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

Guillaume Moroz

TL;DR

The paper tackles the zero-set problem for systems of multivariate polynomials within subdivision frameworks, addressing the high cost of evaluating $F$ on grid boxes. It proposes amortized evaluation on grid boxes using partial Horner-style evaluation, interval arithmetic for robust exclusion/inclusion, and stores the grid subset as a sparse Compressed Sparse Fiber (CSF) structure, enabling the voxelize tool. A main complexity result shows that evaluating a multivariate polynomial on a sparse subset $S$ of a grid $G$ can be done in $O\left(\sum_{i=0}^{k-1} \widetilde{N}_{k-i}(F) N_{i+1}(S)\right)$ arithmetic operations, with favorable bounds when projections are dense, and the approach extends to fast tasks like the discrete Fourier transform with $O(d\log d)$ time. Empirical results demonstrate that voxelize can enclose zero sets of high-degree polynomials (e.g., two trivariate polynomials of degree $100$) and can outperform state-of-the-art subdivision tools such as ibex on selected problems.

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.

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

TL;DR

The paper tackles the zero-set problem for systems of multivariate polynomials within subdivision frameworks, addressing the high cost of evaluating on grid boxes. It proposes amortized evaluation on grid boxes using partial Horner-style evaluation, interval arithmetic for robust exclusion/inclusion, and stores the grid subset as a sparse Compressed Sparse Fiber (CSF) structure, enabling the voxelize tool. A main complexity result shows that evaluating a multivariate polynomial on a sparse subset of a grid can be done in arithmetic operations, with favorable bounds when projections are dense, and the approach extends to fast tasks like the discrete Fourier transform with time. Empirical results demonstrate that voxelize can enclose zero sets of high-degree polynomials (e.g., two trivariate polynomials of degree ) and can outperform state-of-the-art subdivision tools such as ibex on selected problems.

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 , a problem that was previously intractable.
Paper Structure (2 sections, 2 figures)

This paper contains 2 sections, 2 figures.

Table of Contents

  1. Introduction
  2. Amortized Evaluation on a Grid of Boxes

Figures (2)

  • Figure 1: Boxes on the same level of the subdivision tree
  • Figure 2: Enclosing of a curve defined by $2$ trivariate polynomials of degree $100$