Combinatorial analysis of the period mapping: topology of 2D fibers

TL;DR

This paper investigates the topology of fibers in the period mapping from the moduli space of real hyperelliptic curves, with applications to polynomial optimization, using combinatorial and geometric methods.

Contribution

It introduces a combinatorial approach to analyze the topology of fibers in the period mapping for real hyperelliptic curves, linking it to polynomial optimization problems.

Findings

Decomposition of moduli space into polyhedra labeled by planar graphs.

Topological characterization of low-dimensional fibers.

Application to Chebyshev polynomial and rational function optimization.

Abstract

We study the periods mapping from the moduli space of real hyperelliptic curves with marked point on an oriented oval to the euclidean space. The mapping arises in the analysis of Chebyshev construction used in the constrained optimization of the uniform norm of polynomials and rational functions. The decomposition of the moduli space into polyhedra labeled by planar graphs allows to investigate the global topology of low dimensional fibers of the periods mapping.