A. Esterov

L. Lang

We study the geometry of generic spatial curves with a symmetry in order to understand the Galois group of a family of sparse polynomials.

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.

Kouchnirenko-Bernstein-Khovanskii with a symmetry