1 Introduction

A system of n polynomial equations in n variables with indeterminate coefficients is said to be reduced and irreducible, if it cannot be simplified by monomial changes of variables (see below for a formal definition). It was proved in [E18] that the monodromy of such system is the symmetric group, and conjectured, that the monodromy of a non-reduced system equals a certain wreath product of the symmetric group. The aim of the present paper is to verify this conjecture.

We identify a large natural class of systems of equations, for which the conjecture holds. However, for n>1, we also identify a large class of counterexamples. In particular, we completely characterize for what systems the conjecture holds, under the additional assumption that the Newton polytopes of the equations are equal (or homothetic).

Our results are based on the new technique that we call “inductive irreducibility of solution spaces”. We hope it may also prove useful in the context of the Galois theory for other problems of enumerative geometry.

Inductive irreducibility of solution spaces and systems of equations whose Galois group is a wreath product

A. Esterov, L. Lang

1 Introduction

Identifying a point a=(a_{1},\ldots,a_{n})=\mathbb{Z}^{n} with the monomial x^{a}=x_{1}^{a_{1}}\ldots x_{n}^{a_{n}}, every finite set P\subset\mathbb{Z}^{n} gives rise to the vector space of Laurent polynomials \mathbb{C}^{P}=\{\sum_{a\in P}c_{a}x^{a}\} supported at P. Every such polynomial defines a function on the complex torus T:=({\mathbb{C}}\setminus 0)^{n}.

For a tuple of finite sets A=(A_{1},\ldots,A_{n}),\,A_{i}\subset\mathbb{Z}^{n}, a tuple f=(f_{1},\ldots,f_{n}) from the space \mathbb{C}^{A}:=\mathbb{C}^{A_{1}}\oplus\ldots\oplus\mathbb{C}^{A_{n}} can be regarded as a system of polynomial equations f=0. According to the Kouchnirenko–Bernstein theorem [Be75, Theorem A], the number d of solutions of such system in its domain T equals the mixed volume of the convex hulls of A_{1},\ldots,A_{n}, unless the system belongs to a certain proper Zariski closed subset D\subset\mathbb{C}^{A}, called the bifurcation set.

As f travels along a loop S^{1}\to\mathbb{C}^{A}\setminus D, the roots of the system f=0 travel continuously and come back to their original position, up to a permutation. The group of all such permutations will be denoted by G_{A}\subset S_{d} and called the monodromy group (or the Galois group) of the system of equations with indeterminate coefficients supported at A. According to [E18, Theorem 1.5], this group equals the symmetric group S_{d} if A is reduced and irreducible in the following sense.

         Definition 1.1.

A tuple of finite sets A=(A_{1},\ldots,A_{n}),\,A_{i}\subset\mathbb{Z}^{n}, (and the corresponding system of equations with indeterminate coefficients) is said to be non-reduced, if all sets can be shifted to the same proper sublattice, and reducible, if k of them can be shifted to a rank k sublattice for some k<n.

         Example 1.2.

The equation c_{8}x^{8}+c_{4}x^{4}+c_{0}=0 supported at \{0,4,8\} is non-reduced. The system f(x)=g(x,y)=0 is reducible.

         Remark 1.3.

1. In this paper we always assume without loss of generality that 0\in A_{i} for i=1,\ldots,n, because otherwise we can divide every equation of our system by a certain monomial so that the resulting system satisfies this assumption. Since dividing by a monomial does not affect the roots of the system in the complex torus, the resulting system has the same monodromy as the initial one.

2. Under this assumption, we can interpret non-reduced systems as systems that can be simplified by a monomial change of variables, and reducible systems as systems that have a proper square subsystem of equations upon an appropriate monomial change of coordinates, as in the preceding example.

Let \tilde{A}=(\tilde{A}_{1},\ldots,\tilde{A}_{n}) be a non-reduced irreducible tuple, then it has the form \tilde{A}_{i}=L(A_{i}) for some proper linear embedding L:\mathbb{Z}^{n}\to\mathbb{Z}^{n} and some reduced irreducible tuple A=(A_{1},\ldots,A_{n}), which we call a reduction of \tilde{A}. The mixed volumes of the two tuples \tilde{d} and d are related by the equality \tilde{d}=|\mathop{\rm coker}\nolimits L|\cdot d. Although G_{A}=S_{d}, the inclusion G_{\tilde{A}}\subset S_{\tilde{d}} is obviously proper, because G_{\tilde{A}} is imprimitive. It was conjectured in [E18] that the monodromy group G_{\tilde{A}} equals the wreath product of {\mathop{\rm coker}\nolimits}L and S_{d}. Recall its definition.

         Definition 1.4.

The wreath product H\wr S_{d} of a group H and S_{d} is the semidirect product of H^{d} and S_{d} with respect to the natural action of S_{d} on the product H^{d}, permuting the factors. In other words, it is the group of all permutations \sigma of the set H\times\{1,\ldots,d\} such that \forall\,i\in\{1,\ldots,d\},\;\exists\,j\in\{1,\ldots,d\},\;\exists\,h\in H\;:% \;\sigma(\bullet,i)=(h\cdot\bullet,j).

Our main result on the above conjecture requires the following notation. For a linear function \gamma:\mathbb{Z}^{n}\to\mathbb{Z} and a finite set P\subset\mathbb{Z}^{n}, let P^{\gamma}\subset P be the set of all points where \gamma attains its maximal value on P.

         Definition 1.5.

We say that the sets A_{1},\ldots,A_{n} are analogous if, for every \gamma\in\mathbb{Z}^{n}, the respective minimal affine spaces containing A_{1}^{\gamma},\ldots,A_{n}^{\gamma} are all shifted copies of the same vector subspace V_{\gamma}\subset\mathbb{R}^{n}.

         Example 1.6.

If the convex hulls of A_{1},\ldots,A_{n} are equal, or more generally homothetic, then A_{1},\ldots,A_{n} are analogous.

         Remark 1.7.

A tuple of analogous sets is always irreducible. A tuple is analogous if and only if its reduction is analogous.

Let \mathcal{G}_{A}\subset(\mathbb{Z}^{n})^{*} be the (finite) set of all primitive \gamma such that V_{\gamma} is a hyperplane, and let d_{\gamma} be the index in V_{\gamma} of the minimal sublattice to which each of A_{1}^{\gamma},\ldots,A_{n}^{\gamma} can be shifted.

         Definition 1.8.

An analogous tuple \tilde{A} is ample if, for a reduction A,\,\tilde{A}=L(A),\,L:\mathbb{Z}^{n}\to\mathbb{Z}^{n}, the vectors d_{\gamma}\cdot\gamma,\,\gamma\in\mathcal{G}_{A}, together with the lattice L^{*}(\mathbb{Z}^{n}) generate \mathbb{Z}^{n} (here L^{*} is the lattice embedding dual to L).

In particular, a reduced analogous tuple A is always ample, because L^{*}(\mathbb{Z}^{n})=\mathbb{Z}^{n}.

         Theorem 1.9.

Let \Lambda be the sublattice generated by the sets \tilde{A}_{1},\ldots,\tilde{A}_{n}\subset\mathbb{Z}^{n} containing 0, let \tilde{d} be the mixed volume of the convex hulls of these sets, and let d=\tilde{d}/|\mathbb{Z}^{n}/\Lambda|.

1. Assume that \tilde{A}_{1},\ldots,\tilde{A}_{n} are analogous. Then the monodromy G_{\tilde{A}} of the system of equations with indeterminate coefficients supported at \tilde{A} equals (\mathbb{Z}^{n}/\Lambda)\wr S_{d} if \tilde{A} is ample, and is strictly smaller otherwise.

2. Assume that \tilde{A} has a reduction (A_{1},\ldots,A_{n}) such that every A_{i} is contained in the positive quadrant \mathbb{Z}^{n}_{\geqslant 0} and contains the vertices of the standard simplex (i.e. \mathbb{C}^{A_{i}} consists of non-Laurent polynomials and contains the space of affine linear functions). Then the monodromy G_{\tilde{A}} of the system of equations with indeterminate coefficients supported at \tilde{A} equals (\mathbb{Z}^{n}/\Lambda)\wr S_{d}.

The above theorem rely on the notions of inductive irreducibility and solution spaces that we introduce in Section 2 and 3 respectively. In Section 4, we study the inductive irreducibility of solution spaces. This study leads to a more general statement than Theorem 1.9, namely Theorem 4.6. The latter theorem proves that the wreath product conjecture of [E18] fails and holds respectively for many non-analogous tuples. However, there exist non-analogous tuples to which Theorem 4.6 is not applicable. Theorem 1.9 then follows from Theorem 4.6 and Corollary 3.10.

         Example 1.10.

The tuple (P,P) (see the picture below) is not ample, so the monodromy of the tuple (Q,Q) is not the expected wreath product.

This can be seen independently from Theorem 1.9 as follows. The monodromy group consists of permutations of the roots along loops in the set \mathbb{C}^{Q}\times\mathbb{C}^{Q}\setminus D, where \mathbb{C}^{Q}\times\mathbb{C}^{Q} is the space of systems of equations supported at Q\subset\mathbb{Z}^{2}, and D is the bifurcation set (i.e. the closure of the set of all systems with less than 8 isolated roots). Thus the monodromy group is generated by permutations, whose cyclic type is the same as for permutations along small loops around the components D_{i} of the bifurcation set D.

Applying the description of the irreducible components of the bifurcation set (Proposition 1.11/4.10 in the arXiv/journal version of [E11] respectively) to our case, we see that D consists of 5 irreducible components: one component (the discriminant D_{0}) consists of systems with a root of multiplicity 2 (and hence two roots of multiplicity 2, because Q generates an index 2 sublattice in \mathbb{Z}^{2}), and the other four components consist of systems with a root at one of the 4 one-dimensional orbits of the toric variety \mathbb{C}\mathbb{P}^{1}\times\mathbb{C}\mathbb{P}^{1}\supset({\mathbb{C}}% \setminus 0)^{2}. Thus the permutation of roots along a small loop around D_{0} consists of two disjoint transpositions.

Other components D_{i} of D correspond to the edges Q_{i} of the convex hull Q. By the same result from [E11], a generic system of equations from D_{i} has several roots of multiplicity 1 in the complex torus and several roots of multiplicity d at the Q_{i}-orbit of the Q-toric variety, where d is the lattice distance from the line containing Q_{i} to Q\setminus Q_{i}. In our case, d=1 for each of the four edges, so the permutations along small loops around the other four components of D are trivial.

Thus the monodromy group is contained in A_{8}\subset S_{8}, while the wreath product (\mathbb{Z}/2\mathbb{Z})\wr S_{4} is not. Actually, one can manually check that the group G_{Q}\subset S_{8} is the intersection of (\mathbb{Z}/2\mathbb{Z})\wr S_{4} with A_{8}.

         Remark 1.11.

For every analogous tuple A, we can now answer the following question:

\mbox{\it Determine whether the Galois group }G_{A}\mbox{\it equals the % expected wreath product or not.}

Taking this into account, one can distinguish three further key open questions in the study of Galois groups of general systems of polynomial equations.

1) If the answer to the question (*) is negative for an analogous tuple, how to compute the Galois group G_{A} precisely?

2) It is a purely combinatorial, but open and highly non-trivial problem to decide whether the results of this paper actually allow to answer the question (*) for every irreducible tuple (not necessarily analogous). See Remark 4.11 for a precise combinatorial question.

3) We do not see a straightforward way to apply the technique from the present paper to the study of the Galois group for reducible tuples (such that k of the sets can be shifted to the same k-dimensional sublattice).

These questions urge need for new approaches to the topic. One new approach will be presented in an idependent forthcoming paper [BS18].

         Remark 1.12.

We hope that the technique of inductive irreducibility may also prove useful in the context of the Galois theory for other problems of enumerative geometry. This hope comes from the empirical evidence (see for instance [SW13], [Tyom14] or [L19]) that the Galois group of a natural enumerative problem is either S_{n}, A_{n} or it is imprimitive. The latter indicates that the enumerative problem is in a sense a covering over another enumerative problem (the monodromy group of a branched covering f is imprimitive if and only if f non-trivially splits into a composition of two other branched coverings).

Our Theorem 2.1 on inductive irreducibility is intended for the study of such enumerative problems that cover another enumerative problem in a certain strong sense. Having this in mind, we formulate Theorem 2.1 for arbitrary ambient spaces N satisfying the equality H_{1}(N,\mathbb{Z})=\pi_{1}(N) (which holds e.g. for all algebraic groups), rather than just in our current setting N=({\mathbb{C}}\setminus 0)^{n}.

2 Inductive irreducibility

A degree d locally trivial covering of connected spaces \pi:M\to N induces the natural pullback map \pi^{*}:H_{\bullet}(N,\mathbb{Z})\to H_{\bullet}(M,\mathbb{Z}): at the level of chains, it sends every singular simplex to the sum of its d preimages. Accordingly, the element \pi^{*}(\gamma) will be called the preimage of the cycle \gamma\in H_{\bullet}(N,\mathbb{Z}).

         Theorem 2.1.

1) Let \pi:M\to N be a covering of complex algebraic varieties with transitive monodromy action, and let V\subset N be an irreducible subvariety. If the image of H_{1}(V,\mathbb{Z}) in H_{1}(N,\mathbb{Z}) contains an element \gamma whose preimage in H_{1}(M,\mathbb{Z}) is primitive, then U=\pi^{-1}(V) is also irreducible.

2) If moreover M is connected, H_{1}(N,\mathbb{Z})=\pi_{1}(N), and H_{1}(V,\mathbb{Z})+\pi_{*}L generates H_{1}(N,\mathbb{Z}) for some sublattice L\subset H_{1}(M,\mathbb{Z}), then H_{1}(U,\mathbb{Z})+L generates H_{1}(M,\mathbb{Z}).

This motivates the following notion.

         Definition 2.2.

An irreducible algebraic subvariety V of a variety N is said to be L-inductively irreducible (or just inductively irreducible for L=0) for a sublattice L\subset H_{1}(N,\mathbb{Z}), if H_{1}(V,\mathbb{Z})+L generates H_{1}(N,\mathbb{Z}).

Proof.

1) Take y in the smooth part of V, it is enough to prove that the preimages x_{i} of y can be connected through the smooth part of U. Choose a singular 1-cycle c in the smooth part of V passing through y and representing \gamma, then its preimage b is a cycle in the smooth part of U that passes through the x_{i}’s. If the support s of b is connected, the theorem is proved, so assume towards the contradiction that the support s has k>1 connected components s_{j}. This defines the decomposition b=\sum b_{j} with b_{j} supported at s_{j}. Since the components b_{j} are identified with each other by the (transitive) monodromy action of the covering, then the homology cycles \beta_{j} corresponding to b_{j} are equal to the same class \beta, thus k\beta is the homology class of b=\pi^{*}\gamma, which contradicts the primitivity of the latter.

2) It is enough to take an arbitrary loop \alpha in M pointed at x\in U and construct elements in H_{1}(U,\mathbb{Z}) and L whose sum in H_{1}(M,\mathbb{Z}) is represented by the loop \alpha. Since H_{1}(V,\mathbb{Z})+\pi_{*}L generates H_{1}(N,\mathbb{Z}), we can choose a loop \alpha^{\prime} in V pointed at \pi(x) and giving the same element in H_{1}(N,\mathbb{Z}) as \pi_{*}(\alpha)+\pi_{*}(\beta)=\pi_{*}(\alpha+\beta) for some loop \beta in M pointed at x and representing an element of L\subset H_{1}(M,\mathbb{Z}). By the assumption H_{1}(N,\mathbb{Z})=\pi_{1}(N), there exists a homotopy of the loop \pi_{*}(\alpha+\beta) to \alpha^{\prime}. Lifting it, we obtain a homotopy of \alpha+\beta to a certain loop \alpha^{\prime\prime} in U. Thus in H_{1}(M,\mathbb{Z}) we have [\alpha]=[\alpha^{\prime\prime}]-[\beta] with [\alpha^{\prime\prime}]\in H_{1}(U,\mathbb{Z}) and [\beta]\in L. ∎

         Corollary 2.3.

1) If \pi:M\to N is a surjection of complex tori ({\mathbb{C}}\setminus 0)^{n}, and a variety V\subset N is (\pi_{*}L)-inductively irreducible for some sublattice L\subset H_{1}(M,\mathbb{Z})=\mathbb{Z}^{n} (that is, H_{1}(V,\mathbb{Z})+\pi_{*}L=H_{1}(N,\mathbb{Z}), and V is irreducible). Then U:=\pi^{-1}(V) is L-inductively irreducible (that is, H_{1}(U,\mathbb{Z})+L=H_{1}(M,\mathbb{Z}), and U is irreducible).

2) In particular, if H_{1}(V,\mathbb{Z})+\pi_{*}H_{1}(M,\mathbb{Z})=\mathbb{Z}^{n}, then \pi^{-1}(V) is irreducible.

3) Conversely, if H_{1}(V,\mathbb{Z})+\pi_{*}H_{1}(M,\mathbb{Z})\neq\mathbb{Z}^{n}, then \pi^{-1}(V) is reducible.

Proof.

Note first that H_{1}(N,\mathbb{Z})=\pi_{1}(N), because the complex torus is a group, so 2.1.2 is applicable. Denote the standard coordinates in M=({\mathbb{C}}\setminus 0)^{n} by x_{1},\ldots,x_{n}.

We first prove Part 1 in the special case \pi(x_{1},x_{2},\ldots,x_{n})=(x_{1}^{p},x_{2},\ldots,x_{n}) for a prime number p. In this case, \pi^{*}:H_{1}(N,\mathbb{Z})\to H_{1}(M,\mathbb{Z}) multiplies a small circle around the hyperplane x_{i}=0 by 1 for i=1 and by p for other i=2,\ldots,n, and \pi_{*}:H_{1}(M,\mathbb{Z})\to H_{1}(N,\mathbb{Z}) multiplies a small circle around the hyperplane x_{i}=0 by p for i=1 and by 1 for other i=2,\ldots,n. In particular, elements \gamma\in H_{1}(V,\mathbb{Z}) satisfying the condition of Theorem 2.1.1 are exactly those not contained in \pi_{*}H_{1}(M,\mathbb{Z}). Thus, such \gamma exists, otherwise H_{1}(V,\mathbb{Z})\subset\pi_{*}H_{1}(M,\mathbb{Z}), and we would have H_{1}(V,\mathbb{Z})+\pi_{*}L\subset H_{1}(V,\mathbb{Z})+\pi_{*}H_{1}(M,\mathbb% {Z})=\pi_{*}H_{1}(M,\mathbb{Z})\neq\pi_{*}H_{1}(N,\mathbb{Z}). Since Theorem 2.1 is applicable, it implies Part 1 for \pi(x_{1},x_{2},\ldots,x_{n})=(x_{1}^{p},x_{2},\ldots,x_{n}).

Part 1 for an arbitrary \pi reduces to this special case, because an arbitrary \pi can be decomposed into a composition of cyclic covers. The statement now follows from the case \pi(x_{1},x_{2},\ldots,x_{n})=(x_{1}^{p},x_{2},\ldots,x_{n}) by induction on the length of the decomposition (hence the name “inductive irreducibility”).

Part 2 is a special case of Part 2 with L=H_{1}(M,\mathbb{Z}).

In the setting of Part 3, there is no loss of generality in assuming (up to a monomial change of coordinates) that H_{1}(V,\mathbb{Z})+\pi_{*}H_{1}(M,\mathbb{Z}) is contained in p\mathbb{Z}\oplus\mathbb{Z}^{n-1} for some prime p>1. As the map \pi is given by characters, it implies in particular that \pi factors through \tilde{\pi}(x_{1},x_{2},\ldots,x_{n}):=(x_{1}^{p},x_{2},\ldots,x_{n}).

Assuming to the contradiction that \tilde{U}:=\tilde{\pi}^{-1}(V) is irreducible, we can connect two of the preimages of a smooth point y\in V through the smooth part of \tilde{U} with a path \gamma. Then the loop \tilde{\pi}(\gamma) represents a cycle in H_{1}(V,\mathbb{Z}) outside \tilde{\pi}_{*}H_{1}(N,\mathbb{Z}) whose first coordinate is not divisible by p. This is a contradiction. Thus \tilde{\pi}^{-1}(V) is reducible, and so is \pi^{-1}(V), as \pi factors through \tilde{\pi}. ∎

3 The tautological bundle and the solution space

The tautological bundle and its dominant components. Consider the complex torus T:=({\mathbb{C}}\setminus 0)^{n} and define T_{k} to be the set of (ordered) tuples of k distinct points in T. Let B=(B_{1},\ldots,B_{n}) be a tuple of finite sets in the character lattice \mathbb{Z}^{n} of T. Denote the tautological set

\{(x_{1},\ldots,x_{k},f)\,|\,f(x_{1})=\ldots=f(x_{k})=0\}\in T_{k}\times% \mathbb{C}^{B}

by U^{B,k} (or just U^{k}), and its projections to the multipliers by p and \pi respectively. The set U^{k} is not necessarily irreducible.

         Example 3.1.

Let \mathbb{C}^{B_{1}}=\mathbb{C}^{B_{2}} be the space of bivariate quadratic polynomials, then U^{4} is the union of two 12-dimensional sets: the first is U_{1}=\{(x_{1},x_{2},x_{3},x_{4},f_{1},f_{2})\,| all x_{i}’s belong to the same line, and both f_{j}’s vanish on it\}, and the second is the closure of U^{4}\setminus U_{1}.

The decomposition of U^{B,k} into the irreducible components U^{B,k}_{\alpha} (or just U^{k}_{\alpha}) will be of crucial importance for us, because it encodes the sought monodromy.

To study these components, note that every non-empty fiber of the projection p:U^{k}\to T_{k} is a vector space. Thus there exists a stratification T_{k}=\bigsqcup_{S\in\mathcal{S}}S such that the projection p is a vector bundle over every stratum S\in\mathcal{S}.

         Theorem 3.2.

1. The stratification \mathcal{S} can be chosen so that every U^{k}_{\alpha} can be represented as the closure of the vector bundle E_{\alpha}=p^{-1}(S_{\alpha}) over a certain stratum S_{\alpha} in \mathcal{S}.

2. There are two possibilities for every U^{k}_{\alpha}: either the restriction of \pi to U^{k}_{\alpha} has finite degree, or it is not dominant.

         Definition 3.3.

The components of the first kind in the sense of Theorem 3.2.2 will be called dominant components of the tautological set. The closure \Theta\subset T_{k} of the union of the irreducible sets S_{\alpha}, corresponding to the dominant components U^{k}_{\alpha}, will be called the k-solution space of systems supported at B, because it can be informally regarded as the space of k-tuples of roots of generic systems f\in\mathbb{C}^{B}.

         Remark 3.4.

1. The set \Theta is symmetric under permutations, and the study of loops in the quotient \Theta/S_{k} can be regarded as the study of monodromy of the k-tuples of roots of the general system of equations. In order to demonstrate that the monodromy induces a particular permutation of k-tuples of roots, we can do so by constructing a loop in \Theta/S_{k} instead of constructing a particular 1-parametric family of systems f\in\mathbb{C}^{B}\setminus D.

2. When we wish to specify for which tuple B and number k we consider the tautological set U^{k}, its components U^{k}_{\alpha}, the corresponding strata S_{\alpha} and the solution space \Theta, we denote them by U^{B,k}, U^{B,k}_{\alpha}, S^{B,k}_{\alpha} and \Theta^{B,k} respectively.

3. The component U_{2} in Example 3.1 is dominant, while the component U_{1} is not.

The proof of Theorem 3.2 is based on the following observation: by the Kouchnirenko–Bernstein Theorem, the projection \pi:U^{1}\to\mathbb{C}^{B} is a locally trivial covering of the expected degree outside the bifurcation set D\subset\mathbb{C}^{B} of all systems that are degenerate in the sense of Kouchnirenko–Bernstein, see [Be75]. More precisely,

U^{1}\setminus\pi^{-1}(D)\to\mathbb{C}^{B}\setminus D

is a covering, and its degree equals the mixed volume of the convex hulls of B_{1},\ldots,B_{n}, which we denote by V(B). Then the projection \pi:U^{k}\to\mathbb{C}^{B} is also a locally trivial covering of the expected degree outside D\subset\mathbb{C}^{B}, i.e. U^{k}\setminus\pi^{-1}(D)\to\mathbb{C}^{B}\setminus D is a covering of degree \frac{V(B)!}{(V(B)-k)!}, the number of k-tuples of roots of a generic system f\in\mathbb{C}^{B}. The total space of this covering will be denoted by C and called the tautological covering.

Proof.

1. Since every fiber of the projection p restricted to U^{k}_{\alpha} is a vector space, the image of this restriction has a Zariski open subset S_{\alpha} over which p is a vector bundle E_{\alpha}\to S_{\alpha}. The closure of E_{\alpha} is by construction an irreducible set, contained in U^{k} and containing its irreducible component U^{k}_{\alpha}, thus the closure equals U^{k}_{\alpha}. Obviously, S_{\alpha}’s can be chosen to be pieces of the same stratification of T_{k}.

2. If U^{k}_{\alpha} intersects the tautological covering C, then it has finite degree (at most V(B)!) over \mathbb{C}^{B}, otherwise its image is contained in the (proper Zariski closed) bifurcation set D\subset\mathbb{C}^{B}. ∎

         Definition 3.5.

For any tuple B:=(B_{1},\ldots,B_{n}) of finite subset of \mathbb{Z}^{n}, define G_{B} to be the monodromy group of a general system of polynomial equations supported at B, i.e. the monodromy of the covering (\star).

By definition, G_{B} is a subgroup of S_{V(B)}, the group of all permutations of V(B) elements.

         Theorem 3.6.

The monodromy group G_{B} of the general system f\in\mathbb{C}^{B} equals S_{V(B)} if and only if the tautological set U^{V(B)} has a unique dominant component.

Proof.

The dominant component of U^{V(B)} is unique if and only if the total space of the tautological covering C is connected, if and only if the monodromy of the general system f\in\mathbb{C}^{B} equals S_{V(B)}. ∎

The wreath tautological bundle. Consider a reduced irreducible tuple A and denote d:=V(A). Since the monodromy of the general system supported at A equals S_{d} by [E18, Theorem 1.5], Theorem 3.6 implies the following

         Corollary 3.7.

1) The tautological set U^{A,d} has only one dominant component.

2) The solution space \Theta^{A,d} is irreducible.

On the other hand, if the embedding L:\mathbb{Z}^{n}\to\mathbb{Z}^{n} is proper, then the preceding theorem is not applicable to the tuple \tilde{A}=L(A), because its monodromy group G_{\tilde{A}} is at most the wreath product W of {\rm coker}L and S_{d} (see Definition 1.4).

In order to prove that G_{\tilde{A}}=W, we need the following generalization of Theorem 3.6. The embedding L:\mathbb{Z}^{n}\to\mathbb{Z}^{n} induces an epimorphism of tori L^{*}:T\to T. Denote the preimage of T_{d} under the corresponding epimorphism T^{d}\to T^{d} by \tilde{T}_{d}\subset T_{d}\subset T^{d}, and the restriction of the epimorphism to the latter by \tilde{L}:\tilde{T}_{d}\to T_{d}. The intersection of the tautological set U^{\tilde{A},d}\subset T_{d}\times\mathbb{C}^{\tilde{A}} with \tilde{T}_{d}\times\mathbb{C}^{\tilde{A}} will be denoted by \tilde{U}^{\tilde{A},d} (or just \tilde{U}) and called the wreath tautological space. Note that it coincides with the preimage of U^{A,d} under the map (\tilde{L},\,{\rm Id}):\tilde{T}_{d}\times\mathbb{C}^{\tilde{A}}\to T_{d}% \times\mathbb{C}^{A}, and its dominant components are exactly the irreducible components of the preimage of the dominant components of U^{A,d}. Since the latter is unique by Corollary 3.7.1, we have the following.

         Corollary 3.8.

The images of all dominant components of the wreath tautological set by the projection to \tilde{T}_{d} have the same dimension.

Similarly, the intersection of the solution space \Theta^{\tilde{A},d}\subset T_{d} with \tilde{T}_{d} will be denoted by \tilde{\Theta}^{\tilde{A},d} (or just \tilde{\Theta}) and called the wreath solution space. The following is the counterpart to Theorem 3.6 in the non-reduced case.

         Theorem 3.9.

The monodromy group G_{\tilde{A}} equals {\rm coker}L\wr S_{d} if and only if the wreath tautological set U^{V(B)} has a unique dominant component if and only if the wreath solution space \tilde{\Theta} is irreducible.

For the proof, note that the projection of \tilde{U} to \mathbb{C}^{\tilde{A}} is a locally trivial covering outside the same bifurcation set as for the projection of U^{A,d} to \mathbb{C}^{A}. We shall call it the wreath tautological covering and denote it by \tilde{C}.

Proof.

Since the wreath solution space \tilde{\Theta} is the union of the images of the dominant components of \tilde{U} under the projection to \tilde{T}_{d}, and since these components have the same dimension by Corollary 3.8, the wreath solution space \tilde{\Theta} is irreducible if and only if there is only one dominant component of \tilde{U}. This is equivalent to the fact that the total space of the wreath tautological covering \tilde{C} is connected, which, in its turn, is equivalent to the fact that the monodromy of the general system f\in\mathbb{C}^{\tilde{A}} equals the wreath product W. ∎

This theorem allows to reinterpret the sought equality G_{\tilde{A}}=W in terms of the inductive irreducibility of the solution space \Theta^{A,d} of the reduced irreducible tuple A. Denote by \mathcal{L} the image of the map (\mathbb{Z}^{n})^{d}\to(\mathbb{Z}^{n})^{d} dual to (L,\ldots,L):(\mathbb{Z}^{n})^{d}\to(\mathbb{Z}^{n})^{d}.

         Corollary 3.10.

1) If \Theta^{A,d} is \mathcal{L}-inductively irreducible (that is, H_{1}(({\mathbb{C}}\setminus 0)^{n\cdot d},\mathbb{Z})=H_{1}(\Theta^{A,d},% \mathbb{Z})+\mathcal{L}), then \tilde{\Theta}=\tilde{L}^{-1}(\Theta^{A,d}) is irreducible, and thus G_{\tilde{A}}=W.

2) If \Theta^{A,d} is not \mathcal{L}-inductively irreducible, then, for some L, \tilde{\Theta}=\tilde{L}^{-1}(\Theta^{A,d}) is reducible, and thus G_{\tilde{A}}\neq W.

This is a consequence of Corollary 2.3, Corollary 3.7.2 and Theorem 3.9.

4 Inductive irreducibility of solution spaces.

In this section, the tuple A=(A_{1},\ldots,A_{n}) is reduced and irreducible, see Definition 1.1.

Resultants. The lattice \mathbb{Z}^{n} is considered as the lattice of monomials on a complex torus ({\mathbb{C}}\setminus 0)^{n}, so that the first homology group H=H_{1}(({\mathbb{C}}\setminus 0)^{n},\mathbb{Z}) is a dual lattice to \mathbb{Z}^{n}: the composition of a loop S^{1}\to({\mathbb{C}}\setminus 0)^{n}, representing a cycle \gamma\in H, and a monomial m:({\mathbb{C}}\setminus 0)^{n}\to({\mathbb{C}}\setminus 0)^{1} is a map S^{1}\to({\mathbb{C}}\setminus 0)^{1}, and its class d\in\pi_{1}({\mathbb{C}}\setminus 0)^{1}=\mathbb{Z} defines the natural non-degenerate pairing \cdot\,:\,H\times\mathbb{Z}^{n}\to\mathbb{Z},\,(\gamma,m)\mapsto\gamma\cdot m=d.

Let \mathcal{G}\subset H be the set of primitive exterior normal vectors to the facets of the convex hull of A_{1}+\ldots+A_{n}. This set is finite, and every \gamma\in\mathcal{G} considered as a linear function on \mathbb{Z}^{n} attains its maximum on A_{i} at some subset that we denote by A^{\gamma}_{i}. For short, the tuples (A_{1},\ldots,A_{n}) and (A^{\gamma}_{1},\ldots,A^{\gamma}_{n}) will be denoted by A and A^{\gamma} respectively, and the spaces of systems of equations supported at these tuples \mathbb{C}^{A_{1}}\oplus\ldots\oplus\mathbb{C}^{A_{n}} and \mathbb{C}^{A^{\gamma}_{1}}\oplus\ldots\oplus\mathbb{C}^{A^{\gamma}_{n}} – by \mathbb{C}^{A} and \mathbb{C}^{A^{\gamma}}.

The reduced resultant R^{red}_{A^{\gamma}} is the closure of the set of all tuples g=(g_{1},\ldots,g_{n})\in\mathbb{C}^{A^{\gamma}} such that the system g_{1}(x)=\ldots=g_{n}(x)=0 has a root x\in({\mathbb{C}}\setminus 0)^{n}. All A^{\gamma}_{i} by definition can be shifted to the hyperplane \ker\gamma (where \gamma is considered as a linear function on \mathbb{Z}^{n}), so the set of solutions of the system g=0 is invariant under the action of the 1-dimensional subtorus T_{\gamma}\subset({\mathbb{C}}\setminus 0)^{n} whose homology embeds in H as \mathbb{Z}\cdot\gamma.

For a generic tuple g\in R^{red}_{A^{\gamma}}, the quotient \{g=0\}/T_{\gamma} is a finite set, whose cardinality will be denoted by d_{\gamma}. This number should be regarded as a natural multiplicity of the resultant R^{red}_{A^{\gamma}} and will be explicitly computed in Theorem 4.3 below, which requires the following notation.

If k sets in the tuple A^{\gamma} can be shifted to the same (k-2)-dimensional plane, then the resultant R^{red}_{A^{\gamma}} is not a hypersurface, and d_{\gamma} is set to be 0 by convention. Otherwise, there exists a unique (inclusion-wise) minimal subset K\subset\{1,\ldots,n\} such that the sets A^{\gamma}_{i},\,i\in K, can be shifted to the same (|K|-1)-dimensional sublattice, and the minimal such sublattice is denoted by L_{\gamma}\subset\mathbb{Z}^{n}.

In this case, the tuple (B_{1},\ldots,B_{n}) such that B_{k}=\varnothing for k\notin K and B_{k}=A^{\gamma}_{k} otherwise, is said to be the essential tuple defined by \gamma. Note that different \gamma\in\mathcal{G} may give the same essential tuple.

         Example 4.1.

For A_{1} and A_{2} as on the picture below, both (-1,0) and (0,-1) belong to \mathcal{G} and give the same essential tuple \left(\{(0,0)\},\varnothing\right).

         Notation.

We denote by A^{\gamma}_{ess} the essential tuple defined by \gamma\in\mathcal{G}, denote the set of all essential tuples by \mathcal{E}, and the set of maximal-dimensional essential tuples by \mathcal{E}_{0}\subset\mathcal{E} (we say that a tuple is maximal-dimensional if the convex hull of every its element is (n-1)-dimensional).

         Remark 4.2.

1. The map \mathcal{G}\to\mathcal{E},\,\gamma\mapsto A^{\gamma}_{ess}, is one to one over \mathcal{E}_{0}, i.e. every essential tuple from B\in\mathcal{E}_{0} is defined by a unique \gamma\in\mathcal{G}, which we denote by \gamma_{B}.

2. If A_{1},\ldots,A_{n} are analogous (Definition 1.5), then \mathcal{E}_{0}=\mathcal{E}, and A^{\gamma}_{ess}=A^{\gamma}. In particular, \mathcal{G} and \mathcal{E} are in one to one correspondence.

Let d^{\prime}_{\gamma} be the index of the lattice L_{\gamma} in its saturation \bar{L}_{\gamma}. The images of the sets A^{\gamma}_{i},\,i\notin K, under the projection \mathbb{Z}^{n}\to\mathbb{Z}^{n}/\bar{L}_{\gamma} are n-|K|+1 sets in a lattice of the same dimension. Thus the lattice mixed volume of the convex hulls of these images makes sense and is denoted by d^{\prime\prime}_{\gamma}.

         Theorem 4.3 (Proposition 3.5 in [E08]).

The number of solutions d_{\gamma} of a generic consistent system g=0 supported at A^{\gamma} (i.e. the system given by a generic g\in R^{red}_{A^{\gamma}}) equals d^{\prime}_{\gamma}\cdot d^{\prime\prime}_{\gamma}.

         Definition 4.4 ([E08]).

The algebraic resultant of the general system of equations supported at A^{\gamma}, denoted by R_{A^{\gamma}}, is defined as F^{d_{\gamma}}, where F is the equation of the hypersurface R^{red}_{A^{\gamma}} (if the latter is not a hypersurface, then we set R_{A^{\gamma}}=1 by convention). By a harmless abuse of notation, we denote the lift of the polynomial R_{A^{\gamma}} under the natural forgetful projection \mathbb{C}^{A}\to\mathbb{C}^{A^{\gamma}} by the same letter R_{A^{\gamma}}.

The main result. Let \mathcal{R} be the the set of all sets of the form R_{A^{\gamma}}=0 in \mathbb{C}^{A}.

         Remark 4.5.

By the construction, two such equations R_{A^{\gamma}}=0 and R_{A^{\gamma^{\prime}}}=0 define the same set if and only if \gamma and \gamma^{\prime} define the same essential tuple. Thus, \mathcal{R} is in one to one correspondence with the set of essential tuples \mathcal{E} (and, by Remark 4.2, also with \mathcal{G} if A is analogous).

For B\in\mathcal{E}, let R^{red}_{B}\in\mathcal{R} be the corresponding resultant set in \mathbb{C}^{A}, and let \mathcal{G}_{B} be the set of all \gamma\in\mathcal{G} defining this essential tuple. Choose an arbitrary sublattice L\subset\mathbb{Z}^{n}, and denote L\oplus\ldots\oplus L by \mathcal{L}.

         Theorem 4.6.

1) Assume that a tuple A=(A_{1},\ldots,A_{n}) is reduced and irreducible, and the vectors \sum_{\gamma\in\mathcal{G}_{B}}d_{\gamma}\cdot\gamma over all B\in\mathcal{E} together with L do not generate the lattice \mathbb{Z}^{n}. Then the solution space of A is not \mathcal{L}-inductively irreducible.

2) Assume that A is reduced and irreducible, and the vectors d_{\gamma_{B}}\cdot\gamma_{B} over all B\in\mathcal{E}_{0} together with L generate the lattice \mathbb{Z}^{n}. Then the solution space of A is \mathcal{L}-inductively irreducible.

         Remark 4.7.

1. Under the assumption of Part (2), Remark 4.2 assures that the set of vectors from Part (1) coincides with that of Part (2), i.e. \sum_{\gamma\in\mathcal{G}_{B}}d_{\gamma}\cdot\gamma=d_{\gamma_{B}}\cdot\gamma% _{B}. Thus the theorem completely characterizes analogous tuples with inductively irreducible solution spaces.

2. In particular, we have the following three increasing classes of tuples, which coincide for analogous tuples:

– Tuples A, such that the vectors d_{\gamma_{B}}\cdot{\gamma_{B}} over all B\in\mathcal{E}_{0}, generate the lattice;

– Tuples with inductively irreducible solution spaces;

– Tuples A, such that the vectors \sum_{\gamma\in\mathcal{G}_{B}}d_{\gamma}\cdot\gamma over all B\in\mathcal{E} generate the lattice.

We expect that, for general reduced irreducible tuples, the second of these classes is strictly larger than the first one. Regarding the comparison of the latter two classes, see the subsequent Remark 4.11.

Proving inductive irreducibility. In order to prove inductive irreducibility, we should construct loops in the solution space \Theta\subset T^{d} that would generate the lattice H_{1}(T^{d},\mathbb{Z})=H^{\oplus d} together with the lattice \mathcal{L} (see Section 3 for \Theta,d,T and other notation related to solution spaces). More specifically, if we find a loop \alpha in the space of systems of equations \mathbb{C}^{A} such that the roots permute trivially along this loop, then, choosing an arbitrary ordering \sigma:\{roots\}\leftrightarrow\{1,\ldots,d\}, the i-th root travels a loop, representing a certain element \tilde{\alpha}_{\sigma,i}\in H. In this notation, the corresponding loop in the solution space will represent an element \tilde{\alpha}_{\sigma}=(\tilde{\alpha}_{\sigma,1},\ldots,\tilde{\alpha}_{% \sigma,d})\in H^{\oplus d}. We aim at constructing enough loops \alpha to generate H^{\oplus d}/\mathcal{L} with the respective elements \tilde{\alpha}_{\sigma}.

In this respect, the following obvious combinatorial fact will be useful.

         Definition 4.8.

For an element u=(u_{1},\ldots,u_{d})\in H^{\oplus d}, the sum u_{1}+\ldots+u_{d}\in H is denoted by \sum u. The element u is said to be homogeneous if all its non-zero entries are equal to each other.

         Lemma 4.9.

I. Let U\subset H^{\oplus d} be a subset of homogeneous elements such that the vectors \sum u over all u\in U do not generate H/L. Then the set S_{d}\cdot U (in the sense of the natural action of the permutation group S_{d} on the d direct summands of H^{\oplus d}) does not generate the space H^{\oplus d}/\mathcal{L}.

II. Assuming almost the opposite, let U\subset H^{\oplus d} be a subset of homogeneous elements satisfying the following:

1) for any u=(u_{1},\ldots,u_{d})\in U with a non-zero entry u_{i}, there exists \tilde{u}=(\tilde{u}_{1},\ldots,\tilde{u}_{d})\in U

such that \tilde{u}_{j}=u_{i} and \tilde{u}_{k}=0 for some indices j and k, and

2) the vectors \sum u over all u\in U generate H/L.
In this case, the set S_{d}\cdot U generates the space H^{\oplus d}/\mathcal{L}.

Proof.

In the setting of part I, the map \sum sends S_{d}\cdot U and \mathcal{L} to the proper sublattice of H generated by \sum u,\,u\in U, and L, thus S_{d}\cdot U and \mathcal{L} also generate a proper sublattice of H^{\oplus d}.

In the setting of Part II, the assumption 1) allows to obtain, starting from a homogeneous element u\in U with the nonzero entry \delta, the element in S_{d}\cdot U of the form (\delta,\ldots,\delta,0,\ldots,0) with at least one zero, then the element (\delta,\ldots,\delta,0,\delta,0\ldots,0) with the same number of zeroes, then, by permuting the difference of the preceding two vectors, the element (0,\ldots,0,\delta,0,\ldots,0,-\delta,0,\ldots,0) with \delta and -\delta at arbitrary positions, and finally, adding such elements to the initial u, the vector (0,\ldots,0,\sum u,0,\ldots,0) with \sum u at an arbitrary position. By the assumption 2), such vectors together with \mathcal{L} generate H^{\oplus d}. ∎

Note that Part II does not hold without the assumption 1. Eventually by this reason we shall need the following elementary geometric fact.

         Lemma 4.10.

Under the assumptions of Theorem 4.6.2, assume that, for some B\in\mathcal{E}_{0} and the corresponding \gamma=\gamma_{B}, every j=1,\ldots,n and every point a\in A_{j}\setminus A_{j}^{\gamma}, we have V(A)=h_{a}\cdot d_{\gamma}, where h_{a}=|\gamma(a)-\gamma(A_{j}^{\gamma})| is the lattice distance from a to the hyperplane A_{j}^{\gamma}+\ker\gamma. Then \mathop{\rm MV}\nolimits(A)=1 (in particular, all A_{i} are equal to subsets of vertices of the same elementary lattice simplex up to a shift, see [EG12]).

Proof.

By monotonicity of the mixed volume, we have \mathop{\rm MV}\nolimits(A)\geqslant\mathop{\rm MV}\nolimits\big{(}a\cup B_{j}% ,\left\{B_{i}\right\}_{i\neq j}\big{)}=h_{a}\mathop{\rm MV}\nolimits_{j}, where \mathop{\rm MV}\nolimits_{j} is the (n-1)-dimensional lattice mixed volume of the convex hulls of B_{i}=A_{i}^{\gamma},\,i\neq j. Since all of these convex hulls are (n-1)-dimensional (by definition of \mathcal{E}_{0}), the mixed volume \mathop{\rm MV}\nolimits_{j} is a positive multiple of d_{\gamma}. Thus, for every j and a, we have \mathop{\rm MV}\nolimits_{j}=d_{\gamma}, and h_{a}=h does not depend on a (and j). In particular, in the minimal lattice L containing every B_{i} up to a shift, there exists an elementary simplex S containing every B_{i} up to a shift. Thus, up to a shift, every A_{i} consists of the vertices of S and some points at the lattice distance h from L (on the same side from it). In this case, the equality \mathop{\rm MV}\nolimits(A)=h_{a}\cdot d_{\gamma}=h\cdot\mathop{\rm Vol}\nolimits S implies the existence of a point a_{0} such that every A_{i} consists of the vertices of S and possibly a_{0}. This implies \mathop{\rm Vol}\nolimits S=h=1, otherwise A is not reduced.∎

Proof of Theorem 4.6.2. From the very beginning, we assume that A does not satisfy the assumption of Lemma 4.10, otherwise by this lemma we have V(A)=1, and the inductive irreducibility is trivial.

According to Lemma 4.9, we just need loops \alpha in \mathbb{C}^{A}\setminus\{bifurcation set\} such that the corresponding elements \tilde{\alpha}_{\sigma}\in H^{\oplus d} are homogeneous, and the vectors \sum\tilde{\alpha}_{\sigma} generate H/L. We construct such loops explicitly as follows, starting from the primitive covector \gamma=\gamma_{B}, corresponding to an arbitrary essential tuple B\in\mathcal{E}_{0}, an arbitrary number j\in\{1,\ldots,n\} and an arbitrary point a\in A_{j}\setminus B_{j}.

I: constructing the system of equation that will be the center of the sought loop. With no loss of generality, we assume that 0\in A^{\gamma}_{i} for every i=1,\ldots,n (otherwise we can shift A_{i} accordingly). We should now make some consecutive choices to construct the sought loop. First, we choose generic tuples g=(g_{1},\ldots,g_{n}) in R^{red}_{B} and \tilde{g}=(\tilde{g}_{1},\ldots,\tilde{g}_{n}) in \mathbb{C}^{A} (which means that subsequently we shall use certain properties of g and \tilde{g} that are satisfied for all pairs (g,\tilde{g})\in R^{red}_{B}\times\mathbb{C}^{A} outside a certain proper Zariski closed subset).

For a given j\in\{1,\ldots,n\} and a\in A_{j}\setminus A_{j}^{\gamma}, define the tuple

f_{j,t,g}(x)=F_{j}(x,t)=x^{a}+g_{j}(x)+t\cdot\tilde{g}_{j}(x)\mbox{ and }f_{i,% t,g}(x)=F_{i}(t,x)=g_{i}(x)+t\cdot\tilde{g}_{i}(x)\mbox{ for }i\neq j,

mostly omitting subscripts t and g in what follows. We call f_{1,0}=\ldots=f_{n,0}=0 the central system of equations, because the theorem will be proved by travelling an appropriate loop around this system.

II: describing the roots of the central system of equations. The polynomials F_{j} are defined on the complex torus \tilde{T}=({\mathbb{C}}\setminus 0)^{n}\times({\mathbb{C}}\setminus 0)^{1} with the standard coordinates (x,t), and their Newton polytopes are contained in the character lattice \mathbb{Z}^{n}\times\mathbb{Z}^{1} of \tilde{T}. Let X be a smooth toric compactification of \tilde{T} compatible with the Newton polytopes of F_{1},\ldots,F_{n}. The fan of this compactification contains the rays generated by the covectors (0,-1) and (\gamma,0)\in(\mathbb{Z}^{n}\times\mathbb{Z}^{1})^{*}, and we may assume with no loss of generality that it contains the simple two-dimensional cone they generate. Let O be the codimension 2 orbit of X corresponding to this two-dimensional cone, and \tilde{O} be its union with the two adjacent codimension 1 orbits (corresponding to the aforementioned rays).

The coordinate function t on the torus \tilde{T} extends to a meromorphic function on X, and the equation t=0 defines a normal crossing divisor supported at the closure of certain codimension 1 orbits O_{m}\subset X. The polynomial F_{i} extends to a section of the line bundle on X corresponding the Newton polytope of F_{i}. Thus the equation F_{i}=0 defines a Cartier divisor on X. Moreover, since we have assumed 0\in B_{i}, the section F_{i} defines a regular function on \tilde{T}\cup\tilde{O}, extending the Laurent polynomial on \tilde{T}.

Note that the toric compactification X is chosen so that the closure of the hypersurface F_{i}=0 in X is smooth for every i, and the intersection of these closures is smooth as well and equals the closure of the curve F_{1}=\ldots=F_{n}=0, see Theorem 2.2 in [Kh77] for details. In particular, the system of equations F_{1}=\ldots=F_{n}=t=0 has finitely many roots x_{m,k} in the orbits O_{m} and d_{\gamma} multiplicity h_{a} roots x_{k} in the codimension 2 orbit O.

III: describing a good neighborhood of a root x_{m,k} of the central system of equations. By the generic choice of g and \tilde{g} in step (I), every root x_{m,k} admits an open neighborhood U_{m,k} with a local analytic coordinate system such that:

t is locally a monomial of the first coordinate;

F_{i}=0 are locally coordinate planes corresponding to the other coordinates;

U_{m,k} does not intersect orbits of X outside \tilde{T}\cup O_{m}.

This implies the existence of neighbourhoods V_{m,k} of g\in\mathbb{C}^{B} and W_{m,k} of 0\in\mathbb{C}^{1} such that, for every loop g_{s},\,s\in S^{1}, in V_{m,k} around the resultant R^{red}_{B} and every non-zero t_{0}\in W_{m,k}, we have the following:

– the system f_{\bullet,t_{0},g_{s}}=0 has the same number of roots in U_{m,k} as the multiplicity of the root x_{m,k} of the system f_{\bullet,0,g}=0;

– as s\in S^{1} travels a loop, the aforementioned roots permute trivially.

In particular, each of the aforementioned roots travels a loop that is contractible in the torus T.

IV: describing a good neighborhood of a root x_{k} of the central system of equations. Similarly, by the generic choice of g and \tilde{g} above and Lemma 4 from [EG14], every root x_{k} and every choice of j^{\prime}\in\{1,\ldots,n\} admit an open neighborhood U_{k} with a local analytic coordinate system such that:

t and F_{i},\,i\neq j^{\prime}, are n of the n+1 coordinate functions;

– If \ell is the coordinate line defined by them and \varphi is the remaining coordinate function, then the restriction of F_{j^{\prime}} to \ell equals \varphi^{h_{a}};

U_{k} does not intersect orbits of X outside \tilde{T}\cup\tilde{O}.

This implies the existence of neighbourhoods V_{k} of g\in\mathbb{C}^{B} and W_{k} of 0\in\mathbb{C}^{1} such that, for every loop g_{s},\,s\in S^{1}, in V_{k} travelling once around the resultant R^{red}_{B} and every non-zero t_{0}\in W_{k}, we have the following:

– the system f_{\bullet,t_{0},g_{s}}=0 has the same number of roots in U_{k} as the multiplicity h_{a} of the root x_{k} of the system f_{\bullet,0,g}=0;

– as s\in S^{1} travels a loop, the aforementioned roots permute in a cycle of length h_{a};

– the paths that the aforementioned roots travel constitute together a loop whose class in the homology H of the torus T equal \gamma.

For instance, thanks to the assumption 0\in A_{i}, we can take the loop g_{s}=g+\varepsilon\exp(2\pi is) for a small generic vector \varepsilon\in\mathbb{C}^{n}.

V: describing the loop. Now, choosing non-zero t_{0}\in\cap_{k}W_{k}\cap_{m,k}W_{m,k} and a loop g_{s},\,s\in S^{1}, in \cap_{k}V_{k}\cap_{m,k}V_{m,k} that travels once around the resultant R^{red}_{B}, and making s travel h_{a} times around S^{1}, we conclude from steps (III) and (IV) that

– the roots of the system f_{\bullet,t_{0},g_{s}}=0 permute trivially;

– among them, d_{\gamma}\cdot h_{a} roots travel along a loop whose class in the homology H of the torus T equal \gamma, and the loops that the other roots travel vanish in H.

We have constructed a loop \alpha=\alpha_{j,a,\gamma} in the space \mathbb{C}^{A} such that the roots permute trivially, the corresponding element \tilde{\alpha}_{\sigma}=\tilde{\alpha}_{\sigma,j,a,\gamma}\in H^{\oplus d} is homogeneous for every choice of \sigma:\{roots of the base system of equations\}\leftrightarrow\{1,\ldots,d\}, and \sum\tilde{\alpha}_{\sigma}\in H equals h\cdot d_{\gamma}\cdot\gamma.

By the construction, the element \tilde{\alpha}_{\sigma,j,a,\gamma} belongs to the homology of the solution space for an appropriate choice of \sigma. Since the monodromy group G_{A} is symmetric by [E18], the same holds for every choice of \sigma.

VI: generating the homology with the constructed loops. Define U\subset H^{\oplus d} to be the set of homogeneous elements \alpha_{\sigma,j,a,\gamma} for some particular choice of \sigma=\sigma_{0}. We now aim to apply Lemma 4.9.II in order to show that the solution space \Theta^{A,d} is \mathcal{L}-inductively irreducible. The condition (1) of Lemma 4.9.II follows from Lemma 4.10. Since the tuple A is reduced, the numbers h=h_{j,a} for all j and a\in A_{j} with a given \gamma are mutually prime, so an appropriate linear combination of the vectors \sum\tilde{\alpha}_{\sigma,j,a,\gamma}=h_{j,a}\cdot d_{\gamma}\cdot\gamma equals d_{\gamma}\cdot\gamma. Since the vectors d_{\gamma}\cdot\gamma over all \gamma\in\mathcal{G} generate H/L by the assumption of Theorem 4.6.2, the subset U also fulfils the condition (2) of Lemma 4.9.II.

Thus, by this lemma, the set S_{d}\cdot U (which consists of the elements \alpha_{\sigma,j,a,\gamma} for all choices of \sigma) generates H^{\oplus d}/\mathcal{L}. As we have proved in the preceding step (V), all elements \alpha_{\sigma,j,a,\gamma} belong to the homology of the solution space, so the result follows. \hfill\Box

         Remark 4.11.

Actually, the construction from the preceding proof is applicable to B\in\mathcal{E} even if B\notin\mathcal{E}_{0}. Choose an arbitrary tuple C=(C_{1},\ldots,C_{n}),\,C_{i}\subset A_{i}, generic \tilde{f}=(\tilde{f}_{1},\ldots,\tilde{f}_{n})\in\mathbb{C}^{C} and a small loop g_{s}\in\mathbb{C}^{B},\,s\in S^{1}, around the resultant R^{red}_{B}. For every linear function \gamma\in\mathcal{G}_{B}, let h_{\gamma} be \max_{i}(\max\gamma|_{A_{i}}-\max\gamma|_{C_{i}}). Set f_{i,t,s}(x)=g_{i,s}(x)+\tilde{f}_{i}(x)+t\cdot\tilde{g}_{i}(x) and let s\in S^{1} run a loop for a small t\neq 0. Then, as in the preceding proof, the roots of the system f_{\bullet,t,s}=0 permute so that (cf. Proposition 5.2 in [EG14]):

1) among the disjoint cycles of the permutation of the roots, we have d_{\gamma} cycles of length h_{\gamma} for every \gamma\in\mathcal{G}_{B};

2) the paths of the roots from one cycle constitute a loop whose class in the homology H equals \gamma.

As s\in S^{1} runs around the circle sufficiently many times (more specifically, M={\rm LCM}\{h_{\gamma}\,|\,\gamma\in\mathcal{G}_{B}\} times), we obtain a certain element \tilde{\gamma}_{B}\in H^{\oplus d} in the homology of the solution space. According to 1) and 2), this element \tilde{\gamma}_{B} has d_{\gamma} entries equal to \frac{M}{h_{\gamma}}\gamma for every \gamma\in\mathcal{G}_{B}, and the other entries are equal to 0. As in the preceding proof, we now have four increasing classes, extending Remark 4.7.2:

a) Tuples A, such that the vectors d_{\gamma_{B}}\cdot{\gamma_{B}} over all B\in\mathcal{E}_{0} generate the lattice H;

b) Tuples A, such that the vectors \tilde{\gamma}_{B} over all B\in\mathcal{E} generate the lattice H^{\oplus d};

c) Tuples with inductively irreducible solution spaces;

d) Tuples A, such that the vectors \sum_{\gamma\in\mathcal{G}_{B}}d_{\gamma}\cdot\gamma over all B\in\mathcal{E} generate the lattice H.

It is now a purely combinatorial (although highly non-trivial) problem to understand whether the classes (b) and (d) coincide for all reduced irreducible tuples A. If the answer is “yes” (and this is what we expect at least for n=2), then we have (b)=(c)=(d), so Theorem 4.6.1 actually provides a criterion of whether the Galois group of a given tuple equals the expected wreath product. If the answer is “no”, then a more subtle study of solution spaces is required to answer this question.

Proving inductive reducibility. In order to prove that a given tuple A is not inductively irreducible, we need the following Poisson-type formula for the product of roots of a system of polynomial equations. It is the special case of the Poisson-Pedersen-Sturmfels-D’Andrea-Sombra formula [DS13, Theorem 1.1], when one of the n+1 polynomials involved is a monomial x^{b}.

         Theorem 4.12.

For a generic system of equations f=(f_{1},\ldots,f_{n})\in\mathbb{C}^{A}, the product of the values of the monomial x^{b} over the roots of f_{1}=\ldots=f_{n}=0 equals \prod_{\gamma\in\mathcal{G}}[R_{A^{\gamma}}(f)]^{\gamma\cdot b}.

Proof of Theorem 4.6.1. Under the assumptions of Theorem 4.6.1, we can choose b\in\mathbb{Z}^{n} and p>1 that divides d_{\gamma}\cdot(\gamma\cdot b) and l\cdot b for all \gamma\in\mathcal{G} and all l\in L. In this case, the preceding Poisson-type formula implies that the product of the monomial x^{b} over the roots of f equals F^{p} for some polynomial F on \mathbb{C}^{A} that does not vanish at systems of equations that have d isolated roots. Thus, for any loop \alpha in the space of such systems, the corresponding element \tilde{\alpha}_{\sigma}\in H^{\oplus d} will have (\sum\tilde{\alpha}_{\sigma})\cdot b divisible by p. All such elements together with the sublattice \mathcal{L} generate a proper sublattice in H^{\oplus d}, because it is contained in the proper sublattice of all u\in H^{\oplus d} such that b\cdot\sum u is divisible by p. Therefore the solution space of A is not \mathcal{L}-inductively irreducible. \hfill\Box

Acknowledgement. The two authors met during the program “Tropical Geometry, Amoebas and Polytopes” held at the Institute Mittag-Leffler in spring 2018. The authors would like to express their gratitude to the organizers J. Draisma, A. Jensen, H. Markwig, B. Nill and to the institute for providing inspiring working conditions.

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A. Esterov
National Research University Higher School of Economics
Faculty of Mathematics NRU HSE, Usacheva str., 6, Moscow, 119048, Russia
Email: aesterov@hse.ru

L. Lang
Department of Mathematics, Stockholm University, SE - 106 91 Stockholm, Sweden.
Email: lang@math.su.se

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