Complex conjugation and simplicial algebraic hypersurfaces

TL;DR

This paper investigates the action of complex conjugation on the homology of coamoebas of simplicial real algebraic hypersurfaces in complex tori, aiming to understand their topological properties and conditions for Galois maximality.

Contribution

It describes how complex conjugation acts on the homology of coamoebas of simplicial hypersurfaces, extending previous work and proposing conditions for Galois maximality.

Findings

Describes the conjugation action on coamoeba homology.

Identifies conditions for Galois maximality under a conjecture.

Provides a framework for understanding topology of real algebraic hypersurfaces.

Abstract

We call a real algebraic hypersurface in $(\mathbb{C}^*)^n$ simplicial if it is given by a real Laurent polynomial in $n$-variables that has exactly $n+1$ monomials with non-zero coefficients and such that the convex hull in $\mathbb{R}^n$ of the $n+1$ points of $\mathbb{Z} ^n$ corresponding to the exponents is a non-degenerate $n$-dimensional simplex. Such hypersurfaces are natural building blocks from which more complicated objects can be constructed, for example using O. Viro's Patchworking method. Drawing inspiration from related work by G. Kerr and I. Zharkov, we describe the action of the complex conjugation on the homology of the coamoebas of simplicial real algebraic hypersurfaces, hoping it might prove useful in a variety of problems related to topology of real algebraic varieties. In particular, assuming a reasonable conjecture, we identify the conditions under which such a hypersurface is Galois maximal.