Bernstein-Kouchnirenko-Khovanskii with a symmetry

TL;DR

This paper investigates the geometric properties of symmetric spatial curves defined by polynomials with prescribed Newton polytopes, analyzing their components, singularities, and potential generalizations, with applications to Galois groups of polynomial families.

Contribution

It provides a detailed geometric analysis of symmetric spatial curves and explores their properties and generalizations, linking to Galois theory of polynomial families.

Findings

Characterization of the number, degree, and genus of irreducible components.

Analysis of singularities and their types.

Application to Galois groups of polynomial families.

Abstract

A generic polynomial f(x,y,z) with a prescribed Newton polytope defines a symmetric spatial curve f(x,y,z)=f(y,x,z)=0. We study its geometry: the number, degree and genus of its irreducible components, the number and type of singularities, etc. and discuss to what extent these results generalize to higher dimension and more complicated symmetries. As an application, we characterize generic one-parameter families of complex univariate polynomials, whose Galois group is a complete symmetric group.