ∪ C 2 ∪ C 3 ⊂ T (I). These 3 are in fact the cones associated to the edges of N (H) lying in the coordinate axes. When looking at J={1, 2}, we also need to consider α = (1, 1). There is no zero of in (1,1) (f 1 ) that is not a zero of in (1,1) (f 2 ). Consider σ = (ε v (1 + s 1 ), ε v (1 + t 1 )), where v > 0, (s 1 , t 1 ) ∈ L 2 and val(s 1 ), val(t 1 ) > 0. Then

f 1 (σ) = ε 2v (2s 1 -2t 1 + s 2 1 -t 2 1 -ε v (1 + s 1 ) 3 ) f 2 (σ) = -ε 2v (2s 1 -2t 1 + s 2 1 -t 2 1 + ε v (1 + t 1 ) 3 ) f 3 (σ) = ε 2v (4(1 + s 1 )(1 + t 1 ) -ε v (2 + s 1 + t 1 ) 3 ). If 2s 1 -2t 1 + s 2 1 -t 2 1 = 0 and we choose s 1 = aε ω + s 2 , t 1 = bε ω + t 2 with 0 < ω < v = ω + v with v > 0, then val(f 1 (σ)) = val(f 2 (σ)) = 2v + 3ω and val(f 3 (σ)) = 2v + 2ω. Thus, the cone R ≥0 (1, 1, 1) + R ≥0 (3, 3, 2) is contained in T (I).
The union of this cone with C 9 ⊂ T (I) gives the entire cone corresponding to the edge (4, 0, 0)(0, 4, 0). If J = {1}, we analyze those α such that face α (A 1 ) has dimension 1. When α = (0, 1) we obtain the cone R ≥0 (1, 0, 0) ⊂ C 1 which we already have. When α = (-2, -3) there are zeros of in (-2,-3)

(f 1 ) = x 2 2 + x 3 1 that are not zeros of in (-2,-3) (f 2 ) = in (-2,-3) (f 3 ) = x 3 2 or J in (-2,-3) (f1) = (3x 2 1 2x
2 ) and we get the cone C a,1 . Recall that (with the notation of Example 2.13) C a,i ∪ C b,i = C 6+i . Interestingly and as in the generic case, since we already obtained in (2.16) the cone C b,1 we have that C a,1 ∪ C b,1 = C 7 ⊂ T (I). In the same way, for J = {2} we obtain that C 8 ⊂ T (I). These cones correspond to the edges [(9, 0, 0)(0, 0, 9)] and [(0, 9, 0)(0, 0, 9)] of N (H). The set J = {3} is a bit more complicated to analyze. If we consider all faces face α (A 1 ) with dimension 1, both α = (1, 2) and α = (2, 1) gives the cone C a,3 . Via the cones obtained in (2.16), C 9 ⊂ T (I). As we mentioned before, joining C 9 with R ≥0 (1, 1, 1) + R ≥0 (3, 3, 2) we obtain the cone R ≥0 (0, 0, 1) + R ≥0 (3, 3, 2) ⊂ T (I) associated to the edge [(4, 0, 0), (0, 4, 0)].

Finer computations are required to find the other five cones associated to the edges of the two new faces of the Newton polytope of H, as it is necessary to consider higher order terms in σ.

2.6. The multiplicities. We complete the description of the tropicalization of the closure of the image of f with the computation of the multiplicities of the maximal cones in Theorem 2.8.

Recall that as the variety im f is irreducible of dimension d, T (im f ) is a polyhedral set that can be given the structure of a pure d-dimensional fan. Theorem 2.8 gives a description of T (im f ) as a set, and as in [START_REF] Sturmfels | The Newton polytope of the implicit equation[END_REF], this description does not have a natural fan structure. However, every maximal dimensional cone σ in T (im f ) has a multiplicity mult(σ) ∈ Z >0 . This multiplicity can be computed at any regular point w in the cone, by means of [START_REF] Sturmfels | Elimination theory for tropical varieties[END_REF]Theorem 3.12] and [START_REF] Sturmfels | Elimination theory for tropical varieties[END_REF]Theorem 5.1].

Consider again the homomorphism of tori ρ in (2.13). Given a regular point w in a maximal cone σ of T (im f ) such that the preimage f -1 (w) is a finite set of regular points in T (im F), m w = mult(σ) equals the sum over all v ∈ f -1 (w), of the quantities 1 deg(ρ)

m v • index L w ∩ Z n : A(L v ∩ Z d ) ,
where L w denotes the Q vector subspace which coincides with T (im f ) near w, and the multiplicity m v of the regular point v in T (im F) is as computed in [START_REF] Sturmfels | Elimination theory for tropical varieties[END_REF]Theorem 5.1].

Consider the map Ψ ST (α) = (m α (A 0 ), . . . , m α (A n )) defined as in (2.9), but for the rational map F). For every J ⊂ [n] 0 with |J| ≥ d and every coherent collection of faces F = (F 0 , . . . , F n ) of (A 0 , . . . , A n ) adapted to J, define index(F, J) as the index of the lattice Ψ ST (C F ∩ Z d ) + Z J in its saturated lattice, when both have rank d. Otherwise, index(F, J) = 0. With this notation, the multiplicity m v of the regular point v in T (im F) is equal to the sum of

1 deg(F) index(F, J) • MV(F i | i ∈ J)
over all sets J ⊂ [n] 0 and all coherent collection F of faces of A 0 , . . . , A n adapted to J such that Ψ(C F ) + R J ≥0 contains v. Here, MV(F i | i ∈ J) denotes the |J|-dimensional mixed volume. We give a simple example of the multiplicity computation via this formula.

Example 2.15. Let f 0 , f 1 , f 2 ∈ K[x] be the polynomials

f 0 = x 3 + 3x, f 1 = x 5 + 5x 3 and f 2 = x 5 + 22x 3 + 17x.
The rational map f has degree 2 and the cones in the tropicalization of its image are

C {1} = ψ(R ≥0 ) = R ≥0 (1, 0); C {2} = R ≥0 (0, 1) and C {0} = ψ(R ≤0 ) = R ≥0 (-1, -1).
The map F has degree 1 and the tropicalization of its image has 5 cones. The following table shows the multiplicity m w of a regular point w in each of them (and the data used to compute it):

w v ∈ ρ -1 (w) m v index(L w ∩ Z 2 : A(L v ∩ Z 3 )) m w (λ, 0, 0) 2 1 (-λ, -λ) 2 -λ 2 (3, 5, 5) 1 2 (0, λ, 0) 2 1 (λ, 0) 2 λ 2 (1, 3, 1) 1 2 (0, λ) (0, 0, λ) 4 1 2 .
For example, take a regular point w = (λ, 0) ∈ R 2 ≥0 . The set f -1 (w) has two points in T (im F): the point v 1 = (0, λ, 0) which is in a cone of multiplicity 2 and the point v 2 = (λ/2, 3λ/2, λ/2)} which lies in a cone of multiplicity 1. Since A(0, 1, 0) = (1, 0), the index is 1, but since A(1, 3, 1) = 2(1, 0), the corresponding index is 2. By equation (2.6), m w = 2.

Degree of rational varieties under sparse parametrizations

As before, we consider n+1 supports A 0 , . . . , A n which lie in the nonnegative orthant (Z ≥0 ) d and generic polynomials f 0 , . . . , f n with respective supports A 0 , . . . , A n , such that the rational map from (1.2)

f = f 1 f 0 , . . . , f n f 0
is generically finite and thus rationally parameterizes an irreducible variety S of dimension d. We give in Theorem 3.3 an explicit formula for its degree in terms of the supports when the coefficients are generic, and which is an upper bound in any case. Our sharp bound for the degree of S in Theorem 3.3 for generic coefficients is similar to the bound in Proposition 4 in [START_REF] Herrero | On degree bounds for the sparse Nullstellensatz[END_REF] in case f 0 = 1. The upper bound given by the volume of the convex hull of the union of the supports appears in Section 3 in [START_REF] Emiris | Implicit polynomial support optimized for sparseness[END_REF]. We give conditions for equalities in the chain (3.5) of inequalities in Theorem 3.10, and we also show in Example 3.6 that all the inequalities can be strict. Consider the open set

U = {x ∈ (K * ) d : n i=0 f i (x) = 0}, the incidence variety (3.1) W = {(x, y) ∈ U × (K * ) : f 0 (x) y 1 -f 1 (x) = • • • = f 0 (x) y n -f n (x) = 0},
and the projection

(3.2) π : W × (K * ) n → (K * ) n , π (x, y) = y.
Then, S coincides with the closure of the image π(W ). Moreover, this map is finite iff f is finite and both have the same degree. Observe that the polynomials defining W don't have in general generic coefficients with respect to their supports even if f 0 , . . . , f n have generic coefficients (so the results in [START_REF] Esterov | Elimination theory and Newton polytopes[END_REF] do not directly apply). In fact, we compute deg(S) in Theorem 3.3 studying the following similar genericity question: given a subset I of [n] 0 , with cardinality d and generic coefficients (c ij , i = 1, . . . d, j = 0, . . . n), when do the polynomials n j=0 c ij f j , i = 1, . . . , d, have generic coefficients with respect to their support? If not, which is the number of their common zeros in the torus (K * ) d ?

In order to state our main result in this section, we need to introduce two definitions.

Definition 3.1. Given a subset A ⊂ Z d and a natural number j, denote by A j ⊂ Z d+j the finite subset

(3.3) A j = {(α, 0) ∈ Z d+j | α ∈ A} ∪ {e d+1 , . . . , e d+j },
where 0 ∈ Z j and e i denotes the i-th canonical basis vector.

We also define new lattice subsets associated to a collection of finite subsets. We then have Theorem 3.3. Consider Laurent polynomials f 0 , . . . , f n in d variables and coefficients in K with respective supports A 0 , . . . , A n ⊂ Z d , such that the rational map f : (K * ) d (K * ) n defined as in (1.2) by f = f1 f0 , . . . , fn f0 , is generically finite with deg(f ) = δ. Then,

Definition 3.2. Let A 0 , . . . , A n ⊂ Z d with d ≤ n, be a collection of finite lattice subsets. Given I = {i 1 , . . . , i d } ⊂ [n] 0 with |I| = d, for any i ∈ [d] we denote (3.4) A i,I = A i ∪ ∪ j / ∈I A j ⊂ Z d ,
(3.5) deg(S) • δ ≤ MV(A n+1-d 0 , . . . , A n+1-d n ) ≤ min |I|=d d I ≤ max |I|=d d I ≤ vol(conv(∪ n j=0 A j ))
. Moreover, the left equality holds when f 0 , . . . , f n have generic coefficients.

Proof. As we mentioned, we can assume without loss of generality (possibly after a common translation of the supports) that all exponents are nonnegative. The degree of S equals the number of intersection points of S with the zero set of d generic linear forms in n variables with coefficients in K, which we write as follows:

(3.6) i = a i,1 y 1 + • • • + a i,n y n + a i,0 , i ∈ [d].
Since 1 , . . . , d are generic, we can assume that every intersection point lies in f (U ), and it is enough to consider all common zeros of the form z = f (σ) for some σ in the domain of f (that is, such that f i (σ) = 0 for any i ∈ [n] 0 ). Such σ belongs to the variety defined by the following polynomials g 1 , . . . , g d ∈ K[x 1 , . . . , x d ]:

g i = n j=0 a i,j f j , i ∈ [d].
These polynomials have support ∪ n i=0 A i and so, the number of isolated common zeros in the torus is bounded by vol(conv(∪ n j=0 A j )) by Bernstein's theorem, with equality in case g 1 , . . . , g d are generic polynomials with this support. But, depending on the relative positions of A 0 , . . . , A n , this need not be the case.

A first refinement of this bound is given by the following observation. As the coefficients (a i,j

) i∈[n] 0 j∈[d]
are generic, for every set I = {i 1 , . . . , i d } ⊂ [n] 0 with |I| = d, the system g 1 = . . . g d = 0 is equivalent by means of row operations to a system of the form (3.7)

h I 1 = f i1 + j / ∈I µ I 1j f j = 0 . . . . . . h I d = f i d + j / ∈I µ I dj f j = 0,
(3.8) h [d] i = f i + n+1 j=d+1 µ [d] ij x j , 1 ≤ i ≤ d, h [d] j = f j -x n , d + 1 ≤ j ≤ n, h [d] n+1 = f 0 -x n+1 .
Denote by U the complement in (K * ) d of the union of the zeros of f 0 , . . . , f n+1 , that is, the domain of definition of f . Clearly, there is a bijection between common zeros in U × (K * ) n+1-d of h

[d] 1 , . . . , h [d]
n+1 and common zeros in U of g 1 , . . . , g d , so deg(S) deg(f ) is bounded above by the mixed volume of their supports. But, for generic f 0 , . . . , f n and generic linear forms 1 , . . . , d , these new polynomials h

[d] 1 , . . . , h [d]
n+1 have generic coefficients. Moreover, they do not have common zeros in common in (K * ) n+1 with any f i . Indeed, no f i with i = d + 1, . . . , n can vanish if (x d+1 , . . . , x n+1 ) ∈ (K * ) n+1-d . Take any i ∈ [d], for instance i = 1. Then, if x = (x 1 , . . . , x n+1 ) is a solution in the torus of system (3.8) and f 1 (x 1 , . . . , x d ) = 0, then x is also a solution of the system of n + 2 generic polynomials in n + 1 variables: ). If the vertices are rational numbers, we could scale the supports by a natural number k to turn them into integers. Therefore, our inequalities apply and we can then divide by d k everywhere. Moreover, as the (mixed) volumes vary continuously, one could approximate the (mixed) volumes by considering approximations of the convex sets by rational polytopes. . But in general, even for generic polynomials, the inequalities may be strict, as we show in the next example.

f 1 = n+1 j=d+1 µ [d] 1j x j = h [d] 2 = • • • = h [d] n+1 = 0,
(i) MV(B 1 , . . . , B n-1 ) = vol(conv(∪ n-1 i=1 B i )). (ii) For every r ∈ [n-1] and every face F of ∪ n-1 i=1 B i of codimension r, the cardinality of the set {i | B i ∩F = ∅} is at least n -r. (iii) For every proper face F of ∪ n-1 i=1 B i there exists a nonempty subset

J ⊂ {i | B i ∩ F = ∅} such that dim j∈J (B j ∩ F) < |J|.
Proof. The equivalence between items (i) and (ii) follows from applying [2, Corollary 3.7] using the polytope conv(∪ n-1 i=1 B i ). Item (iii) follows from negation of item (iii) in Lemma 3.7.

Example 3.6, continuation. Recall that by Definition 3.2, for any I = {j, k} ⊂ [3] with 2 elements and i the index in its complement, d I = MV(A j ∪ A 0 ∪ A i , A k ∪ A 0 ∪ A i ) and both supports are contained in the union ∪ 3 j=0 A i . It can be checked that item (iii) in Corollary 3.8 holds and so d I < vol(conv(∪ 3 j=0 A j )) for all I. The following example shows that the conditions given in items (ii) and (iii) in Lemma 3.7 are different.

Example 3.9. Consider the finite sets in (Z ≥0 ) 3 B 1 = {(0, 0, 0), (1, 0, 0)}, B 1 = {(0, 0, 0), (2, 0, 0), (0, 2, 0), (0, 0, 2)}, B 2 = {(0, 1, 0), (1, 1, 0), (0, 2, 0)}, B 2 = {(0, 0, 0), (2, 0, 0), (0, 2, 0)}, B 3 = {(0, 1, 0), (1, 1, 0), (0, 2, 0)} and B 3 = {(0, 0, 0), (2, 0, 0), (0, 2, 0)}.

x y z B 1 x y z B 2 x y z B 3 x y z B 1 x y z B 2 x y z B 3 Clearly 0 = MV(B 1 , B 2 , B 3 ) < MV(B 1 , B 2 , B 3
). If we take the coherent collection of faces of B 1 , B 2 , B 3 associated to the interior normal vector (1, 1, 0), the collection from item (ii) in Lemma 3.7 is essential. However, the collection from item (iii) is not. On the other hand, if we consider the coherent collection of faces associated to the interior normal vector (0, 1, 1), the collection from item (iii) Lemma 3.7 is essential but the collection from item (ii) is not.

Therefore, it is not straightforward to prove in a direct way the equivalence of items (ii) and (iii) in Lemma 3.7. It would be interesting to see a direct combinatorial proof without going through item (i).

The following theorem provides a necessary and sufficient condition for the equality d I = vol(conv(∪ n j=0 A j )) for some I ⊂ [n] 0 with |I| = n -1. We denote by I c the complement of I in [n] 0 . Theorem 3.10. Let A 0 , . . . , A n be finite sets in (Z ≥0 ) n-1 . Fix I ⊂ [n] 0 with |I| = n -1. The following statements are equivalent:

(i) For every r ∈ [n -1] and every face F of ∪ n j=0 A j of codimension r and every J ⊂

[n] 0 such that |J| = r + 2 that contains I c , we have (∪ j∈J A j ) ∩ F = ∅. (ii) vol(conv(∪ n j=0 A j )) = d I .
Proof. For every set J ⊂ [n] 0 that contains I c , we can define J = J\I c ⊂ I. Then item (i) happens if and only if for every face F of ∪ n j=0 A j of codimension r > 0 and every J ⊂ I with | J| = r, (∪ j∈ J (A j ∪ (∪ i∈I c A i )) ∩ F = ∅. Using item (ii) of Corollary 3.8, this is equivalent to item (ii).

The next theorem provides a sufficient condition for the equality deg(S)

• deg(f ) = d I for some I ⊂ [n] 0 with |I| = n -1.
Theorem 3.11. Let A 0 , . . . , A n be finite sets in (Z ≥0 ) n-1 . Let f 0 , . . . , f n be generic polynomials with support A 0 , . . . , A n and coefficients in K such that the rational map f from (1.2) is generically finite. Assume I ⊂ [n] 0 with |I| = n -1 satisfies that for every coherent collection of proper faces F of (A i ∪ (∪ j∈I c A j )) i∈I , there exists J nonempty subset of

P F = {i ∈ I | A i ∩ F i = ∅} such that dim i∈J A i ∩ F i < |J|. Then, deg(S) • deg(f ) = d I = MV(A i1 , . . . , A in-1 ).
Proof. Denote I = {i 1 , . . . , i n-1 } and I c = {i 0 , i n }. As in the proof of Theorem 3.3, take generic coefficients {µ j,I , ν j,I } n-1 j=1 so that the system h j,

I = f ij + µ j,I f i0 + ν j,I f in = 0 for j ∈ [n -1] has deg(S) deg(f ) common isolated zeros, all of them in the open set {x ∈ (K * ) n-1 | n j=0 f j (x) = 0}.
Consider the polynomials {f ij + t µ j,I f i0 + t ν j,I f in , j = 1, . . . , n -1, t ∈ K}. For all but finitely many values of t, the system given by these polynomials has deg(S) • deg(f ) isolated common zeros with all non-zero coordinates, and MV(A i1 , . . . , A in-1 ) when t = 0. Then, using Theorem

3.3, MV(A i1 , . . . , A in-1 ) ≤ deg(S)•deg(f ) ≤ d I = MV(A i1 ∪A i0 ∪ A in , . . . , A in-1 ∪ A i0 ∪ A in ) common
zeros with all non-zero coordinates. By Lemma 3.7, MV(A i1 , . . . , A in-1 ) = d I .

Example 3.12. Let d ∈ N be an even number. Consider the hypersurface S parameterized by the rational map f as in (1.2), where f 0 , . . . , f 3 are generic polynomials with two variables, coefficients in K and supports A0 = {(0, 0), (1, 0), (0, 1), (d/2, d/2)} = A3, A1 = {(0, 0), (1, 0), (0, d)}, A2 = {(0, 0), (d, 0), (0, 1)}.

Then the conditions of Theorem 3.11 are satisfied for I = {1, 2} and we have that deg(f ) = 1, so the degree of the hypersurface equals deg

(S) = d 2 = M V (A 0 ∪ A 1 ∪ A 3 , A 0 ∪ A 2 ∪ A 3 ).

Orders of rational hypersurfaces under sparse parametrizations

In this section, we assume that d = n -1 and S is a hypersurface in (K * ) n . We study its order at the origin ord 0 (S), which is defined as follows. Let H be a (reduced) polynomial such that S = (H = 0). Then, ord 0 (S) = max{m ∈ Z ≥0 | ∂ β (H)(0) = 0, for all β ∈ (Z ≥0 ) n , |β| < m}.

We compute ord 0 (S) in Theorem 4.2 under an additional condition on the family of supports. As we remarked in the Introduction, depending on the relative positions of the supports A 0 , . . . , A n it could happen that ord 0 (S) > 1 (that is, the origin is a singular point of the closure of S in K n ) even for generic coefficients, We end with two examples where our proofs can be extended to other cases not falling into the hypotheses of Theorem 4.2.

Before stating Theorem 4.2, we need to introduce a definition.

Definition 4.1. Let A 0 , . . . , A n be a family of finite sets in (Z ≥0 ) n-1 . Given α ∈ Q n-1 , we call J α = {j ∈

[n] 0 | m α (A j ) < max i∈[n]0 {m α (A i )}}. We say that the family is tame if for any α such that J α = ∅ and |J α | = n, the family {face α (A j )} j∈Jα in (Z ≥0 ) n-1 is not essential (see Definition 2.6).

Note that in case all A i are equal for all i ∈ [n] 0 or if all A i are equal for i ∈ [n] and A 0 consists of a single point (thus f i /f 0 are Laurent polynomials with the same support, the case studied in [START_REF] Sturmfels | Minimal polynomials and sparse resultants[END_REF]), the family is always tame.

Recall that by Theorem 3.3 the degree of S for generic coefficients satisfies deg(S) • deg(f ) = MV(A 2 0 , . . . , A 2 n ), where the notation A 2 i ⊂ Z n+1 is given in Definition 3.1. Theorem 4.2. Let A 0 , . . . , A n be a tame family of finite sets in (Z ≥0 ) n-1 . Let f 0 , . . . , f n be generic polynomials with respective supports A 0 , . . . , A n and coefficients in K such that the map f from n j=0 f j (x) = 0}. We can assume that H, 1 -a 1,0 ε, . . . , n-1 -a n-1,0 ε have D = MV(A 2 0 , . . . , A 2 n )/ deg(f ) common roots, and that they all lie in f L (U ). To compute the order at the origin of H, we consider the line in L n defined by the generic affine linear forms 1 -a 1,0 ε, . . . , n-1 -a n-1,0 ε. Notice that this line can be parameterized as {x ∈ L n | 1 (x) = a 1,0 ε, . . . , n-1 (x) = a n-1,0 ε} = {λ u + ε v | λ ∈ L}, with u, v ∈ K n . The common zeros of the polynomials H, 1 -a 1,0 ε, . . . , n-1 -a n-1,0 ε in f L (U ) are exactly the points

ω k = f 1 (σ k (ε)) f 0 (σ k (ε)) , . . . , f n (σ k (ε)) f 0 (σ k (ε)) = λ k u + v ε, k ∈ [D],
for which H(λ k u + v ε) = 0. If we consider the polynomial H(λ U + ε V ) ∈ K[U, V, ε][λ], we have (4.1)

H(λ U + ε V ) = H 0 (U )λ D + H 1 (U, V, ε)λ D-1 + • • • + H D (V, ε),
where for all j ∈ [D] 0 , H j (U, V, ε) is a polynomial of degree at most j in ε. By genericity of i and a i,0 , we can assume that u, v ∈ (K * ) n , H 0 (u) = 0 and H D (v, ε) = H(v ε) = 0. Note that val(H 0 (u)) = 0 and val(H j (u, v, ε)) ≥ 0 for all j ∈ [D -1] 0 . Hence, if we consider the coefficients of H(uλ + vε) = D i=0 H D-i (u, v, ε)λ i ∈ L[λ] as elementary symmetric functions of the roots {λ k } D k=1 ⊂ L, we can easily see that val(λ k ) ≥ 0 for all k ∈ [D]. Indeed, assume that val(λ k ) < 0 for some k and consider the non-empty set J = {k ∈ [D] | val(λ k ) < 0}. Then, the elementary symmetric function H #J (u, v, ε) has the same negative valuation as H 0 (u) k∈J λ k , which is a contradiction. Moreover, the minimum power of λ with a coefficient of valuation 0 is D -#{k | val(λ k ) = 0}. On the other hand, if we now consider H as a sum of homogeneous terms of degree ord 0 (S) to D and we evaluate it at uλ + vε, the minimum power of λ with a coefficient of valuation 0 is ord 0 (S). Therefore,

2. 3 .

 3 Two auxiliary results. We first recall Hensel's lemma. The valuation ring of L is O = {σ ∈ L | val(σ) ≥ 0}, with maximal ideal m. Here O * = O m denotes the units in O, i.e. those elements with valuation 0. Given a sequence of polynomials h = (h 1 , . . . , h r ) ∈ L[x 1 , . . . , x d ], let us denote by J h (x) ∈ L(x) d×r its associate Jacobian matrix. Observe that the field L is algebraically closed, and so Hensel's lemma [14] holds in L: Hensel's lemma: Let h = (h 1 , . . . , h d ) ∈ O[x 1 , . . . , x d ] be a family of non-zero polynomials. Assume that there exists b = (b 1 , . . . , b d ) ∈ O d such that val(h i (b)) > 0 for 1 ∈ [d] and val(det J h (b)) = 0. Then, there exists a unique σ ∈ O d such that h i (σ) = 0 and val(σ i -b i ) > 0 for all i ∈ [d].

Lemma 2 . 10 .

 210 Let B ⊂ Z d ≥0 be a finite set and h ∈ L[x 1 , . . . , x d ] be a non-zero polynomial supported on B. For any σ

  and λ ∈ R ≥0 }, where τ J := {α ∈ R d | dim( j∈K face α (conv(A j ))) ≥ |K|, for all K ⊆ J}. By the Fundamental Theorem of Tropical Geometry, it suffices then to show that val(f L (σ)) lies in one of the cones C J for σ in (L * ) d \{ n i=0 f i = 0}, where we simply denote f iL = f i . Let α = val(σ), with σ = (b 1 ε α1 +h.o.t.(ε), . . . , b d ε α d + h.o.t.(ε)) and b = (b 1 , . . . , b d ) ∈ (K * ) d . Consider the initial forms in α (f i ). If in α (f 0 )(b) = 0, take J as the subset of [n]

3 Example 2 . 13 (

 3213 Generic coefficients). Consider the parametrization f : (C * ) 2 (C * ) 3 defined by the polynomials

  and we define d I as the mixed volume d I = MV(A i1,I , . . . , A i d ,I ).

1 ,Remark 3 . 4 .

 134 which is a contradiction. So the number of common zeros in U × (K * ) n+1-d is the mixed volume of their supports, MV n+1 (A n+1-d 1 , . . . , A n+1-d d , (A d+1 , 0) ∪ {e d+1 }, . . . , (A n , 0) ∪ {e n }, (A 0 , 0) ∪ {e n+1 }), where 0 ∈ Z n+1-d as in Definition 3.1. Note that replacing (x d+1 , . . . , x n+1 ) by a generic linear combination, we would get the same number of solutions in the torus, and thus, this mixed volume coincides with the mixed volume MV(A n+1-d 0 , . . . , A n+1-d n ), as stated in(3.5), and the first inequality follows.To prove that MV(A n+1-d 0, . . . , A n+1-d n ) ≤ min |I|=d d I , observe that MV(A n+1-d 0 , . . . , A n+1-d n) is the number of solutions of a generic system of equations (3.8), which coincides with deg(S) deg(f ). As proved using system (3.7), we have deg(S) deg(f ) ≤ d I for all I ⊂ [n] with |I| = d, we deduce the second inequality in (3.5), which concludes the proof.We deduce from the proof of Theorem 3.3 that the generic value MV(A n+1-d 0, . . . , A n+1-d n ) of deg(S) • deg(f ) equals MV n+1 A n+1-d . . . , A n+1-d d , (A d+1 , 0) ∪ {e d+1 }, . . . , (A n , 0) ∪ {e n }, (A 0 , 0) ∪ {e n+1 } .Moreover, we could replace the choice of indices [d] by any subset I of [n] 0 of cardinal d. Note that the inequalities involving mixed volumes in(3.5) in the statement of Theorem 3.3 that we proved for lattice polytopes, are also valid for arbitrary convex sets. All the terms are homogeneous of degree d, in the sense that if we multiply all the supports A i by a factor of k, the (mixed) volumes are multiplied by k d , including MV(A n+1-d 0 , . . . , A n+1-d n

Example 3 . 5 . 3 . 1 .

 3531 Consider the mapping f given by the generic polynomials from Example 2.13. Theorem 3.3 proves the known fact that deg(S) = 9. Moreover, in this case the first inequality in (3.5) deg(S) • deg(f ) = MV A 2 0 , . . . , A 2 3 ≤ min |I|=2 d I in the statement is actually an equality. In fact, we can consider the steps in the proof of Theorem 3.3 to verify that deg(S) • deg(f ) = d {1,2} . We see below that for general supports the equalities in (3.5) can be strict. Sufficient conditions for equality. With the notations from Theorem 3.3, if A 0 = A 1 = • • • = A n = A and f 0 , . . . , f n are generic, the inequalities in (3.5) are all equalities and deg(S)•deg(f ) equals vol(conv(A))

( 1 . 2 )

 12 is generically finite and parameterizes a hypersurface S in K n . Then,ord 0 (S) • deg(f ) = MV(A 2 0 , . . . , A 2 n ) -MV(A 1 1 , . . . A 1 n ) ≥ MV(A 2 0 , . . . , A 2 n ) -min j∈[n] {MV({A i ∪ A j } i∈[n] i =j )}. Proof. Consider i = a i,1 y 1 + • • • + a i,n y n generic linear forms with coefficients in K, and generic constants a i,0 ∈ K * for all i ∈ [n -1]. As before, consider the open set U = {x ∈ (L * ) n-1 |

(4. 2 )

 2 ord 0 (S) = D -#{k | val(λ k ) = 0}.We focus in what follows on bounding the number of points {ω k } D k=1 for which val(λ k ) = 0. Let ω k = f1(σ k ) f0(σ k ) , . . . , fn(σ k ) f0(σ k )be one of them. Since m = val(f i (σ k )) is fixed for all i ∈ [n] 0 and the supports are tame, let us prove that this implies that ifσ k = b k ε α + h.o.t.(ε) with b k ∈ (K * ) n, then necessarily α = 0. Assume first that α = 0 and recall the notation in (2.5): m α (A j ) = min p∈Aj α, p . Since in α (f 1 ), . . . , in α (f n ) are generic, they do not all vanish on b k . Also, m = m α (A j ) if and only if in α (f j )(b k ) = 0 for all j ∈ [n] 0 . However, since the supports are tame, {in α (f j )} j∈Jα can only have common zeros with all non-zero coordinates when J α = ∅, and so inα (f j )(b k ) = 0 for all j ∈ [n] 0 . As a i,1 f 1 (σ k ) + • • • + a i,n f n (σ k ) -a i,0 f 0 (σ k )ε = 0 for all i ∈ [n -1], b k ∈ (K * ) n-1is a common zero of the polynomials

  we denote by MV(Q 1 , . . . , Q d ) their d-dimensional mixed volume , . . . , B d are finite sets, the mixed volume MV(B 1 , . . . , B d ) of B 1 , . . . , B d is defined as the mixed volume of their convex hulls conv(B 1 ), . . . , conv(B d ). Recall that, by Bernstein's Theorem[START_REF] Bernstein | The number of roots of a system of equations[END_REF], this is an upper bound for the number of isolated common zeros in (L * ) d of any system of polynomials with support set B 1 , . . . , B d . Moreover, equality holds for generic polynomials with those supports.We denote by [n] 0 the set [n] 0 = {0, . . . , n} and by [n] the set [n] = {1, . . . , n}.

	(2.7)	MV(Q 1 , . . . , Q d ) =	(-1) n-#I vol e (	Q j ).
		I⊂{1,...,d}	j∈I
	If B 1

  5. An example. We apply Theorem 2.8 to an example with generic coefficients (Example 2.13 below) and we then consider in Example 2.14 polynomials with the same supports but non-generic coefficients. These examples are addressed in terms of geometric tropicalization in Examples 4.3 and 5.3 in [4].

  where the coefficients µ I ij ∈ K are generic. These polynomials h I j have supports A ij ,I for any j ∈[d]. By the BKK bound, the system (3.7) has at most d I isolated common zeros in (K * ) d and by the monotony of the mixed volume,d I ≤ MV(∪ n j=0 A j , . . . , ∪ n j=0 A j )) = vol(conv(∪ n j=0 A j )). But still, depending on the original supports A 0 , . . . , A n , h I 1 , . . . , h I d need not have generic coefficients with those supports, even if f i are generic for their support A i . Take I = [d] and consider the following polynomials in K[x 1 , . . . , x n+1 ]:

Example 3.6. Let d = 2, n + 1 = 4, so S is a hypersurface in dimension 3. Consider the generic polynomials in K[x 1 , x 2 ]:

1 + c 2 x 1 x 2 and f 3 (x 1 , x 2 ) = a 3 x 1 x 2 + b 3 x 3 1 x 2 + c 3 x 1 x 3 2 . Let A 0 , A 1 , A 2 , A 3 be their supports sets. We draw their respective convex hulls P i , i ∈ [START_REF] Bieri | The geometry of the set of characters induced by valuations[END_REF]

The closure S ⊂ K 3 of the image of the associated rational map f = f 1 f 0 , f 2 f 0 , f 3 f 0 is a hypersurface. Let H be a reduced polynomial defining S. We observe that the degree of the map f is 1, and we can compute deg(S) = deg(H) = MV(A 2 0 , . . . , A 2 3 ) = 5, vol(conv(∪ 3 j=0 A j )) = 11 and 5

In fact, there exists a unique monomial in H of maximal degree 5. In what follows we present conditions on the supports A 0 , . . . , A n for which some of the inequalities in the statement of Theorem 3.3 are actually equalities in case d = n -1. We need some preliminary results, which include equivalences due to Bihan and Soprunov (see [START_REF] Bihan | Criteria for Strict Monotonicity of the Mixed Volume of Convex Polytopes[END_REF]), and the notion of an essential collection of subsets in Definition 2.6, which was originally introduced in [START_REF] Sturmfels | The Newton polytope of the resultant[END_REF]. Conditions for strict monotonicity were previously determined in [START_REF] Rojas | A convex geometric approach to counting the roots of a polynomial system[END_REF], but the approach in [START_REF] Bihan | Criteria for Strict Monotonicity of the Mixed Volume of Convex Polytopes[END_REF] is different (see Example 3.9).

Then, the following statements are equivalent:

.

(ii) There exists a coherent collection of proper faces F = (F 1 , . . . , F n-1 ) of the collection B 1 , . . . , B n-1 such that the collection B 1,F , . . . , B n-1,F is essential, where

(iii) There exists a coherent collection of proper faces F = (F 1 , . . . , F n-1 ) of the collection B 1 , . . . , B n-1 such that the collection ) if and only if there exists a coherent collection of proper faces F = (F 1 , . . . , F n-1 ) of (B 1 , . . . , B n-1 ) such that the polynomials { α∈Bi∩Fi c iα x α } n-1 i=1 have a common zero with all non-zero coordinates. But, since g 1 , . . . , g n-1 are generic, this is equivalent to the set

Then, the following statements are equivalent:

The convex hull of the support of any linear combination of in α (f 1 (x)), . . . , in α (f n (x)) has dimension at most n -2 because all supports are contained in the hyperplane with equation α, p = m. Then, the variety W α ⊂ (K * ) n parameterized for all x ∈ U by (in α (f 1 (x)), . . . , in α (f n (x)) has dimension at most n -2. The space of lines from a point in W α through the origin has then codimension at least one in the space of lines in K n through the origin. Thus, the intersection of W α with a generic line 1 = 0, . . . , n-1 = 0 is empty. Therefore, α = 0.

Consider then

. The polynomials

have finitely many common zeros in (K * ) n-1 , b k being one of those zeros. As in the proof of Theorem 3.3, by taking linear combinations, we can tranform them into polynomials of the form f 1 + µ 1 f n , . . . , f n-1 + µ n-1 f n with generic coefficients µ 1 , . . . , µ n-1 , and thus its number of common zeros is bounded above by min j∈[n] {MV({A i ∪ A j } i =j,0 )}. Moreover, one can take an extra variable x n and define an associated generic system h 1 = • • • = h n = 0 where

By the genericity of their coefficients, f 1 , . . . , f n do not have common zeros, and hence common zeros of g 1 , . . . , g n-1 in (K * ) n-1 correspond to common zeros of the polynomials h 1 , . . . , h n in (K * ) n . By Bernstein's Theorem, the number of common roots of g 1 , . . . g n-1 equals MV({A 1 i }} n i=1 ). Since U is smooth, [10, Corollary 6.7.2] ensures that the Jacobian matrix of g 1 , . . . , g n-1 has nonzero determinant when evaluated at a common zero b ∈ U ⊂ (K * ) n-1 . This implies that the Jacobian matrix of i (f 1 (x), . . . , f n (x)) -a i,0 f 0 (x)ε has valuation zero when evaluated at the common roots b ∈ (K * ) n-1 of g 1 , . . . , g n-1 and, by Hensel's Lemma, each of these common zeros b can be lifted to a unique common zero σ of a i,1

) and the result follows from (4.2). Remark 4.3. As in Remark 3.4, the inequality about mixed volumes in the statement of Theorem 4.2 that we proved for lattice polytopes, also holds for arbitrary convex sets.

The tameness condition of Theorem 4.2 is sufficient but not necessary as shown by the following examples. Indeed, we expect that the equality in the statement of Theorem 4.2 holds in general even if the supports are not tame.

with the following supports:

The implicit equation of the parametric surface S: is of degree 3 and order ord 0 (H) = 1. We have deg

3 ) = 1. However, the family is not tame since for α = (1, 0), J α = {0} and the family {face α (A 0 )} is essential.

Example 4.5. Consider the family of lattice sets

It is easy to check that this family of supports is not tame: for α = (-1, 1, 1), the set J α from Definition 4.1 is J α = {3, 4}. However, the family {face α (A 3 ), face α (A 4 )} is essential. Even in this case, for generic f 0 , f 1 , f 2 , f 3 , f 4 with those supports, we still have (as in the statement of Theorem 4.2) that ord 0 (S)

{MV({A i ∪ A j } i∈ [START_REF] Cueto | Implicitization of Surfaces via Geometric Tropicalization[END_REF] i =j

)}.

Moreover, the inequality above is an equality. It can be computed that for generic polynomials, deg(f ) = 1, deg(S) = 9, ord 0 (S) = 4, and MV({A 1 i } 4 i=1 ) = min i∈ [START_REF] Cueto | Implicitization of Surfaces via Geometric Tropicalization[END_REF] {MV({A j ∪ A i } j =i )} = 5. Example 4.6. Consider again Example 2.14 with non-generic coefficients. As in Example 4.8, by applying Bernstein's Theorem to the polynomial systems in the proof of Theorem 3.3, we can see that all inequalities in Theorem 3.3 are actually equalities and the degree of the surface S is 9 = vol(conv(∪ 3 j=0 A j )). The polytope conv(∪ 3 j=1 A j ) with vertices (2, 0), (0, 2), (3, 0), (0, 3) ∈ (Z ≥0 ) 2 has volume vol(conv(∪ 3 j=1 A j )) = 5. Theorem 4.2 predicts the lower bound 4 for generic systems with those supports, which is also true in this case. To see that, we can follow the steps in its proof, and first verify that there are no σ k = b k ε α + h.o.t.(ε) ∈ C{{ε R }} with α = 0 and val(λ k ) = 0. Then, we can check that the polynomials i (f 1 , f 2 , f 3 ) for i ∈ {1, 2} have exactly 5 common zeros in (K * ) 2 by means of the genericity conditions in Bernstein's Theorem. Finally, applying [START_REF] Jouanolou | Théorèmes de Bertini et applications[END_REF]Corollary 6.7.2] and Hensel's Lemma, the order at the origin is exactly 9 -5 = 4.

The following proposition presents a sufficient condition for the inequality in the statement Theorem 4.2 to be an equality. This allows us to compute the order in an easier way. Proposition 4.7. Under the hypotheses and notation of Theorem 4.2, if there exists j 0 ∈ [n] such that for all coherent collection of proper faces F of (A i ∪ A j0 ) i =j0 , there exists a nonempty subset

Proof. Denote MV j0 = MV(A 1 , . . . , A j0-1 , A j0+1 , . . . , A n ). As the polynomials f 0 , . . . , f n are generic with respect to their supports, we have by Theorem 3.3 that deg(S)

Because the supports are tame, we saw in the proof of Theorem 4.2 that ord 0 (S)

). Now, as in the proof of Theorem 3.11, we introduce a new variable t and consider the polynomials:

For almost all t, their number of common zeros in (K * ) n-1 is N . As f 1 , . . . , f n are generic polynomials, they have M V (A 1 , . . . , A j0-1 , A j0+1 , . . . , A n ) common zeros with all non-zero coordinates. Since these zeros are isolated, each is in a germ of curve of zeros of (4.5) not contained in {t = 0}. This implies that M V (A 1 , . . . , A j0-1 , A j0+1 , . . . , A n ) ≤ N . Therefore, MV j0 ≤ N ≤ MV(A 1 ∪ A j0 , . . . , A j0-1 ∪ A j0 , A j0+1 ∪ A j0 , . . . , A n ∪ A j0 ).

But now, it follows from Lemma 3.7 that MV(A 1 , . . . , A j0-1 , A j0+1 , . . . , A n ) = MV j0 , and hence ord 0 (S) deg(f ) = M V (A 2 0 , . . . , A 2 n ) -M V j0 , as claimed. The remaining equality in (4.4) follows from the computation of ord 0 (S) in the statement of Theorem 4.2.

Example 4.8. Consider again the mapping f given by the generic polynomials from Example 2.13. Since f 0 , f 1 , f 2 , f 3 are generic, the conditions in Proposition 4.7 are realized for any j 0 ∈ [3] and we deduce that ord 0 (S) = 4 because deg(f ) = 1.