An arithmetic Bernštein–Kušnirenko inequality

We present an upper bound for the height of the isolated zeros in the torus of a system of Laurent polynomials over an adelic field satisfying the product formula. This upper bound is expressed in terms of the mixed integrals of the local roof functions associated to the chosen height function and to the system of Laurent polynomials. We also show that this bound is close to optimal in some families of examples. This result is an arithmetic analogue of the classical Bernštein–Kušnirenko theorem. Its proof is based on arithmetic intersection theory on toric varieties.


Introduction
The classical Bernštein-Kušnirenko theorem bounds the number of isolated zeros of a system of Laurent polynomials over a field, in terms of the mixed volume of their Newton polytopes Martínez and Sombra were partially supported by the MINECO research projects MTM2012-38122-C03-02 and MTM2015-65361-P. Martínez was also partially supported by the CNRS project PICS 6381 "Géométrie diophantienne et calcul formel". Martinez was also partially supported by the Deutsche Forschungsgemeinschaft collaborative research center SFB 1085 "Higher Invariants".. [2,17]. This result, initiated by Kušnirenko and put into final form by Bernštein, is also known as the BKK theorem to acknowledge Khovanskiȋ's contributions to this subject. It shows how a geometric problem (the counting of the number of solutions of a system of equations) can be translated into a combinatorial, simpler one. It is commonly used to predict when a given system of equations has a small number of solutions. As such, it is a cornerstone of polynomial equation solving and has motivated a large amount of work and results over the past 25 years, see for instance [14,21,24] and the references therein.
When dealing with Laurent polynomials over a field with an arithmetic structure like the field of rationals, it might be also important to control the arithmetic complexity or height of their zero set. In this paper, we present an arithmetic version of the BKK theorem, bounding the height of the isolated zeros of a system of Laurent polynomials over such a field. It is a refinement of the arithmetic Bézout theorem that takes into account the finer monomial structure of the system.
Previous results in this direction were obtained by Maillot [20] and by the second author [23]. Our current result improves these previous upper bounds, and generalizes them to adelic fields satisfying the product formula, and to height functions associated to arbitrary nef toric metrized divisors.
Let K be a field and K its algebraic closure. Let M Z n be a lattice and set which is an equality when the f i 's are generic with respect to their Newton polytopes [2,17], see also Theorem 2.9. Now suppose that K is endowed with a set of places M, so that the pair (K, M) is an adelic field (Definition 3.1). Each place v ∈ M consists of an absolute value | · | v on K and a weight n v > 0. We assume that this set of places satisfies the product formula, namely, for all α ∈ K × , v∈M n v log |α| v = 0.
The classical examples of adelic fields satisfying the product formula are the global fields, that is, number fields and function fields of regular projective curves.
Let X be toric compactification of T M and D 0 a nef toric metrized divisor on X , see Sects. 4 and 5 for details. This data gives a notion of height for 0-cycles of X , see [3,Chapter 2] or Sect. 4. The height is a nonnegative real number, and it is our aim to bound this quantity in terms of the monomial expansion of the f i 's.
The toric Cartier divisor D 0 defines a polytope 0 ⊂ M R . Following [8], we associate to D 0 an adelic family of continuous concave functions ϑ 0,v : 0 → R, v ∈ M, called the local roof functions of D 0 .
For i = 1, . . . , n, write R n be the dual space and, for each place v ∈ M, consider the concave function ψ i,v : N R → R defined by u if v is non-Archimedean. (1. 2) The Legendre-Fenchel dual ϑ i,v = ψ ∨ i,v is a continuous concave function on i . We denote by MI M the mixed integral of a family of n + 1 concave functions on convex bodies of M R (Definition 5.6). It is the polarization of (n + 1)! times the integral of a concave function on a convex body. It is a functional that is symmetric and linear in each variable with respect to the sup-convolution of concave functions, see [21,Sect. 8] for details.
The following is the main result of this paper. Using the basic properties of the mixed integral, we can bound the terms in the right-hand side of (1.3) in terms of mixed volumes. From this, we can derive the bound (Corollary 6.8) where ( f i ) denotes the (logarithmic) length of f i (Definition 6.6). This bound might be compared with the one given by the arithmetic Bézout theorem (Corollary 6.9).
The following example illustrates a typical application of these results. It concerns two height functions applied to the same 0-cycle. Our upper bounds are close to optimal for both of them and, in particular, they reflect their very different behavior on this family of Laurent polynomials. Example 1.2 Take two integers d, α ≥ 1 and consider the system of Laurent polynomials The 0-cycle Y := Z ( f 1 , . . . , f n ) of G n m,Q is the single point (α, α d+1 , . . . , α d n−1 +···+d+1 ) with multiplicity one.
Let P n Q be the n-dimensional projective space over Q and E can the divisor of the hyperplane at infinity, equipped with the canonical metric. Its associated height function is the Weil height. We consider two toric compactifications X 1 and X 2 of G n m . These are given by compactifying the torus via the equivariant embeddings ι i : G n m → P n Q , i = 1, 2, respectively defined, for p = ( p 1 , . . . , p n ) ∈ G n m (Q) = (Q × ) n , by ι 1 ( p) = (1 : p 1 : · · · : p n ) and ι 2 ( p) = (1 : p 1 : p 2 p −d 1 : · · · : p n p −d n−1 ).
, which are nef toric metrized divisors on X i . By an explicit computation, we show that On the other hand, the upper bounds given by Theorem 1.1 are To the best of our knowledge, the first arithmetic analogue of the BKK theorem was proposed by Maillot [20,Corollaire 8.2.3], who considered the case of canonical toric metrics. Another result in this direction was obtained by the second author for the unmixed case and also canonical toric metrics [23, Théorème 0.3]. Theorem 1.1 improves and refines these previous upper bounds, and generalizes them to adelic fields satisfying the product formula and to height functions associated to arbitrary nef toric metrized divisors, see Sect. 7 for details.
The key point in the proof of Theorem 1.1 consists of the construction, for each Laurent polynomial f i , of a nef toric metrized divisor D i on a proper toric variety X , such that f i corresponds to a small section of D i (Proposition 6.2 and Lemma 6.4). The proof then proceeds by applying the constructions and results of [4,8] and basic results from arithmetic intersection theory.
Trying to keep our results at a similar level of generality as those in [8], we faced difficulties to define and study global heights of cycles over adelic fields. This lead us to a more detailed study of these notions. In particular, we give a new notion of adelic field extension that preserves the product formula (Proposition 3.7) and a well-defined notion of global height for cycles with respect to metrized divisors that are generated by small sections (Proposition-Definition 4.15).
As an application of Theorem 1.1, we give an upper bound for the size of the coefficients of the u-resultant of the direct image under a monomial map of the solution set of a system of Laurent polynomial equations.
, m 0 ∈ (Z n ) r +1 and α 0 ∈ (Z\{0}) r +1 with r ≥ 0. Set 0 = conv(m 0,0 , . . . , m 0,r ) ⊂ R n and let ϕ be the monomial map associated to m 0 and α 0 as in (1.4). For i = 1, . . . , n, let i ⊂ R n be the Newton polytope of f i , and α i the vector of nonzero coefficients of f i . Then The paper is organized as follows. In Sect. 2 we recall some preliminary material on intersection theory and on the algebraic geometry of toric varieties. In Sect. 3 we study adelic fields satisfying the product formula. In Sect. 4 we recall the notions of metrized divisors and its associated measures and heights, with an emphasis on the 0-dimensional case. In Sect. 5 we explain the notation and basic constructions of the arithmetic geometry of toric varieties. In Sect. 6 we prove Theorem 1.1, whereas in Sect. 7 we give examples illustrating the applications of our bounds, and prove Theorem 1.3.

Intersection theory and toric varieties
In this section, we recall the proof of the Bernštein-Kušnirenko theorem using intersection theory on toric varieties, which is the model that we follow in our treatment of the arithmetic version of this result. Presenting this proof also allows us to introduce the basic definitions and results on the intersection of Cartier divisors with cycles, and on the algebraic geometry of toric varieties. For more details on these subjects, we refer to [12,19] and to [13].
Let K be an infinite field and X a variety over K of dimension n. For 0 ≤ k ≤ n, the group of k-cycles, denoted by Z k (X ), is the free abelian group on the k-dimensional irreducible subvarieties of X . Thus, a k-cycle is a finite formal sum where the V 's are k-dimensional irreducible subvarieties of X and the m V 's are integers. The support of Y , denoted by |Y |, is the union of the subvarieties V such that The k-cycle associated to Z is then defined as the sum being over the irreducible components of Z . Let V be an irreducible subvariety of X of codimension one and f a regular function on an open subset U of X such that U ∩ V = ∅. The order of vanishing of f along V is defined as For a Cartier divisor D on X , the order of vanishing of D along V is defined as This definition does not depend on the choice of U , g and h. Moreover, ord V (D) = 0 for all but a finite number of V 's. The support of D, denoted by |D|, is the union of these subvarieties V such that ord V (D) = 0. The Weil divisor associated to D is then defined as the sum being over the irreducible components of |D|. Now let W be an irreducible subvariety of X of dimension k. If W ⊂ |D|, then D restricts to a Cartier divisor on W . In this case, we define D · W as the Weil divisor of W obtained by restricting (2.1) to W . This gives a (k − 1)-cycle of X . If W ⊂ |D|, then we set D · W = 0, the zero element of Z k−1 (X ). We extend by linearity this intersection product to a morphism with the convention that Z −1 (X ) = 0, the zero group.
For 0 ≤ r ≤ n and Cartier divisors D i on X , i = 1, . . . , r , we define inductively the intersection product If D 1 , . . . , D r intersect X properly, then the cycle r i=1 D i does not depend on the order of the D i 's [12,Corollary 2.4.2]. This conclusion does not necessarily hold if these divisors do not intersect properly.

Example 2.2 Let X = A 2
K and consider the principal Cartier divisors D 1 = div(x 1 x 2 ) and D 2 = div(x 1 ). Then where the sum is over the closed points p of X and m p ∈ Z. This 0-cycle is effective and, for each isolated closed point p of the intersection n i=1 |div(s i )|, where U is a trivializing neighborhood of p, and f i is a defining function for s i on U , i = 1, . . . , n.
Proof The fact that the cycle in (2.2) is effective follows from the hypothesis that the s i 's are global sections. For the second statement, by possibly replacing U with a smaller open neighborhood of p, we can assume that div(s 1 ), . . . , div(s n ) intersect X properly on U , and so this intersection is of dimension 0. By [12, Proposition 7.1 and Example 7.1.10], By [12, Example A.1.1], we have the equality completing the proof.
For the rest of this section, we assume that the variety X is projective. With this hypothesis, Chow's moving lemma allows to construct, given a cycle and a family of Cartier divisors, another family of linearly equivalent Cartier divisors intersecting the given cycle properly, in the sense of Definition 2.1.

Definition 2.4
Let Y be a k-cycle of X and D 1 , . . . , D k Cartier divisors on X . The degree of Y with respect to D 1 , . . . , D k , denoted by deg D 1 ,...,D k (Y ), is inductively defined by the rules: The degree of a cycle with respect to a family of Cartier divisors does not depend on the choice of the rational section s k in (2), see for instance [12,Sect. 2.5] Proof Since Y is effective and s k is a global section, div(s k ) · Y is also effective. Since For the upper bound, we reduce without loss of generality to the case when Y = V is an irreducible subvariety of dimension k. If V ⊂ |div(s k )|, then div(s k ) · Y = 0 ∈ Z k−1 (X ). Hence deg(div(s k ) · Y ) = 0 and the bound follows from the nefness of the D i 's. Otherwise, from the definition of the degree, which completes the proof.
We now turn to toric varieties. Let M Z n be a lattice and set for its group K -algebra and algebraic torus over K , respectively. The elements of M correspond to the characters of T and, given m ∈ M, we denote by χ m ∈ Hom(T, G m,K ) the corresponding character. Set also M R = M ⊗ R. Let N = M ∨ Z n be the dual lattice and set N R = N ⊗ R. Given a complete fan in N R , we denote by X the associated toric variety with torus T. It is a proper normal variety over K containing T as a dense open subset. When the fan is regular, in the sense that it is induced by a piecewise linear concave function on N R , the toric variety X is projective.
Set X = X for short. Let D be a toric Cartier divisor on X , and denote by D its associated virtual support function on . This is a piecewise linear function D : N R → R satisfying that, for each cone σ ∈ , there exists m ∈ M such that, for all u ∈ σ , The condition that D is concave is both equivalent to the conditions that D is nef and that Now let 1 , . . . , r be lattice polytopes in M R . For each i , we consider its support function, which is the piecewise linear concave function with lattice slopes i : N R → R given by Let be a regular complete fan in N R compatible with the collection 1 , . . . , r , in the sense that the i 's are virtual support functions on . Such a fan can be constructed by taking any regular complete fan in N R refining the complex of cones that are normal to the faces of i , for all i. Let X be the toric variety corresponding to this fan and D i the toric Cartier divisor on X corresponding to these virtual support functions. By construction, i is concave. Hence D i is nef and O(D i ) is globally generated, and its associated polytope coincides with i . Let vol M be the Haar measure on M R such that M has covolume 1, and take r = n. The mixed volume of 1 , . . . , n is defined as the alternating sum A fundamental result in toric geometry states that the degree of a toric variety with respect to a family of nef toric Cartier divisors is given by the mixed volume of its polytopes [13,Sect. 5.4]. In our present setting, this amounts to the formula (2.7) We turn to 0-cycles of the torus defined by families of Laurent polynomials.
The 0-cycle associated to f 1 , . . . , f n is defined as be a Laurent polynomial. Its support is defined as the finite subset of M of the exponents of its nonzero terms, that is supp( f ) = {m | α m = 0}. The Newton polytope of f is the lattice polytope in M R given by the convex hull of its support, that is N( f ) = conv(supp( f )).
Let be a regular complete fan in N R compatible with the Newton polytopes of the f i 's and, for i = 1, . . . , n, let D i be the Cartier divisor on X associated to N( f i ) and s i the global section of O(D i ) corresponding to f i as in (2.5).
where the sum is over the closed points of X . Then Since T is Cohen-Macaulay, Proposition 2.3 gives the first statement. Since the sections s i are global, the 0-cycle n i=1 div(s i ) is effective. Hence, the second statement follows directly from the first one.
Finally, we prove the version of the Bernštein-Kušnirenko theorem in (1.1).

Adelic fields and finite extensions
In this section, we consider adelic fields following [8]. We also give a new notion of adelic field extension that behaves better than the one in loc. cit.. With this definition, the product formula is preserved when passing to finite extensions. Definition 3.1 Let K be an infinite field and M a set of places. Each place v ∈ M is a pair consisting of an absolute value | · | v and a positive real weight n v . We say that (K, M) is an adelic field if (1) for each v ∈ M, the absolute value |·| v is either Archimedean or associated to a nontrivial discrete valuation; (2) for each α ∈ K × , we have that |α| v = 1 for all but a finite number of v ∈ M.
Let (K, M) be an adelic field. For each place v ∈ M, we denote by K v the completion of K with respect to the absolute value | · | v . By a theorem of Ostrowski, if v is Archimedean, then K v is isomorphic to either R or C [10, Chapter 3, Theorem 1.1]. In particular, an adelic field has only a finite number of Archimedean places.  To each closed point v ∈ C we associate the absolute value and weight given, where ord v ( f ) denotes the order of vanishing of f at v and The set of places M K(C) is indexed by the closed points of C, and consists of these absolute values and weights. The pair (K(C), M K(C) ) is an adelic field which satisfies the product formula.

Lemma 3.4 Let F be a finite extension of K and v ∈ M. Then
where the sum is over the absolute values | · | w on F whose restriction to K v coincides with | · | v , and where the E w 's are local Artinian K v -algebras with maximal ideal p w . For each w, we have E w /p w F w .
Proof Since K → F is a finite extension, the tensor product F⊗K v is an Artinian K v -algebra. By the structure theorem for Artinian algebras, where I is a finite set and the E i 's are local Artinian K v -algebras. Let p i be the maximal ideal of E i , for each i. These are the only prime ideals of F ⊗ K v , and so rad(F ⊗ K v ) = i∈I p i . Each w in the decomposition (3.3) corresponds to an absolute value | · | w on F extending |·| v , and there is a natural inclusion F → F w . The diagonal morphism F → w F w extends to a map of K v -vector spaces By [5, Chapitre VI, Sect. 8.2 Proposition 11(b)], this morphism is surjective and its kernel is The summands in both extremes of (3.4) are fields over K v , and so local Artinian K v -algebras. By the uniqueness of the decomposition in the structure theorem for Artinian algebras, there is a bijection between the elements in I and the w's, identifying each i ∈ I with the unique The following definition was introduced by Gubler in the context of M-fields, see [15, Remark 2.5].
Definition 3.5 Let (K, M) be an adelic field and F a finite extension of K. For every place v ∈ M, we denote by N v the set of absolute values | · | w on F that extend | · | v with weight given by where the E w 's are the local Artinian K v -algebras in the decomposition of is an adelic field. The adelic fields of this form are called adelic field extensions of (K, M).
Remark 3. 6 With notation as in Lemma 3.4, where l E w (E w ) is the length of E w as a module over itself. This follows from [12, Hence, the weights in Definition 3.5 can be alternatively written as Proof From the definition of adelic field extension and Lemma 3.4, which proves statement (1). To prove the second statement, let α ∈ F × and consider the multiplication map η α : F → F given by η α (x) = αx. The norm N F/K (α) ∈ K × is defined as the determinant of this K-linear map. Moreover, η α extends to the K v -linear map which has the same determinant. Using the decomposition in . Moreover, by [18,VI Proposition 5.6], where the product is over the different embeddings σ of F w in an algebraic closure of K v , and [F w : concluding the proof.

Example 3.8
Let F be a number field. This is a separable extension of Q. By [5, Chapitre VI, Sect. 8.5, Example 3.9 Let (K(C), M K(C) ) be the function field of a regular projective curve C over a field κ with the structure of adelic field as in Example 3.3. The places of K(C) correspond to the closed points of C with absolute values and weights given by (3.1). Let F be a finite extension of K(C) and N the set of places of F as in Definition 3.5. There is a regular projective curve B over κ and a finite map π : B → C such that the extension K(C) → F identifies with the morphism π * : With notation as in Lemma 3.4, by [5, Chapter VI, Sect. 8.5, Corollaire 3], we have E w F w for all w ∈ N v . Hence, the weight of w is given by Let e(w/v) denote the ramification index of w over v. By [5, Chapter VI, Sect. 8.5, . Therefore, for each place w ∈ N v , the weight of w can also be expressed as Following [8], a global field is a finite extension of the field of rational numbers or of the function field of a regular projective curve, with the structure of adelic field described in Examples 3.8 and 3.9. For these fields, Proposition 3.7 is already a known result, see for instance [7,Proposition 2.1].
By a result of Artin and Whaples, global fields can be characterized as the adelic fields having an absolute value that is either Archimedean or associated to a discrete valuation whose residue field has finite order over the field of constants [1, Theorems 2 and 3].
Function fields of varieties of higher dimension provide examples of adelic fields satisfying the product formula, and that are not global fields.

Example 3.10
Let K(S) be the function field of an irreducible normal variety S over a field κ of dimension s ≥ 1, and E 1 , . . . , E s−1 nef Cartier divisors on S. Set S (1) for the set of irreducible hypersurfaces of S. For each V ∈ S (1) , the local ring O V ,S is a discrete valuation ring. We associate to V the absolute value and weight given, for f ∈ K(S), by with c κ as in (3.2). The set of places M K(S) is indexed by S (1) , and consists of these absolute values and weights. For because the Cartier divisor div( f ) is principal. Hence (K(S), M K(S) ) satisfies the product formula.

Height of cycles
In this section, we introduce a notion of global height for cycles of a variety over an adelic field, with respect to a family of metrized divisors generated by small sections. We also recall the notion of local height of cycles from [8, Chapter 1] and give a more explicit description of this construction in the 0-dimensional case.
Let (K, M) be an adelic field satisfying the product formula, and X a normal projective variety over K. For each place v ∈ M, we denote by X an v the v-adic analytification of X . In the Archimedean case, if K v C, then X an v is an analytic space over C whereas, if K v R, then X an v is an analytic space over R, that is, an analytic space over C together with an antilinear involution, as explained in [8, Remark 1.1.5]. In the non-Archimedean case, X an v is a Berkovich space over K v as in [8,Sect. 1.2].
Fix v ∈ M and set Given a 0-cycle Y of X v , a usual construction in Arakelov geometry associates a signed measure on X an v , denoted by δ Y , that is supported on |Y | an and has total mass equal to deg(Y ), see for instance [8,Definition 1.3.15] for the non-Archimedean case. In what follows, we explicit this construction. Let q be a closed point of X v . The function field K(q) is a finite extension of K v and deg(q) = [K(q) : If v is Archimedean, then deg(q) is either equal to one or two. In the first case, the analytification of q is a point of X an v whereas, in the second case, it is a pair of conjugate points. If v is non-Archimedean, choose an affine open neighborhood U = Spec(A) of q and A → K(q) the corresponding morphism of K v -algebras. The analytification of q is the point q an ∈ U an ⊂ X an v corresponding to the multiplicative seminorm given by the composition where | · | is the unique extension to K(q) of the absolute value | · | v .
Since the measure δ q is supported on the point q an and has total mass deg(q), it follows that where δ q an denotes the Dirac delta measure on q an . For an arbitrary 0-cycle Y of X v , the signed measure δ Y is obtained from (4.1) by linearity. It is a discrete signed measure of total mass equal to deg(Y ).
Let D be a Cartier divisor on X . A metric on the analytic line bundle O(D) an v is an assignment that, to each open subset U ⊂ X an v and local section s on U , associates a continuous function We now study the behavior of these objects with respect to adelic field extensions. Let (F, N) be an extension of the adelic field (K, M) (Definition 3.5) and fix a place w ∈ N v , so that F w is a finite extension of the local field K v . Let q be a closed point of X v and consider the subscheme q w of X w = X × Spec(F w ) obtained by base change. Decompose as a finite sum of local Artinian F w -algebras and, for each j ∈ I , denote by q j the corresponding closed point of X w . Thus the associated cycle is given by [q w ] = j∈I l G j (G j ) q j . Hence, by (4.1) and Remark 3.6, The inclusion K v → F w induces a map of the corresponding analytic spaces In the non-Archimedean case, this map of Berkovich spaces is defined locally by restricting seminorms.
The following proposition gives the behavior of the measure associated to a 0-cycle with respect to field extensions. Proposition 4.2 With notation as above, let Y be a 0-cycle of X v and set Y w for the 0-cycle of X w obtained by base change. Then Proof By the compatibility of the map π with restriction to subschemes, we have that π(q an j ) = q an for all j ∈ I . It follows that Let D be a Cartier divisor on X and · v a metric on O(D) an v . The extension of this metric to a metric · w on the analytic line bundle O(D) an w on X an w is obtained by taking the inverse image with respect to the map π in (4.2), that is To define global heights of cycles over an adelic field, we consider adelic families of metrics on the Cartier divisor D satisfying a certain compatibility condition.  By Proposition 1.5.14 in loc. cit., a quasi-algebraic metrized divisor D is adelic in the sense of Definition 4.4. The converse is not true, as it is easy to construct toric adelic metrized divisors that are not quasi-algebraic (Remark 5.4).
For a 0-cycle Y of X and a place v ∈ M, we denote by Y v the 0-cycle of X v defined by base change. When Y = p is a closed point of X , by Lemma 3.4 applied to the finite extension K( p) of K, the 0-dimensional subscheme p v = p × Spec(K v ) of X v decomposes as where the E i 's are the local Artinian K v -algebras in (3.3). Let q w , w ∈ N v , be the irreducible components of this subscheme. Then, the associated 0-cycle of X v writes down as For an arbitrary Y , the 0-cycle Y v is obtained by linearity.
Let D = (D, ( · v ) v∈M ) be a metrized divisor on X , Y a 0-cycle of X and s a rational section of O(D) that is is regular and non-vanishing on the support of Y . For each place where Y v is the 0-cycle of X v obtained by base change. The condition that D is adelic implies that h D,v (Y ; s) = 0 for all but a finite number of places. If s is another rational section of O(D) that is regular and non-vanishing on |Y |, then where Y = p μ p p and γ = p f ( p) μ p ∈ K × .

Definition 4.6
Let D be a metrized divisor on X and Y a 0-cycle of X . The global height of Y with respect to D is defined as with s a rational section of O(D) that is is regular and non-vanishing on |Y |.
The local heights in (4.6) are zero for all but a finite number of places, and so this sum is finite. The equality (4.5) together with the product formula imply that this sum does not depend on the rational section s.
Given a metrized divisor D on X and an adelic field extension (F, N), we denote by D F the metrized divisor on X F obtained by extending the v-adic metrics of D as in (4.3).
Proof Let s be a rational section of O(D) that is is regular and non-vanishing on |Y | and v ∈ M. By Propositions 4.3 and 3.7(1), The statement follows by summing over all the places of K.
Since the global height is invariant under field extension, it induces a notion of global height for algebraic points, that is, a well-defined function When K is a global field, this notion coincides with the one in [7, Definition 2.2]. Now we turn to cycles of arbitrary dimension. Let V be a k-dimensional irreducible subvariety of X and D 0 , . . . , D k−1 a family of k semipositive metrized divisors on X . For each place v ∈ M, we can associate to this data a measure on X an v denoted by It is supported on |Y v | an and has total mass equal to the degree deg D 0 ,...,D k−1 (Y ). We and the convention that the local height of the cycle 0 ∈ Z −1 (X ) is zero. The following notion is the arithmetic analogue of global sections of a line bundle, and Proposition 4.11 below is an analogue for local heights of Proposition 2.5.  Proof Since the cycle Y is effective and the metrized divisors D i are semipositive, their v-adic Monge-Ampere measure is a measure, that is, it takes only nonnegative values. Since the global section s k is D k -small, log s k (q) k,v ≤ 0 for all q ∈ X an v . The inequality follows then from the inductive definition of the local height. Our next step is to define global heights for cycles over an adelic field. We first state an auxiliary result specifying the behavior of local heights with respect to change of sections, extending (4.5) to the higher dimensional case. The following lemma and its proof are similar to [15,Corollary 3.8].
intersects Y properly. We proceed to prove the formula (4.7) with the s i 's in the place of the s i 's. Hence, we want to prove that there is γ ∈ K × such that, for every v ∈ M, (4.9) To this end, consider first the particular case when (3)], the equality (4.7) holds with γ k ∈ K × given by By [8, Theorem 1.4.17(1)], the local height is symmetric in the pairs (D i , s i ). By the hypothesis (4.8), we can reorder the metrized line bundles and rational sections, and iterate the above construction for every i = 0, . . . , k. This proves (4.9) with γ = k i=0 γ i . Assuming that the s i 's are generic enough so that the condition in (4.8) also holds with the s i 's instead of the s i 's, similarly there exists γ ∈ K × such that, for every v ∈ M, (4.10) The statement follows by combining (4.9) and (4.10).
We consider the following notions of positivity of metrized divisors.

Definition 4.13
Let D be a metrized divisor on X .
(1) D is nef if D is nef, D is semipositive, and h D ( p) ≥ 0 for every closed point p of X .
(2) D is generated by small sections if, for every closed point p ∈ X , there is a D-small section s such that p / ∈ |div(s)|.
Proof For k = 0, the statement is obvious, so we only consider the case when k ≥ 1. By Lemma 4.12, it is enough to prove the statement for any particular choice of rational sections s i , provided that their associated Cartier divisors intersect Y properly. We can also reduce without loss of generality to the case when Y = V is an irreducible variety of dimension k. We can then choose rational sections s i , i = 0, . . . , k, such that s i is D i -small. By Proposition 4.11, Since div(s k ) · V is an effective (k − 1)-cycle, the statement follows by induction on k. (4.11)

This sum converges to an element in R ∪ {+∞}, and its value does not depend on the choice of the s i 's.
Proof The existence of rational sections s i such that div(s 0 ), . . . , div(s k ) intersects Y properly follows from the moving lemma, with the hypothesis that X is projective. By Lemma 4.14 and the fact that the local heights of 0-cycles are zero for all but a finite number of places, the local heights in (4.11) are nonnegative, except for a finite number of v's. Hence, the sum converges to an element in R ∪ {+∞}. Lemma 4.12 and the product formula imply that the value of this sum does not depend on the choice of the s i 's. This definition generalizes the notion of global height of cycles of varieties over global fields in [8,Sect. 1.5], to cycles of varieties over an arbitrary adelic field, in the case when the considered metrized divisors are generated by small sections.
In principle, the sum in (4.11) might contain an infinite number of nonzero terms. Nevertheless, we are not aware of any example where this phenomenon actually happens. Moreover, for varieties over global fields, the local heights of a given cycle are zero for all but a finite number of places [8, Proposition 1.5.14], and so their global height is a real number given as a weighted sum of a finite number local heights.
In this context, we propose the following question. A positive answer would imply that, for a variety over an adelic field and a family of semipositive metrized divisors, the global height of a cycle is a well-defined real number, given as a weighted sum of a finite number local heights.
The following results are arithmetic analogues of Proposition 2.5 and Corollary 2.6.

Proposition 4.17
Let Y be an effective k-cycle of X , and D 0 , . . . , D k semipositive metrized divisors on X such that D 0 is nef and D 1 , . . . , D k are generated by small sections. Let s k be a D k -small section. Then Proof We reduce without loss of generality to the case when Y = V is an irreducible subvariety of dimension k. If V ⊂ |div(s k )|, the first inequality is clear. For the second inequality, we choose rational sections s i , i = 0, . . . , k − 1, and s k such that div(s 0 ), . . . , div(s k−1 ), div(s k ) intersect Y properly. Using Lemmas 4.12 and 4.14, the product formula and the fact that D 0 is nef, we deduce that h D 0 ,...,D k (Y ) ≥ 0. Otherwise, V ⊂ |div(s k )| and we choose rational sections s i , i = 0, . . . , k − 1, such that div(s 0 ), . . . , div(s k ) intersect Y properly. The first inequality follows by applying the argument above to div(s k ) · Y , whereas the second one is given by Proposition 4.11.

Metrics and heights on toric varieties
In this section, we recall the necessary background on the arithmetic geometry of toric varieties following [4,8]. In the second part of Sect. 2, we presented elements of the algebraic geometry of toric varieties over a field. In the sequel, we will freely use the notation introduced therein.
Let (K, M) be an adelic field satisfying the product formula. Let M Z n be a lattice and T G n m,K its associated torus over K as in (2.4). For v ∈ M, we denote by T an v the v-adic analytification of T, and by S v its compact torus. In the Archimedean case, S v is isomorphic to the polycircle (S 1 ) n , whereas in the non-Archimedean case, it is a compact analytic group, see [8,Sect. 4.2] for a description. Moreover, there is a map val v : T an v → N R defined, in a given splitting, as This map does not depend on the choice of the splitting, and the compact torus S v coincides with its fiber over the point 0 ∈ N R . Let X be a projective toric variety with torus T given by a regular complete fan on N R , and D a toric Cartier divisor on X given by a virtual support function D on . Recall that X contains T as a dense open subset. Let · v be a toric v-adic metric on D, that is, a metric on the analytic line bundle O(D) an v that is invariant under the action of S v . The associated v-adic metric function is the continuous function ψ · v : N R → R given by for any p ∈ T an v with val v ( p) = u and where s D is the distinguished rational section of O(D). This function satisfies that |ψ · v − D | is bounded on N R and moreover, this difference extends to a continuous function on N , the compactification of N R induced by the fan . Indeed, the assignment is a one-to-one correspondence between the set of toric v-adic metrics on D and the set of such continuous functions on N R [8, Proposition 4.3.10]. In particular, the toric v-adic metric on D associated to the virtual support function D is called the canonical v-adic toric metric of D and is denoted by · v,can . Furthermore, when · v is semipositive, ψ · v is a concave function and it verifies that |ψ · v − D | is bounded on N R , and the assignment in (5.2) gives a one-to-one correspondence between the set of semipositive toric v-adic metrics on D and the set of such concave functions on N R .
When · v is semipositive, we also consider a continuous concave function on the polytope ϑ · v : D → R defined as the Legendre-Fenchel dual of ψ · v , that is The assignment · v → ϑ · v is a one-to-one correspondence between the set of semipositive toric v-adic metrics on D and that of continuous concave functions on D . Under this assignment, the canonical v-adic toric metric of D corresponds to the zero function on D . Let D be a toric metrized divisor on X . For each v ∈ M, we set

Definition 5.1 An (adelic) toric metric on D is a collection of toric v-adic metrics
for the associated v-adic metric function and v-adic roof function, respectively.

Proposition 5.3 Let D = (D, ( · v ) v∈M ) be toric divisor together with a collection of toric v-adic metrics. If D is adelic in the sense of Definition 5.1, then it is also adelic in the sense of Definition 4.4. Moreover, both definitions coincide in the semipositive case.
Proof Let p ∈ X (K) and choose an adelic field extension (F, N) such that p ∈ X (F). Then p F is a rational point of X F and the inclusion is an equivariant map. Hence the inverse image ι * D is an adelic toric metric on p F and so, for w ∈ N, and this quantity vanishes for all but the finite number of w ∈ N such that · w is not the canonical metric. Since this holds for all p ∈ X (K), we conclude that D is adelic in the sense of Definition 4.4. For the second statement, assume that D is semipositive and adelic in the sense of Definition 4.4. Let x i ∈ M, i = 1, . . . , s, be the vertices of the lattice polytope D . By [8, Example 2.5.13], for each i there is an n-dimensional cone σ i ∈ corresponding to x i under the Legendre-Fenchel correspondence, i = 1, . . . , s. Each of these n-dimensional cones corresponds to a 0-dimensional orbit p i of X . Denote by ι i : p i → X the inclusion of this orbit.
Fix 1 ≤ i ≤ s. Modulo a translation, we can assume without loss of generality that x i = 0. By [8,Proposition 4.8.9], for v ∈ M, Hence ϑ D,v (x i ) = 0 for all but a finite number of v's.
On the other hand, let x 0 be the distinguished point of X , which coincides with the neutral element of T, and denote by ι 0 : x 0 → X its inclusion. By [8,Proposition 4.8.10], Hence max x∈ D ϑ D,v (x) = 0 for all but a finite number of v's. For every v ∈ M such that ϑ D,v (x i ) = 0 for all i and max x∈ D ϑ D,v (x) = 0, we have that ϑ D,v ≡ 0 because this local roof function is a concave function on D . Hence, · v coincides with the v-adic canonical metric of D for all these places. for all but a finite number of places. In the absence of convexity, these conditions do not imply that ψ D,v = D for all but a finite number of places.
A classical example of toric metrized divisors are those given by the inverse image of an equivariant map to a projective space equipped with the canonical metric on its universal line bundle. Below we describe this example and we refer to [8,Example 5.1.16] for the technical details.
Let m = (m 0 , . . . , m r ) ∈ M r +1 and α = (α 0 , . . . , α r ) ∈ (K × ) r +1 , with r ≥ 0. The monomial map associated to this data is defined as For a toric variety X with torus T corresponding to fan that is compatible with the polytope = conv(m 0 , . . . , m r ) ⊂ M R , this extends to an equivariant map X → P r K , also denoted by ϕ m,α . Example 5.5 With notation as above, let E can be the divisor of the hyperplane at infinity of P r K , equipped with the canonical metric at all places. Then D = ϕ * m,α E is the nef toric Cartier divisor on X corresponding to the translated polytope − m 0 . We consider the semipositive toric metrized divisor D = ϕ * m,α E on X . For each v ∈ M, the v-adic metric function of D is given by The polytope corresponding to D is − m 0 and, for each v ∈ M, the v-adic roof function of D is given by the maximum being over the vectors λ = (λ 0 , . . . , λ r ) ∈ R r +1 ≥0 with r j=0 λ j = 1 such that In other words, this is the piecewise affine concave function on − m 0 parametrizing the upper envelope of the extended polytope

Definition 5.6
For i = 0, . . . , n, let g i : i → R be a concave function on a convex body i ⊂ M R . The mixed integral of g 0 , . . . , g n is defined as where i 0 + · · · + i j denotes the Minkowski sum of polytopes, and g i 0 · · · g i j the sup-convolution of concave function, which is the function on i 0 + · · · + i j defined as where the supremum is taken over The mixed integral is symmetric and additive in each variable with respect to the supconvolution. Moreover, for a concave function g : → R on a convex body , we have MI M (g, . . . , g) = (n + 1)! g d vol M , see [21,Sect. 8]  global section of O(D) associated to f is D-small. We obtain this metrized divisor as the inverse image of a metrized divisor on a projective space. For r ≥ 0, let P r K be the r -dimensional projective space over K and E the divisor of the hyperplane at infinity. We denote by E this Cartier divisor equipped with the 1 -norm at the Archimedean places, and the canonical one at the non-Archimedean ones. This metric is defined, for p = ( p 0 : · · · : p s ) ∈ P s K (K v ) and a global section s of O(E) corresponding to a linear form ρ s ∈ K[x 0 , . . . , x s ], by The projective space P r K has a standard structure of toric variety with torus G r m,K , included via the map (z 1 , . . . , z r ) → (1 : z 1 : · · · : z r ). Thus E is a toric metrized divisor. It is a particular case of the weighted p -metrized divisors on toric varieties studied in [9,Sect. 5.2].
The following result summarizes the basic properties of this toric metrized divisor and its combinatorial data.

Proposition 6.1 The toric metrized divisor E on P r
K is semipositive and generated by small sections. For v ∈ M, its v-adic metric function is given, for u = (u 1 , . . . , u r ) ∈ R r , by if v is Archimedean, The polytope corresponding to E is the standard simplex r of R r . For v ∈ M, the v-adic roof function of E is given, for x = (x 1 , . . . , x r ) ∈ r , by Proof which gives the expression in (6.2) for this case. The non-Archimedean case is done similarly. We can easily check that these metric functions are concave. In the Archimedean case, this can be done by computing its Hessian and verifying that it is nonpositive and, in the non-Archimedean case, it is immediate from its expression. Hence, E is semipositive. Set s j for the global sections corresponding to the linear forms x j ∈ K[x 0 , . . . , x r ], j = 0, . . . , r . We have that r j=0 |div(s j )| = ∅, and so this is a set of generating global sections. It follows from the definition of the metric in (6.1) that these global sections are E-small. Hence, E is generated by small sections.
The fact that the polytope corresponding to E is the standard simplex is classical, see for instance [13, page 27]. When v is Archimedean, the v-adic roof function can be computed similarly as the one for the Fubini-Study metric in [8,Example 2.4.3]. When v is non-Archimedean, v-adic roof function is zero, because the metric · v is canonical.
Set r ≥ 0. Take m = (m 0 , . . . , m r ) ∈ M r +1 and α = (α 0 , . . . , α r ) ∈ (K × ) r +1 , and consider the polytope = conv(m 0 , . . . , m r ) ⊂ M R . Let X be a projective toric variety over K given by a fan on N R that is compatible with . Let ϕ m,α : T → P r K be the monomial map in (5.3) and set which coincides with the Cartier divisor on X corresponding to . For each v ∈ M, we consider the metric Since ϕ m,α is an equivariant map and E is toric, this is a toric metrized divisor on X .

Proposition 6.2 The toric metrized divisor D = D m,α on X is semipositive and generated by small sections.
For v ∈ M, its v-adic metric is given, for p ∈ T(K v ), by if v is non-Archimedean.

(6.5)
The v-adic metric function of D is given, for u ∈ N R , by (6.6) and the v-adic roof function of D is given, for x ∈ , by the maximum being over the vectors λ = (λ 0 , . . . , λ r ) ∈ R r +1 ≥0 with r j=0 λ j = 1 such that r j=0 λ j m j = x.
Proof Set D = ϕ * m,α E for short. This is a toric metrized divisor on X that is semipositive and generated by small sections, due to Proposition 6.1 and the preservation of these properties under inverse image. Since the v-adic metrics of D are homothecies of those of D , it follows that D is semipositive too. Moreover, a global section ς of O(D ) O(D) is D -small if and only if the global section α 0 ς is D-small. It follows that D is also generated by small sections.
Using (6.1) and the definition of the monomial map ϕ m,α , for v ∈ M, the v-adic metric of D is given, for p ∈ T(K v ), by if v is non-Archimedean.
Since D = div(χ −m 0 ) + D , their distinguished rational sections are related by s D = χ −m 0 s D . It follows from (6.3) that which implies the formulae in (6.5). As a consequence, we obtain also the expressions for the v-adic metric functions of D.
For its roof function, consider first the linear map H : N R → R r +1 given, for u ∈ N R , by H (u) = ( m 0 , u , . . . , m r , u ). For each place v, consider the concave function g v : R r +1 → R given, for ν ∈ R r +1 , by Notice that ψ D,v = H * g v . The domain of the Legendre-Fenchel dual of g v is the simplex S given as the convex hull of the vectors in the standard basis of R r +1 . This Legendre-Fenchel dual is given, for λ ∈ S, by For the Archimedean case, this formula follows from [9,Proposition 5.8], whereas in the non-Archimedean case, it is given by Example 5.5.
By [8,Proposition 2.3.8(3)], the v-adic roof function ϑ D,v is the direct image under the dual map H ∨ of the Legendre-Fenchel dual g ∨ v , which gives the stated formulae in (6.7). Definition 6.3 Let f ∈ K[M] be a Laurent polynomial and X be a projective toric variety over K given by a fan on N R that is compatible with the Newton polytope N( f ). Write f = r j=0 α j χ m j with m j ∈ M and α j ∈ K × . The toric metrized divisor on X associated to f is defined as It follows from (6.5) that s v ≤ 1 on T(K v ), and so s is D-small.
The following result corresponds to Theorem 1.1 in the introduction.
Proof Let be the complete fan corresponding to the proper toric variety X . By taking a refinement, we can assume without loss of generality that is regular and compatible with the Newton polytopes i , i = 1, . . . , n. Hence X is a projective toric variety and D 0 a nef toric metrized divisor, and there are nef toric Cartier divisors D i , i = 1, . . . , n, corresponding to these Newton polytopes.
For i = 1, . . . , n, we denote by D i the toric metrized divisor associated to f i (Definition 6.3). By Proposition 6.2, each D i is semipositive and generated by small sections and, by Lemma 6.4, the global sections s i of O(D i ) corresponding to f i are D i -small. Applying Corollary 4.18 and Theorem 5.7, Due to Proposition 2.8(2), the inequality Z ( f 1 , . . . , f n ) ≤ n i=1 div(s i ) holds. By the linearity of the global height and the nefness of D 0 , which concludes the proof.
For a Laurent polynomial f ∈ K[M], we define its v-adic (logarithmic) length, denoted by v ( f ), as the v-adic length of its vector of coefficients, v ∈ M. We also define its (logarithmic) length, denoted by ( f ), as the length of its vector of coefficients. giving the stated inequality.
Corollary 6.8 With notation as in Theorem 6.5, In particular, for the canonical metric on D 0 (Example 5.2), Proof For 1 ≤ i ≤ n and v ∈ M, let ϑ i,v be the v-adic roof function of the toric semipositive metric associated to f i . Using (6.7), we compute the value of The first statement follows then from Theorem 6.5 and Lemma 6.7. The second statement is a particular case of the first one, using the fact that the v-adic roof functions of D can 0 are the zero functions on 0 .
We readily derive from the previous corollary the following version of the arithmetic Bézout theorem.
where deg( f i ) denotes the total degree of the polynomial f i .

Comparisons, examples and applications
In this section, we first compare our main results (Theorem 6.5 and Corollary 6.8) with the previous ones. Next, we compute the bounds given by these results in two families of examples, and compare them with the actual height of the 0-cycles. The first family of examples illustrates a case in which these bounds do approach the height of the 0-cycle, while the second one shows a situation where the bound of Theorem 6.5 is sharp and that of  (Z ( f 1 , . . . , f n ) . . . , i−1 , i+1 , . . . , n ), (7.1) where m( f i ) denotes the logarithmic Mahler measure of f i , and L( i ) a constant associated to the polytope i . This result is similar to Corollary 6.8 specialized to a system of Laurent polynomials with integer coefficients, and the toric divisor D 0 associated to the polytope given by the Minkowski sum n i=1 i , equipped with the canonical metric. The factors m( f i ) + L( i ) in (7.1)  and that the polytope 0 associated to the nef toric divisor D 0 contains i , i = 1, . . . , n. Then This result is equivalent to the specialization of the upper bound in (6.8) to a system of Laurent polynomials with integer coefficients and Newton polytopes contained in the polytope 0 .
We next turn to the computation of the bounds given by Theorem 6.5 and Corollary 6.8 in two families of examples.
We keep the notation of Sect. 6. We need the the following auxiliary computation of mixed volumes. For its proof, we recall that the mixed volume of a family of polytopes i ⊂ R n , i = 1, . . . , n, can be decomposed in terms of mixed volumes of their lower dimensional faces as MV n ( 1 , . . . , n ) = − u∈S n−1 where S n−1 is the unit sphere of R n , 1 is the support function of 1 as in (2.6), u i is the unique face of i that minimizes the functional u on this polytope, and MV n and MV n−1 denote the mixed volume functions associated to the Lebesgue measure of R n and u ⊥ R n−1 , respectively. In fact, the sum ranges through the normal vectors of the facets of each polytope. We refer to [22, formula (5.1.22)] for more details. where u is the Euclidean norm. We have that The result follows then from (7.3), (7.4) and (7.5).
Let X be a proper toric variety over Q, and D can 0 a nef toric Cartier divisor on X equipped with the canonical metric. Let 0 ⊂ R n be the polytope corresponding to D 0 and, for i = 1, . . . , n, set u i = e i + de i+1 + · · · + d n−i e n ∈ Z n , where the e j 's are the vectors in the standard basis of Z n . The height of p with respect to D By adding these contributions, h D can 0 ( p) = log(α) max m∈ 0 ∩Z n m, u − min m∈ 0 ∩Z n m, u , which gives the formula in (7.6).
Next we compare the value of the height of p with the bounds given by Corollary 6.8. We have ( f i ) = log(α + 1) for all i. Consider the dual basis of the u i 's, given by m 1 = e 1 , m 2 = e 2 − de 1 , . . . , m n = e n − de n−1 ∈ Z n .
For i = 1, . . . , n, the Newton polytope i of f i is a translate of the segment 0 m i , and u i is the smallest lattice point in the line ( j =i Rm j ) ⊥ . Moreover the sublattice j =i Zm i is saturated. By Lemma 7.1 MV Z n ( 0 , . . . , i−1 , i+1 , . . . , n ) = vol Z 0 , u i . Therefore, the bound given by Corollary 6.8 is vol Z 0 , u i log(α + 1). Example 1.2 in the introduction consists of the particular cases corresponding to the polytopes 0 = n , the standard simplex of R n , and 0 = conv(0, m 1 , . . . , m n ).
In the following example, we exhibit a situation where the difference between the bounds given by the results in Sect. 6 is noticeable. Recall that passing from Theorem 6.5 to Corollary 6.8 amounts to replacing the local roof functions by constant functions on the polytope bounding them from above. Hence, to maximize the discrepancy between these two concave functions, we look for local roof functions that are tent-shaped, which is the situation where the difference between the mean value and the maximum value of these functions is the greatest possible.

Example 7.3
Let α ≥ 1 be an integer and consider the system of Laurent polynomials 1 , . . . , x ±1 n ], i = 1, . . . , n, Its zero set in G n m,Q is the rational point p = (α, . . . , α) ∈ (Q × ) n . Let X = P n Q and let E can be the divisor of the hyperplane at infinity equipped with the canonical metric. Then the height of p with respect to E can is h E can ( p) = log(α).
Next we compare the value of this height with the bound given by Theorem 6.5. Since the explicit computation of the mixed integrals appearing in this bound is somewhat involved, instead of giving its exact value we are going to approximate them with an upper bound that is easier to compute.
The polytope associated to the toric Cartier divisor E is 0 = n , the standard simplex of R n . For each v ∈ M Q , the v-adic roof function ϑ 0,v of E can is the zero function on this simplex. For each i = 1, . . . , n, let i = N( f i ) ⊂ R n be the Newton polytope of f i , which coincides with the segment 0 e i . For v ∈ M Q , let ϑ i,v be the v-adic roof function associated to f i (Definition 6.3). This function is given, for t e i ∈ i = 0 e i , by For the Archimedean place, the v-adic roof functions are nonnegative, and so their mixed integral can be expressed as a mixed volume MI Z n (ϑ 0,∞ , . . . , ϑ n,∞ ) = MV Z n+1 ( 0 , . . . , n ), (7.7) with i = conv graph(ϑ i,∞ ), i × {0}) ⊂ R n × R. Consider the concave function ϑ : n → R defined by x i dx = (n + 1) log(2) + log(α). (7.8) When v is non-Archimedean, we have that |α| v ≤ 1 because α is an integer. Hence ϑ i,v ≤ 0, and so the mixed integral of these concave functions is nonpositive. Theorem 6.5 together with (7.7) and (7.8) gives the upper bound h E can ( p) ≤ (n + 1) log(2) + log(α).
As an application of our results, we bound the size of the coefficients of the u-resultant of the direct image under an equivariant map of the 0-cycle defined by a family of Laurent polynomials. As in the previous sections, let (K, M) be an adelic field satisfying the product formula, K an algebraic closure of K, and M Z n a lattice.

Definition 7.4
Let W ∈ Z 0 (P r K ) be a 0-cycle of a projective space over K and u = (u 0 , . . . , u r ) a set of r + 1 variables. Write W K = q μ q q ∈ Z 0 P r K for the 0-cycle obtained from W by the base change K → K. The u-resultant (or Chow form) of W is defined as Res(W ) = q (q 0 u 0 + · · · + q r u r ) μ q ∈ K(u) × , the product being over the points q = (q 0 : · · · : q r ) ∈ P r K (K) in the support of W K . It is well-defined up to a factor in K × .
The length of a Laurent polynomial (Definition 6.6) is invariant under adelic field extensions and multiplication by scalars. It is also submultiplicative, in the sense that it satisfies the inequality ( f g) ≤ ( f ) + (g), for f , g ∈ K[M]. The following result corresponds to Theorem 1.3 in the introduction.