Factorization of bivariate sparse polynomials

We prove a function field analogue of a conjecture of Schinzel on the factorization of univariate polynomials over the rationals. We derive from it a finiteness theorem for the irreducible factorizations of the bivariate Laurent polynomials in families with fixed set of complex coefficients and varying exponents. Roughly speaking, this result shows that the truly bivariate irreducible factors of these sparse Laurent polynomials, are also sparse. The proofs are based on a variant of the toric Bertini's theorem due to Zannier and Fuchs, Mantova and Zannier.


Introduction
A polynomial is said to be sparse (or lacunary) if it has few terms compared with its degree. The factorization problem for sparse polynomials can be vaguely stated as the question of whether the irreducible factors of a sparse polynomial are also sparse, apart from obvious exceptions. Aspects of this problem have been studied in various settings and for dierent formalizations of the notion of sparsenness, see for instance [Len99,Sch00,KK06,AKS07,FGS08,Gre16,ASZ17]. Several of these studies were based on tools from Diophantine geometry like lower bounds for the height of points and subvarieties, and unlikely intersections of subvarieties and subgroups of a torus.
In this text, we consider families of bivariate Laurent polynomials given as the pullback of a xed regular function on a torus by a varying 2-parameter monomial map. Precisely, let t, z be variables, x = (x 1 , . . . , x n ) a set of other n variables, and F ∈ C[x ±1 , z ±1 ] = C[x ±1 1 , . . . , x ±1 n , z ±1 ] a Laurent polynomial. For each vector a = (a 1 , . . . , a n ) ∈ Z n , we consider the bivariate Laurent polynomial given as the pullback of F by the monomial map (t, z) → (t a , z) = (t a 1 , . . . , t an , z), that is The number of coecients of each F a is bounded by those of F , and so these Laurent polynomials can be considered as sparse when F is xed and a is large.
A rst question concerns the irreducibility of F . It has been addressed in [Zan10], as we next describe. Let us assume F irreducible. Under which assumptions F a stays irreducible for a generic a? Let us consider the following example. The Laurent ]. is irreducible. However given (a 1 , a 2 ) ∈ Z 2 \ {(0, 0)} with a 1 even, the Laurent polynomial F a is reducible. This show that the sole assumption that F is irreducible is not enough to get an irreducibility statement for a generic specialization.
Zannier's result can be stated as follows. For a, b ∈ Z n , we denote by a, b = Theorem 1.1 ([Zan10, Theorem 3]). Let F ∈ C[x ±1 , z ±1 ] \ C[x ±1 ] be an irreducible Laurent polynomial, that is monic in z and such that F (x d 1 , . . . , x d n , z) is irreducible for d = deg z (F ). There is a nite subset Σ ⊂ Z n \ {0} such that, for each a ∈ Z n , either there is c ∈ Σ with c, a = 0, or F (t a , z) is irreducible.
Previously, Schinzel [Sch90] proved a similar result in the same direction, for Laurent polynomials over Q satisfying the strong additional assumption that F is not self-inversive. More recently, Fuchs, Mantova and Zannier [FMZ18,Addendum to Theorem 1.5] showed that the set Σ can be chosen independently of the coecients of F .
In the present paper we are interested in the factorization of F a . Our motivation is an old conjecture of Schinzel [Sch65] on the factorisation of sparse polynomial with rational coecients (Conjecture 2.1).
This conjecture implies the statement below. A Laurent polynomial in Q[x ±1 ] is cyclotomic if it can be written as a unit times the composition of a univariate cyclotomic polynomial with a monomial.
There is a nite set of matrices Ω ⊂ Z n×n satisfying the following property. For each a ∈ Z n , there are M ∈ Ω and b ∈ Z n with a = M b such that if P is an irreducible factor of F (x M ), then P (t b ) is either a product of cyclotomic Laurent polynomials, or an irreducible factor of F (t a ).
Our main result in this text is the following function eld analogue.
There is a nite set of matrices Ω ⊂ Z n×n satisfying the following property. For each a ∈ Z n , there are M ∈ Ω and b ∈ Z n with a = M b such that if P is an irreducible factor of F (x M , z), then P (t b , z) is, as an element of C(t)[z ±1 ], either a unit or an irreducible factor of F (t a , z).
Moreover, we also obtain in Theorem 5.1 the function eld analogue of Conjecture 2.1.
Theorem 1.3 shows that for each a ∈ Z n , there is a matrix M within the nite set Ω ⊂ Z n×n and a vector b ∈ Z n with a = M b such that, unless F x M , z = 0, the irreducible factorization for F a as in (1.1), the product being over the irreducible factors P in (1.2) such that Hence, the irreducible factorizations in C(t)[z ±1 ] of the F a 's can be obtained by specializing the irreducible factorizations of the Laurent polynomials F (x M , z) for a nite number of matrices M . These irreducible factors of the F a 's are sparse, in the sense that they are all represented as the pullback of a nite number of regular functions on the (n + 1)-dimensional torus G n+1 m by 2-parameter monomial maps. In particular, both the number of these irreducible factors and of their coecients are bounded above independently of a.
The proof of Theorem 1.3 relies on a variant of the aforementioned result of Zannier.
To state it, we rst introduce some further notation. Let t = (t 1 , . . . , t k ) be a set of k variables. A matrix A = (a i,j ) i,j ∈ Z n×k denes the family of n monomials in the variables t given by t a n,j j . Given a = (a 1 , . . . , a n ) ∈ Z n , we can consider it as a row vector or as a column vector. Thus x a j j and t a = (t a 1 , . . . , t an ).
and G ∈ C[x ±1 ] the coecient of the term of highest degree in the variable z. There are nite subsets Φ ⊂ Z n×n of nonsingular matrices and Σ ⊂ Z n of nonzero vectors such that, for a ∈ Z n , one of the next alternatives holds: Back to the factorization problem for sparse polynomials, it is natural to consider the more general setting of pullbacks of regular functions on G n m by arbitrary monomial maps, instead of only those appearing in (1.1). Let y = (y 1 , . . . , y n ) and t = (t 1 , . . . , t k ) be groups of n and k variables, respectively. For a Laurent polynomial H ∈ C[y ±1 ], consider the family of k-variate Laurent polynomials given by the pullback of H by Denote by S the multiplicative subset of C[t ±1 ] generated by the Laurent polynomials of the form f (t d ) for f ∈ C[z ±1 ] and d ∈ Z k .
We propose the following conjecture which, as explained in Remark 5.2, partially generalizes Theorem 1.3. Conjecture 1.5. Let H ∈ C[y ±1 ] and k ≥ 2. There is a nite set of matrices Ω ⊂ Z n×n satisfying the following property. For each A ∈ Z n×k , there are N ∈ Ω and B ∈ Z n×k with A = N B such that if P is an irreducible factor of H(y N ), then P (t B ) is, as an element of C[t ±1 ] S , either a unit or an irreducible factor of H(t A ).
The validity of this conjecture would imply that the irreducible factors of the H A 's that truly depend on more than one variable, are also the pullback of a nite number of regular functions on G n m by k-parameter monomial maps. The possible univariate irreducible factors of the H A 's split completely, and so they cannot be accounted from a nite number of such regular functions. This conjecture might follow from a suitable toric analogue of the classical Bertini's theorem that we propose in Conjecture 3.13.
Plan of the paper. In Section 2 we state Schinzel's conjecture and our function eld analogue (Theorem 2.4). In Section 3 we recall some facts on ber products and prove a variant of the Fuchs-Mantova-Zannier theorem concerning the irreducibility of pullbacks of cosets by a dominant maps W → G n m (Theorem 3.4). In Section 4 we prove Theorem 1.4, wereas in Section 5 we apply this result to prove Theorem 2.4 and then Theorem 1.3. anonymous referee for his/her useful comments. Part of this work was done while the authors met the Universitat de Barcelona and the Université de Caen. We thank these institutions for their hospitality.

A conjecture of Schinzel and its function field analogue
In [Sch65], Schinzel proposed the conjecture below on the factorization of univariate polynomials over Q.
] be a non-cyclotomic irreducible Laurent polynomial.
There are nite sets Ω 0 ⊂ Z n×n of nonsingular matrices and Γ ⊂ Z n of nonzero vectors satisfying the following property. Let a ∈ Z n ; then one of the next conditions holds: (1) there is c ∈ Γ verifying c, a = 0; (2) there are M ∈ Ω 0 and b ∈ Z n with a = M b such that if For the validity of this statement, in its condition (2) it is necessary to take out the cyclotomic part of F (t a ) and of the P (t b )'s, as shown by the example below.
. Let a ∈ Z 2 and choose a nonsingular matrix M ∈ Z 2×2 and a vector b ∈ Z 2 with a = M b. We have that Schinzel proved this conjecture when n = 1 in loc. cit. and, under the restrictive hypothesis that F is not self-inversive, when n ≥ 2 [Sch70], see also [Sch00, 6.2]. The general case when n ≥ 2 remains open.
In Section 5, we prove the function eld analogue for Laurent polynomials over the Remark 2.3. The analogy between cyclotomic Laurent polynomials over Q and irreducible constant Laurent polynomials over C(z) stems from height theory. Let K denote either Q or C(z), and h the canonical height function on subvarieties of the torus G n m,K , induced by the standard inclusion G n m,K → P n K .
Then the condition that h(V (F )) = 0 is equivalent to the fact that F is cyclotomic when K = Q, and to the fact that F is constant when K = C(z).
There are nite sets Ω 0 ⊂ Z n×n of nonsingular matrices and Γ ⊂ Z n of nonzero vectors satisfying the following property. Let a ∈ Z n ; then one of the next conditions holds: Similarly as for Conjecture 2.1, for the validity this statement it is is necessary to take out in its condition (2) the constant part of F (t a ) and of the P (t b )'s. As before, let x = (x 1 , . . . , x n ) be a set of n variables and denote by G n m = Spec(C[x ±1 ]) the n-dimensional torus over C.
Let W be a variety, that is, a reduced separated scheme of nite type over C. We assume that W is irreducible and quasiprojective of dimension n ≥ 0, and equipped with a dominant (regular) map π : W −→ G n m of degree e ≥ 1 that is nite onto its image. Given an isogeny λ of G n m , that is, an endomorphism of G n m with nite kernel, we denote by λ * W the bered product G n m × λ,π W , and by the corresponding bered product square.
Denition 3.1. The map π satises the property PB (pullback) if, for every isogeny λ of G n m , we have that λ * W is an irreducible variety. Theorem 3.2. Let W be an irreducible quasiprojective variety of dimension n and π : W → G n m a dominant map that is nite onto its image and that satises the property PB. There is a nite union E of proper subtori of G n m such that, for every subtorus When the property PB is not veried, the conclusion of this theorem does not necessarily hold because the map π factors through a nontrivial isogeny, as it was already pointed out in [Zan10].
and consider the isogeny The variety W is irreducible and, since F is monic in z, the map π is nite. However, it does not satisfy the property PB, since for the isogeny λ of G 2 m dened by λ(x 1 , and so this pullback is reducible. Indeed, this map does neither satisfy the conclusion of Theorem 3.2: given (a 1 , a 2 ) ∈ Z 2 \{(0, 0)} with a 1 even, let T ⊂ G 2 m be the 1-dimensional subtorus given as the image of the map t → (t a 1 , t a 2 ). Then and so this ber is not irreducible.
Here we need a variant of Theorem 3.2 that can be also applied in the situation when the map π does not verify the property PB. In this more general situation, the conclusion of that theorem does not necessarily hold. Instead, this conclusion is replaced by an alternative that explains the possibility that a ber is reducible by its factorization through a reducible pullback of the variety W by an isogeny of G n m within a nite set.
Theorem 3.4. Let W be an irreducible quasiprojective variety of dimension n and π : W → G n m a dominant map that is nite onto its image. There is a nite union E of proper subtori of G n m and a nite set Λ of isogenies of G n m such that, for each subtorus T ⊂ G n m and each point p ∈ G n m (C) = (C × ) n , one of the next conditions holds: (1) T ⊆ E; (2) there is λ ∈ Λ with λ * W reducible and a subtorus T ⊂ G n m with λ inducing an Remark 3.5. When the condition (2) above is satised, there is a diagram with λ * W reducible and λ : T → T an isomorphism, and where ι denotes the inclusion of the subtorus T into G n m .
Both inner squares in this diagram are bered products, and so is the outer square.
This implies that the bers π −1 (T ) and π −1 (T ) are isomorphic. Thus π −1 (T ) can be identied with the ber of a subtorus for the reducible cover π : λ * W → G n m , and so this ber is expected to be reducible as well.
Example 3.6. We keep the notation from Example 3.3. In particular, subtorus given as the image of the map t → (t a 1 , t a 2 ). These vectors satisfy the condition (2) in Theorem 3.4 for the isogeny λ : . Indeed, λ * W is reducible, and this isogeny induces an isomorphism T → T with the subtorus T ⊂ G 2 m given as the image of the map t → (t a 1 /2 , t a 2 ).
We prove this theorem by reducing it to the previous toric Bertini's theorem, through a variation (Proposition 3.8) of a factorization result for rational maps from [Zan10].
We give the proof after some auxiliary results. We rst study the reducibility of pullbacks of varieties with respect to isogenies of tori.
Lemma 3.7. Let π : W → X be a map of varieties and λ : X → X an étale map.
Then X × λ,π W is a variety.
In particular, for a map π : W → G n m and an isogeny λ of G n m , we have that λ * W is a variety.
Proof. Since λ : X → X is étale, the map is an isomorphism. Since W is a variety, the local ring O p is reduced and, by a theorem of Chevalley [ZS75,8.13], the completion O p is reduced too.
By the isomorphism in (3.3), the completed ring O q is reduced. Since this is the completion of a ring with respect to a maximal ideal, the map O q → O q is injective, and so the local ring O q is also reduced. Since the condition of being reduced is local, this implies that X × λ,π W is a variety.
The last statement comes from the fact that the isogenies of algebraic groups over C are étale maps.
Thanks to this result, λ * W can be identied with its underlying algebraic subset in the Cartesian product G n m (C) × W (C), namely Proposition 3.8. Let π : W → G n m be a map from an irreducible variety W , and λ an isogeny of G n m . The following conditions are equivalent: (1) the pullback λ * W is reducible; (2) there is a factorization λ = µ • τ with µ, τ isogenies of G n m such that µ is not an isomorphism, and a map ρ : W → G n m such that π = µ • ρ.
In other terms, the condition (2) in the proposition above amounts to the existence of the commutative diagram extending (3.1) of the form Proof. Suppose that the condition (2) holds. In this case, for p ∈ G n m (C) and w ∈ W (C), the fact that λ(p) = π(w) is equivalent to µ(τ (p)) = µ(ρ(w)), and so this holds if and only if there is ζ ∈ ker(µ) with τ (p) = ζ · ρ(w). From (3.4), the pullback decomposes into disjoints subvarieties as Since µ is not an isomorphism, this pullback is reducible, giving the condition (1). Conversely, suppose that the condition (1) holds. Then λ * W has a decomposition into irreducible components Similarly as in (3.2), the map λ * W → W is étale, and so the U i 's are disjoint. Since λ is an isogeny, the map λ * W → W is also nite. The nite subgroup ker(λ) of G n m (C) acts on λ * W via the maps (p, w) → (ζ · p, w) for ζ ∈ ker(λ), and this action respects the bers of λ. The action is transitive on the bers, and so it is also transitive on the U i 's. Let H ⊂ ker(λ) be the stabilizer of the irreducible component U 1 , and U 1 /H the quotient variety. We have that H acts on U 1 transitively on the bers and without xed points. The induced map terms and identifying G n m /H G n m , this map is dened, for w ∈ W , as ρ(w) = τ (p · H) for any p ∈ G n m such that (p, w) ∈ U 1 . Both G n m /H and G n m / ker(λ) are isomorphic to G n m , and so there is a factorization λ = µ • τ, with τ and µ corresponding to the projections G n m → G n m /H and G n m /H → G n m / ker(λ), respectively. For w ∈ W and (p, w) ∈ U 1 , we have that µ • ρ(w) = µ • τ (p) = π(w). Since the action of ker(λ) on the U i 's is transitive and k ≥ 2, we have that H = ker(λ) and so µ is not an isomorphism, giving the condition (2). Suppose that there is a further isogeny ν such that ν * W = G n m × ν,ρ W is reducible. By Proposition 3.8, there would be an isogeny µ that is not an isomorphism and a map ρ : By construction, this is not possible. Hence ν * W is irreducible for every isogeny ν of G n m , and so ρ satises the property PB.
The next result gives a criterion to detect if the inclusion of a subtorus can be factored through a given isogeny as in Proposition 3.8(2).
Lemma 3.11. Let T ⊂ G n m be a subtorus and λ an isogeny of G n m . The following conditions are equivalent: (1) there is a subtorus T ⊂ G n m such that λ induces an isomorphism T → T ; (2) λ −1 (T ) is the union of deg(λ) distinct torsion cosets. Proof. First suppose that λ −1 (T ) is the union of deg(λ) = # ker(λ) distinct torsion cosets, and denote by T the one that contains the neutral element. Then T is a subtorus and T ∩ ker(λ) = {1}. It follows that λ| T : T → T is an isogeny of degree 1 and hence an isomorphism, giving the rst condition.
Conversely, let T ⊂ G n m be a subtorus such that λ| T : T → T is an isomorphism. contained in E and every point p ∈ G n m (C), the ber ρ −1 (p · T ) is irreducible. Set E = λ(E ).
We next show that the pair (Λ, E) satises the requirements of Theorem 3.4. Let T be a subtorus of G n m that is not contained in E and write λ −1 (T ) = k i=1 T i as a disjoint union of torsion cosets T i of G n m .
When k = 1, we have that λ −1 (T ) = T 1 is a subtorus of G n m that is not contained Otherwise, k ≥ 2. Let H ⊂ ker(λ) be the stabilizer of the (unique) subtori in this decomposition, say T 1 . This is a proper subgroup, because ker(λ) acts transitively on this collection of torsion cosets and k ≥ 2.
Consider the factorization λ = µ H • τ H as in (3.5). Then µ H ∈ Λ and µ −1 H (T ) splits as an union of k = [ker(λ) : H] = deg(µ H ) distinct torsion cosets. By Lemma 3.11, µ H induces an isomorphism between a subtorus T of G n m and T . Moreover, Proposition 3.8(2) applied to the map τ H • ρ and the isogeny µ H shows that the pullback µ * H W is reducible, completing the proof.
It seems interesting to extend these results to maps that are not necessarily domi- Conjecture 3.12. Let W be an irreducible quasiprojective variety and ϕ : W → G n m a map that is nite onto its image and satises the property PB. There is a nite union E of proper subtori of G n m such that, for every subtorus T of G n m with dim(T ) ≥ codim(ϕ(W )) + 1 that is not contained in E and every point p ∈ G n m (C), we have that ϕ −1 (p · T ) is an irreducible subvariety of W .
Similarly, we propose the following conjectural extension of Theorem 3.4. Conjecture 3.13. Let W be an irreducible quasiprojective variety and ϕ : W → G n m a map that is nite onto its image. There is a nite union E of proper subtori of G n m and a nite set Λ of isogenies of G n m such that, for each subtorus T ⊂ G n m with dim(T ) ≥ codim(ϕ(W )) + 1 and each point p ∈ G n m (C), one of the next conditions holds: (1) T ⊆ E; (2) there is λ ∈ Λ with λ * W reducible and a subtorus T ⊂ G n m with λ inducing an isomorphism T → T ; (3) ϕ −1 (p · T ) is irreducible.

Pullbacks of Laurent polynomials by monomial maps
We next prove Theorem 1.4 stated in the introduction. To this end, we rst recall some notation and introduce some auxiliary results.
Let t = (t 1 , . . . , t k ) be a set of k variables. A matrix A = (a i,j ) i,j ∈ Z n×k denes the family of n monomials in the variables t given by t A = k j=1 t a 1,j j , . . . , k j=1 t a n,j j . The rule t → t A denes a k-parameter monomial map G k m → G n m . This is a group morphism and indeed, every group morphism from G k m to G n m is of this form. The isogenies of G n m correspond to the nonsingular matrices of Z n×n . Given a = (a 1 , . . . , a n ) ∈ Z n , we can consider it as a row vector, that is, as a matrix in Z 1×n . In this case, is an n-variate monomial. Row vectors give characters of G n m , that is, group morphisms G n m → G m . When a is primitive, the kernel of its associated character is a subtorus of G n m of codimension 1, and every such subtorus arises in this way.
Else, we can consider a as a column vector, that is, as a matrix in Z n×1 . Then t a = (t a 1 , . . . , t an ) is a collection of n univariate monomials in a variable t. Column vectors give group morphisms G m → G n m . When a = 0, the image of such a morphims is a subtorus of G n m of dimension 1, that we denote by T a . When a is primitive, the associated group morphism G m → G n m gives an isomorphism between G m and T a .
For subvarieties of tori, bered products like those in (3.1) can be expressed in more concrete terms. The next lemma gives such an expression for the case of hypersurfaces. z). Proof. The maps π and λ correspond to the morphisms of C-algebras and the bered product G k m × λ,π W is the scheme associated to the tensor product This tensor product is isomorphic to the C-algebra which gives the statement.
The group µ m of m-th roots of the unity acts on the set of these irreducible factors by p i (t, z) → p i (ζ · t, z), i = 1, . . . , k, for ζ ∈ µ m . Let P ⊂ {p 1 , . . . , p k } be a nonempty orbit of this action. The polynomial p∈P p is invariant under the action of µ m , and so it is of the form g(t m , z) with g ∈ C(t)[z]. This product is a nontrivial factor of f (t m , z), and so g coincides with f up to a scalar. It follows that P = {p 1 , . . . , p k } and so the action is transitive. In particular, all the p i 's have the same degree in the variable z, and so this degree is positive and k|d.
The stabilizer of an irreducible factor p i is a subgroup of µ m , hence it is of the form µ l with l|m. Since the action is transitive and µ m is abelian, this subgroup does not depend on the choice of p i . Moreover, m/l is equal to k, the number of irreducible factors of f (t m , z), also because of the transitivity of the action.
By the invariance of each p i under the action of µ l , there is q i ∈ C(t)[z] \ C(t) with p i = q i (t l , z). It follows from (4.1) that with e = m/l. Clearly e|m and as explained, e = k, and so this quantity also divides d,  (2) There is a nite subset ∆ F of Z n such that for a ∈ Z n with c, a = 0 for all c ∈ ∆ F , the polynomial F (t a , z) has degree d in the variable z.
This morphism is an integral extension because the leading term G is invertible in ] G , and so the map W → U is nite and, a fortiori, surjective.
For the second statement, write G = r j=1 G j x c j with G j ∈ C × and c j ∈ N n , j = 1, . . . , r, and consider the nite subset of Z n given by For a ∈ Z n with c, a = 0 for all c ∈ ∆ F , we have that G(t a ) = 0 and so deg z (F (t a , z)) = d.
As for Lemma 4.2, the proof of the last assertion relies on the action of torsion points on irreducible factors, and so we only sketch it. Using Lemmas 4.1 and 3.7, we the decomposition of this Laurent polynomial into distinct irreducible factors. The action of the nite group {x ∈ G n m | x A = 1} on the the sets of these irreducible factors is transitive, and so the P i 's have the same degree with respect to the variable z. Hence for i = 1, . . . , k, we have that k deg z (P i ) = d ≥ 1. In particular, deg z (P i ) ≥ 1, proving the statement.
Proof of Theorem 1.4. The statement of this result, restricted to primitive vectors a ∈ Z n , is a specialization of Theorem 3.4. To see this, rst reduce, multiplying by a suitable monomial, to the case when F is an irreducible polynomial in C[x, z] of degree d ≥ 1 in the variable z.
Let Λ be a nite subset of isogenies of G n m and E a nite union of proper subtori of G n m satisfying the conclusion of Theorem 3.4 applied to this map. Set then Φ 1 for the nite subset of nonsingular matrices in Z n×n corresponding to the isogenies in Λ, and Σ 1 for a nite subset of nonzero vectors of Z n such that For a primitive vector a ∈ Z n , set T a for the 1-dimensional subtorus dened as the image of the group morphism G m → G n m . This map gives an isomorphism between G m and T a . By Lemma 4.1, the ber π −1 (T a ) is isomorphic to the subscheme of G 2 m \ V (G(t a )) dened by F (t a , z). For the isogeny λ associated to a nonsingular matrix M ∈ Φ 1 , the same result shows that λ * W is isomorphic to the subscheme of G n+1 m \ V (G) dened by F (x M , z). The three alternatives from Theorem 3.4 applied to the map π, the subtorus T a and the point p = (1, . . . , 1) ∈ G n m (C), then boil down to those in the theorem under examination, as explained below.
(2) Else suppose that there is an isogeny λ ∈ Λ with λ * W reducible and a subtorus T of G n m with λ inducing an isomorphism between T and T a . For M ∈ Φ 1 the nonsingular matrix associated to λ, we have that a ∈ im(B) and, by Lemma 4.1, F (x M , z) is reducible.
We next enlarge these nite sets to cover the rest of the cases. Let d ≥ 1 be the degree of F in the variable z, and let e be a divisor of d. If F (x e , z) is irreducible, we respectively denote by Φ e and Σ e the nite subsets of nonsingular matrices in Z n×n and of nonzero vectors of Z n given by the application of Theorem 3.4 to this polynomial. Otherwise, we set Φ e = {I n } with I n the identity matrix of Z n×n , and Σ e = ∅. Set also ∆ for the nite subset of nonzero vectors in Z n associated to F by Given an arbitrary vector a ∈ Z n , write a = mb with m ∈ N and b ∈ Z n primitive, and set Suppose that neither (1) nor (2) hold for a. Let e ∈ N be a common divisor of d and m. A fortiori, these conditions do neither hold for e b and, as explained before, , giving the condition (3) for a and concluding the proof.
Remark 4.4. Using the toric Bertini's theorem 3.4 for cosets of arbitrary dimension, the present polynomial version in Theorem 1.4 might be extended to k-parameter monomial maps for any k, and also to arbitrary translates of these monomial maps.
We have kept the present more restricted statement for the sake of simplicity, and also because it is sucient for our application.

Factorization of sparse polynomials
Here we prove the results on the factorization of Laurent polynomials announced in the introduction and in Section 2. Theorem 2.4 is easily seen to be implied by the following statement.
]. There are nite sets Ω 0 ⊂ Z n×n of nonsingular matrices and Γ ⊂ Z n of nonzero vectors satisfying the property that, for a ∈ Z n \ {0}, one of the next alternatives holds: (1) there is c ∈ Γ with c, a = 0; (2) there are M ∈ Ω 0 and b ∈ Z n with a = M b such that if Proof. We proceed by induction on deg z (F ). When deg z (F ) = 0, the statement is trivial, and so we assume that deg z (F ) ≥ 1.
If F is irreducible, we respectively denote by Φ and Σ the nite sets of nonsingular matrices in Z n×n and of nonzero vectors in Z n from Theorem 5.1 applied to this Laurent polynomial. If F is reducible, we set Φ = {I n } and Σ = ∅.
Let a ∈ Z n . When F is irreducible, if the condition (1) in Theorem 1.4 holds, then the condition (1) in Theorem 5.1 also holds by taking Γ as any nite set containing Σ.
Still in the irreducible case, if the condition (3) in Theorem 1.4 holds, the Laurent polynomial F (t a , z) is irreducible, and the condition (1) in Theorem 5.1 holds provided that Ω 0 contains I n .
Else, suppose that the condition (2) in Theorem 1.4 holds, that is, there are M ∈ Φ and b ∈ Z n with a = M b and F (x M , z) is reducible. Let i and Γ i respectively denote the nite sets of nonsingular matrices in Z n×n and of nonzero vectors in Z n whose existence is assured by this theorem.
By construction, either there is a vector c ∈ Γ 1 ∪ Γ 2 with c, b = 0, or we can nd M i ∈ Ω i and b i ∈ Z n with b = M i b i and a decomposition Consider the lattices K i = im(M i ), i = 1, 2, and set K = K 1 ∩ K 2 . Since K is also a lattice, there is a nonsingular matrix M ∈ Z n×n with K = im(M ) and, since K ⊆ K i , there are nonsingular matrices N i , i = 1, 2, with M = is irreducible in C(t)[s ±1 ] for all i, j. The statement follows by taking Ω 0 as any nite set containing all the matrices of the form M M for M ∈ Φ, and Γ as in (5.1).
We conclude by giving the proof of our main result.
Proof of Theorem 1.3. We proceed by induction on n. When n = 0 the statement is trivial, and so we assume that n ≥ 1. Let F ∈ C[x ±1 , z ±1 ] and write F = CF with C ∈ C[x ±1 ] and F ∈ C[x ±1 , z ±1 ] without nontrivial factors in C[x ±1 ]. By Lemma 4.3(2), there is a nite subset ∆ ⊂ Z n such that C(t b ) = 0 for all b ∈ Z n with c, b = 0 for all c ∈ ∆. Let also Ω 0 ⊂ Z n×n and Γ ∈ Z n be the nite subsets given by Theorem 5.1 applied to F . Let a ∈ Z n . When c, a = 0 for all c ∈ Γ ∪ ∆, Theorem 5.1(2) implies the statement, provided that we choose any nite subset Ω ⊂ Z n×n containing Ω 0 .
Otherwise, suppose that there is c ∈ Γ ∪ ∆ with c, a = 0. If C(t a , z) = 0, we add to the nite set Ω the matrix M ∈ Z n×n made by adding to n − 1 zero columns to the vector a. Otherwise, choose a matrix L ∈ Z n×(n−1) dening a linear map Z n−1 → Z n whose image is the submodule c ⊥ ∩ Z n , and a vector d ∈ Z n−1 with a = Ld. Let u = (u 1 , . . . , u n−1 ) be a set of n − 1 variables and set By the inductive hypothesis, there is a nite subset Ω c ⊂ Z (n−1)×(n−1) satisfying the statement of Theorem 1.3 applied to this Laurent polynomial. In particular, there are N ∈ Ω c and e ∈ Z n−1 with d = N e such that, for an irreducible factor Q of G(u N , z), we have that Q(t e , z) is, as a Laurent polynomial in C(t)[z ±1 ], either a unit or an irreducible factor of G(t d , z).
We have that G(u N , z) = F (u LN , z), and so Q is an irreducible factor of this latter Laurent polynomial. Moreover, a = LN e. Enlarging the matrix LN ∈ Z n×(n−1) to a matrix M ∈ Z n×n by adding to it a zero column at the end, and similarly enlarging the vector e to a vector b ∈ Z n by adding to it a zero entry at the end, the previous equalities are preserved with M and b in the place of LN and e. Hence, a = M b and, if Q(x 1 , . . . , x n−1 ) is an irreducible factor of F (x M , z), then Q(t e , z) = Q(t b , z) is, as a Laurent polynomial in C(t)[z ±1 ], either a unit or an irreducible factor of G(t d , z) = F (t a , z).
The statement then follows by also also adding to Ω all the matrices M ∈ Z n×n constructed in this way.
Remark 5.2. In the setting of Theorem 1.3, the bivariate Laurent polynomials F a can be dened as the pullback of the multivariate Laurent polynomial F by the 2-parameter monomial map (t, z) → (t, z) A given by the matrix In Conjecture 1.5 applied to F and k = 2, one can consider all matrices in Z (n+1)×2 , and so its setting is more general than that of Theorem 1.3.
On the other hand, the conclusion of Conjecture 1.5 in this situation is slightly weaker than that of Theorem 1.3, since it does not give the irreducible factorization of the F a in C(t)[z ±1 ], but rather its irreducible factorization modulo the Laurent polynomials of the form f (t d 1 z d 2 ) for a univariate f and d 1 , d 2 ∈ Z.