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Galois theory for general systems of polynomial equations

Part of: Curves Permutation groups Polytopes and polyhedra Theory of singularities and catastrophe theory

Published online by Cambridge University Press:  07 January 2019

A. Esterov
A. Esterov*
Affiliation:
National Research University Higher School of Economics Faculty of Mathematics NRU HSE, Usacheva str., 6, Moscow, 119048, Russia email [email protected]
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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. 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.

Keywords

topological Galois theorymonodromydiscriminantNewton polytopedual defectmixed volume

MSC classification

Primary: 14H05: Algebraic functions; function fields 14H30: Coverings, fundamental group 20B15: Primitive groups 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) 58K10: Monodromy
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 2 , February 2019 , pp. 229 - 245
Copyright
© The Author 2019 

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Footnotes

Research supported by the Russian Science Foundation grant, project 16-11-10316.

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