Matrix factorizations of generic polynomials

TL;DR

This paper proves a conjecture about the minimal rank of matrix factorizations for generic polynomials, introducing a new concept of secondary strength and employing ultraproduct techniques.

Contribution

It establishes the Buchweitz-Greuel-Schreyer Conjecture for generic polynomials, advancing understanding of matrix factorizations in algebraic geometry.

Findings

Conjecture holds for generic polynomials of given degree and strength

Introduces secondary strength as a new polynomial invariant

Utilizes ultraproduct techniques to prove the result

Abstract

We prove that the Buchweitz-Greuel-Schreyer Conjecture on the minimal rank of a matrix factorization holds for a generic polynomial of given degree and strength. The proof introduces a notion of the secondary strength of a polynomial, and uses a variant of the ultraproduct technique of Erman, Sam, and Snowden.