A Linear Bound on the Projective Dimension of Height 3 Quadratic Ideals

TL;DR

This paper establishes a nearly optimal linear upper bound on the projective dimension of height 3 quadratic ideals, advancing understanding of algebraic complexity in polynomial ideals.

Contribution

It provides a new, nearly optimal linear bound specifically for height 3 quadratic ideals, improving upon previous bounds and addressing a special case of Stillman's Question.

Findings

Derived a nearly optimal linear bound for height 3 quadratic ideals.

Improved understanding of projective dimension bounds in polynomial ideal theory.

Advances in algebraic complexity related to polynomial generators.

Abstract

In 2016, Ananyan and Hochster gave the first proof of a positive answer to Stillman's Question, which asked for a bound on the projective dimension of a graded polynomial ideal purely in terms of the number and degrees of its generators. Explicit formulas for such a bound are limited and often not optimal. In this paper, we give a nearly optimal linear upper bound on the projective dimension of height $3$ ideals generated by any number of degree $2$ homogenous polynomials.