Computing all Space Curve Solutions of Polynomial Systems by Polyhedral Methods

TL;DR

This paper introduces a hybrid symbolic-numeric polyhedral approach to compute Puiseux series solutions for space curves of polynomial systems, addressing challenging cases with tropical geometry techniques.

Contribution

It presents a novel hybrid method combining polyhedral and symbolic-numeric techniques to compute space curve solutions, including handling complex cases with tropical geometry.

Findings

Method effectively computes Puiseux series for space curves.

Addresses difficult cases with tropical prevariety analysis.

Polyhedral end games recover hidden tropisms.

Abstract

A polyhedral method to solve a system of polynomial equations exploits its sparse structure via the Newton polytopes of the polynomials. We propose a hybrid symbolic-numeric method to compute a Puiseux series expansion for every space curve that is a solution of a polynomial system. The focus of this paper concerns the difficult case when the leading powers of the Puiseux series of the space curve are contained in the relative interior of a higher dimensional cone of the tropical prevariety. We show that this difficult case does not occur for polynomials with generic coefficients. To resolve this case, we propose to apply polyhedral end games to recover tropisms hidden in the tropical prevariety.