Galois extension

Monomial

Pure mathematics

Discrete mathematics

Irreducible polynomial

Differential equation

Hypersurface

Monomial basis

Differential algebraic equation

Computer science

System of linear equations

Generic polynomial

Polynomial

System of polynomial equations

Polynomial ring

Simultaneous equations

Mathematics

Alternating polynomial

Matrix polynomial

Discriminant

Mathematical analysis

Artificial intelligence

Ordinary differential equation

Stable polynomial

Galois group

Monodromy

Theory of equations

Differential Galois theory

Compositio Mathematica

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. 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.

Galois theory for general systems of polynomial equations