Newton polytopes and numerical algebraic geometry

TL;DR

This paper introduces numerical algorithms connecting polyhedral and algebraic geometry, including a Newton polytope oracle, a tropical membership test, and a recursive solver for sparse polynomial systems, advancing computational algebraic geometry.

Contribution

It presents novel numerical algorithms for Newton polytopes, tropical membership, and solving sparse polynomial systems, integrating polyhedral and algebraic geometry techniques.

Findings

Developed a numerical oracle for Newton polytopes of hypersurfaces.

Created a tropical membership algorithm using the polytope oracle.

Designed a recursive solver for lacunary or triangular sparse polynomial systems.

Abstract

We develop a collection of numerical algorithms which connect ideas from polyhedral geometry and algebraic geometry. The first algorithm we develop functions as a numerical oracle for the Newton polytope of a hypersurface and is based on ideas of Hauenstein and Sottile. Additionally, we construct a numerical tropical membership algorithm which uses the former algorithm as a subroutine. Based on recent results of Esterov, we give an algorithm which recursively solves a sparse polynomial system when the support of that system is either lacunary or triangular. Prior to explaining these results, we give necessary background on polytopes, algebraic geometry, monodromy groups of branched covers, and numerical algebraic geometry.