Galois theory for general systems of polynomial equations

Abstract

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.

Classification of Triples of Lattice Polytopes with a Given Mixed Volume

“…Classification Problem 1.3 is of particular interest for m ≤ 4, since by a result of Esterov [14], it allows to describe all generic systems f 1 = • • • = f d = 0 as in Theorem 1.1 that are solvable by radicals. The family of d-tuples of lattice polytopes with mixed volume m is invariant under application of a common unimodular transformation to all polytopes of the tuple, independent translations of the polytopes by lattice vectors, and permutations of the polytopes of the d-tuple.…”

“…, P d ) of polytopes in R d is called irreducible if the Minkowski sum of any k polytopes in the tuple is at least (k + 1)dimensional for every 1 ≤ k ≤ d − 1. In [14] Esterov showed that the number of irreducible d-tuples of lattice polytopes is finite up to equivalence for a fixed mixed volume and dimension (see Theorem 2.4 below).…”

Monodromy of rational curves on toric surfaces

“…Proposition 5.2. Let C ∈ V Δ,1 be parametrised by φ := φ {a} as in (6). Then, we have the following bijective correspondence P :…”

“…Any node ν ∈ C corresponds to an unordered pair {t, t } ⊂ C * such that φ(t) = φ(t ). By (6), the latter equality is equivalent to (t/t )…”

Inequalities Between Mixed Volumes of Convex Bodies: Volume Bounds for the Minkowski Sum

“…, P d are full-dimensional. Esterov [8] has shown that in this case the volume of the Minkowski sum (P) := P 1 + • • • + P d has the asymptotic order at most O(m 2 d ), as m → ∞. This bound allows to control the sizes of the P i in the tuple P. In particular, it implies that the number of possible tuples P of d-dimensional lattice polytopes with a given value of the mixed volume m is finite, up to the natural equivalence consisting of permutation of the polytopes within the tuple, independent lattice translations of the polytopes, and a common unimodular transformation of all the polytopes of the tuple.…”