For a system of polynomial equations, whose coefficients depend on parameters, the Newton polyhedron of its discriminant is computed in terms of the Newton polyhedra of the coefficients. This leads to an explicit formula (involving Euler obstructions of toric varieties) in the unmixed case, suggests certain open questions in general, and generalizes similar known results ( Introduction.Let F 0 , . . . , F l be Laurent polynomials on the complex torus (C \ 0) k , whose coefficients are Laurent polynomials on the parameter space (C \ 0) n . Consider the set Σ ⊂ (C \ 0) n of all values of the parameter, such that the corresponding system of polynomial equations F 0 = . . . = F l = 0 defines a singular set in (C \ 0) k . In most cases (see below for details), the closure of Σ is a hypersurface, and its defining equation is called the discriminant of F 0 = . . . = F l = 0.In this paper, we compute the Newton polyhedron of the discriminant in terms of Newton polyhedra of the coefficients of the polynomials F 0 , . . . , F l . The answer is known in many special cases, and we give a number of references as examples of various approaches to this problem: the universal special case for l = 0 and l = k was studied in [GKZ], [S94] and [DFS] (universal case means that (C \ 0) n parameterizes all collections of polynomials F 0 , . . . , F l , whose monomials are contained in a given finite set of monomials), the general case for l = k − 1, for l = k − 1 = 0 and for l = k was studied in [McD], [G] and [EKh].To formulate the answer in general, we need the following notation: we denote the Minkowski sum {a + b | a ∈ A, b ∈ B} of polyhedra A and B by A + B, and denote the mixed fiber polyhedron of the polyhedra ∆ 0 , . . . ,Definition 1.11, or Appendix). To a set A ⊂ Z k and a face B of its convex hull we associate its Euler obstruction e B,A ∈ Z, whose combinatorial definition is given in Subection 1.5, and whose geometrical meaning is (−1) k−dim B times the Euler obstruction of the A-toric variety at its orbit, corresponding to B (see also [MT] and a remark at the end of Subsection 4.4).Considering F i as a polynomial on (C\0) k with polynomial coefficients, denote the set of its monomials by A i ⊂ R k ; considering the same F i as a polynomial on (C \ 0) n × (C \ 0) k with complex coefficients, denote its Newton polyhedron by ∆ i ⊂ R n ⊕ R k . Denote the preimage of a set A ′ under the natural projection ∆ i → (convex hull of A i ) by ∆ i (A ′ ). For simplicity, we assume here that A 0 = . . . = A l = A, and pairwise differences of its points generate Z k .Theorem. If F 0 , . . . , F l are generic polynomials with Newton polyhedra ∆ 0 , . . . , ∆ l , then the Newton polyhedron of the discriminant equals

We study confluent A-hypergeometric functions introduced by Adolphson [1]. In particular, we give their integral representations by using rapid decay homology cycles of Hien [17] and [18]. The method of toric compactifications introduced in [27] and [31] will be used to prove our main theorem. Moreover we apply it to obtain a formula for the asymptotic expansions at infinity of confluent A-hypergeometric functions.

on the occasion of his 70th birthday. AbstractWe prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables.In particular, our result proves the multivariate version of the Abel-Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive.The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables.

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We generalize the Abel-Ruffini theorem to arbitrary dimension, i.e. classify general square systems of polynomial equations solvable by radicals. In most cases, they reduce to systems whose tuples of Newton polytopes have mixed volume not exceeding 4. The proof is based on topological Galois theory, which ensures non-solvability by any formula involving quadratures and single-valued functions, and the computation of the monodromy group of a general system of equations, which may be of independent interest.