On the Complexity of Computing Gr\"obner Bases for Quasi-homogeneous Systems

TL;DR

This paper develops algorithms for computing Gr"obner bases of quasi-homogeneous polynomial systems, demonstrating that under certain conditions, the complexity is polynomial in the weighted Be9zout bound, and shows practical improvements in solving such systems.

Contribution

It adapts existing homogeneous algorithms to quasi-homogeneous systems and analyzes their complexity, providing new strategies and experimental validation.

Findings

Complexity is polynomial in the weighted Be9zout bound for generic systems.

Quasi-homogeneous structure enables solving larger systems more efficiently.

Experimental results confirm the practical advantage of the proposed methods.

Abstract

Let $\K$ be a field and $(f_1, \ldots, f_n)\subset \K[X_1, \ldots, X_n]$ be a sequence of quasi-homogeneous polynomials of respective weighted degrees $(d_1, \ldots, d_n)$ w.r.t a system of weights $(w_{1},\dots,w_{n})$. Such systems are likely to arise from a lot of applications, including physics or cryptography. We design strategies for computing Gr\"obner bases for quasi-homogeneous systems by adapting existing algorithms for homogeneous systems to the quasi-homogeneous case. Overall, under genericity assumptions, we show that for a generic zero-dimensional quasi-homogeneous system, the complexity of the full strategy is polynomial in the weighted B\'ezout bound $\prod_{i=1}^{n}d_{i} / \prod_{i=1}^{n}w_{i}$. We provide some experimental results based on generic systems as well as systems arising from a cryptography problem. They show that taking advantage of the quasi-homogeneous structure of the systems allow us to solve systems that were out of reach otherwise.