Sparse Tensors and Subdivision Methods for Finding the Zero Set of Polynomial Equations

TL;DR

This paper introduces a novel method for solving polynomial equations by partially evaluating polynomials using sparse tensor subdivision, enabling efficient computation reuse and solving previously intractable problems.

Contribution

It presents a new partial evaluation approach with sparse tensor subdivision that improves efficiency in solving polynomial systems compared to traditional black-box methods.

Findings

Efficiently enclose curves defined by high-degree polynomials

Successfully applied to intractable polynomial systems of degree 100

Demonstrated practical efficiency of the proposed method

Abstract

Finding the solutions to a system of multivariate polynomial equations is a fundamental problem in mathematics and computer science. It involves evaluating the polynomials at many points, often chosen from a grid. In most current methods, such as subdivision, homotopy continuation, or marching cube algorithms, polynomial evaluation is treated as a black box, repeating the process for each point. We propose a new approach that partially evaluates the polynomials, allowing us to efficiently reuse computations across multiple points in a grid. Our method leverages the Compressed Sparse Fiber data structure to efficiently store and process subsets of grid points. We integrated our amortized evaluation scheme into a subdivision algorithm. Experimental results show that our approach is efficient in practice. Notably, our software \texttt{voxelize} can successfully enclose curves defined by two trivariate polynomial equations of degree $100$, a problem that was previously intractable.