On minimal free resolutions of sub-permanents and other ideals arising in complexity theory

TL;DR

This paper computes minimal free resolutions for ideals generated by sub-permanents and square-free monomials, providing algebraic tools to analyze complexity theory problems.

Contribution

It offers explicit calculations of minimal free resolutions and Hilbert functions for ideals relevant to algebraic complexity theory, advancing the algebraic approach.

Findings

Computed the linear strand of the minimal free resolution for sub-permanent ideals.

Determined the full minimal free resolution for square-free monomial ideals.

Provided Hilbert functions relevant to complexity theory.

Abstract

We compute the linear strand of the minimal free resolution of the ideal generated by k x k sub-permanents of an n x n generic matrix and of the ideal generated by square-free monomials of degree k. The latter calculation gives the full minimal free resolution by work of Biagioli-Faridi-Rosas. Our motivation is to lay groundwork for the use of commutative algebra in algebraic complexity theory. We also compute several Hilbert functions relevant for complexity theory.