# Solving multivariate polynomial systems and an invariant from commutative algebra

## TL;DR

This paper provides a rigorous method to estimate the solving degree of multivariate polynomial systems, especially in cryptography, by relating it to algebraic invariants like Castelnuovo-Mumford regularity, and clarifies assumptions for bounds.

## Contribution

It establishes a connection between solving degree and algebraic invariants, offering new bounds and clarifying conditions under which these bounds hold.

## Key findings

- Solving degree often equals Castelnuovo-Mumford regularity.
- Systems with field equations are in generic coordinates.
- Comparison between solving degree and degree of regularity.

## Abstract

The complexity of computing the solutions of a system of multivariate polynomial equations by means of Groebner bases computations is upper bounded by a function of the solving degree. In this paper, we discuss how to rigorously estimate the solving degree of a system, focusing on systems arising within public-key cryptography. In particular, we show that it is upper bounded by, and often equal to, the Castelnuovo-Mumford regularity of the ideal generated by the homogenization of the equations of the system, or by the equations themselves in case they are homogeneous. We discuss the underlying commutative algebra and clarify under which assumptions the commonly used results hold. In particular, we discuss the assumption of being in generic coordinates (often required for bounds obtained following this type of approach) and prove that systems that contain the field equations or their fake Weil descent are in generic coordinates. We also compare the notion of solving degree with that of degree of regularity, which is commonly used in the literature. We complement the paper with some examples of bounds obtained following the strategy that we describe.