On the Termination of the General XL Algorithm and Ordinary Multinomials

TL;DR

This paper analyzes the termination behavior of the XL algorithm for polynomial systems of degree three or higher, establishing specific bounds and proving properties of ordinary multinomials that influence algorithm performance.

Contribution

It proves the XL algorithm terminates at certain bounds for systems with specific equation-variable relationships and demonstrates ordinary multinomials are strongly unimodal.

Findings

XL terminates at a specific D when equations exceed variables by 1 or 2

The termination value is likely optimal and smaller than the degree of regularity

Ordinary multinomials are strongly unimodal, aiding the analysis

Abstract

The XL algorithm is an algorithm for solving overdetermined systems of multivariate polynomial equations, which was initially introduced for quadratic equations. However, the algorithm works for polynomials of any degree, and in this paper we will focus on the performance of XL for polynomials of degree $\geq3$, where the optimal termination value of the parameter $D$ is still unknown. We prove that the XL algorithm terminates at a certain value of $D$ in the case that the number of equations exceeds the number of variables by 1 or 2. We also give strong evidence that this value is best possible, and we show that this value is smaller than the degree of regularity. Part of our analysis requires proving that ordinary multinomials are strongly unimodal, and this result may be of independent interest.