Polynomial Systems Solving by Fast Linear Algebra

TL;DR

This paper introduces faster algorithms for solving polynomial systems by leveraging advanced linear algebra techniques, reducing computational complexity from cubic to near-linear in the number of solutions, especially for generic systems.

Contribution

It presents novel algorithms that significantly improve the complexity of polynomial system solving using fast matrix multiplication, applicable to generic and Bézout-bound systems.

Findings

Complexity reduced from ^3 to ^ in solving systems.

New algorithms rely on fast linear algebra techniques.

Applicable to both bounded and unbounded degree systems.

Abstract

Polynomial system solving is a classical problem in mathematics with a wide range of applications. This makes its complexity a fundamental problem in computer science. Depending on the context, solving has different meanings. In order to stick to the most general case, we consider a representation of the solutions from which one can easily recover the exact solutions or a certified approximation of them. Under generic assumption, such a representation is given by the lexicographical Gr\"obner basis of the system and consists of a set of univariate polynomials. The best known algorithm for computing the lexicographical Gr\"obner basis is in $\widetilde{O}(d^{3n})$ arithmetic operations where $n$ is the number of variables and $d$ is the maximal degree of the equations in the input system. The notation $\widetilde{O}$ means that we neglect polynomial factors in $n$. We show that this complexity can be decreased to $\widetilde{O}(d^{\omega n})$ where $2 \leq \omega < 2.3727$ is the exponent in the complexity of multiplying two dense matrices. Consequently, when the input polynomial system is either generic or reaches the B\'ezout bound, the complexity of solving a polynomial system is decreased from $\widetilde{O}(D^3)$ to $\widetilde{O}(D^\omega)$ where $D$ is the number of solutions of the system. To achieve this result we propose new algorithms which rely on fast linear algebra. When the degree of the equations are bounded uniformly by a constant we propose a deterministic algorithm. In the unbounded case we present a Las Vegas algorithm.