Polyhedral Methods for Space Curves Exploiting Symmetry Applied to the Cyclic n-roots Problem
For researchers solving polynomial systems with symmetry, this algorithm reduces computational redundancy, but the improvement is incremental over existing tropical methods.
The paper presents a polyhedral algorithm for manipulating positive dimensional solution sets of polynomial systems, exploiting symmetry to reduce redundant computations. Applied to cyclic n-roots problems, it demonstrates efficiency gains through systematic filtration of group orbits.
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.