Galois theory for general systems of polynomial equations

TL;DR

This paper establishes that the monodromy group of general polynomial systems is symmetric, advancing Galois theory for such systems and linking solvability by radicals to lattice polytope classifications.

Contribution

It proves the monodromy group of general polynomial systems is symmetric and connects solvability by radicals to lattice polytope classifications, extending Galois theory to multivariate cases.

Findings

Monodromy group equals the symmetric group for general systems.

Solvability by radicals relates to lattice polytopes of mixed volume 4.

Discriminant is a hypersurface unless the system is linear after monomial change.

Abstract

We 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, similarly to what Sottile and White conjectured in Schubert calculus. 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.