Polyhedral Methods for Space Curves Exploiting Symmetry Applied to the Cyclic n-roots Problem

TL;DR

This paper introduces a polyhedral algorithm that leverages symmetry to efficiently analyze positive dimensional solution sets of polynomial systems, demonstrated on cyclic n-roots problems.

Contribution

It develops a symmetry-exploiting polyhedral method for manipulating solution sets, reducing redundant computations in polynomial system solving.

Findings

Effective handling of symmetry in polynomial systems.

Successful application to cyclic n-roots problems.

Integration with existing computational tools.

Abstract

We present a polyhedral algorithm to manipulate positive dimensional solution sets. Using facet normals to Newton polytopes as pretropisms, we focus on the first two terms of a Puiseux series expansion. The leading powers of the series are computed via the tropical prevariety. This polyhedral algorithm is well suited for exploitation of symmetry, when it arises in systems of polynomials. Initial form systems with pretropisms in the same group orbit are solved only once, allowing for a systematic filtration of redundant data. Computations with cddlib, Gfan, PHCpack, and Sage are illustrated on cyclic $n$-roots polynomial systems.