Explicit Stillman bounds for all degrees

TL;DR

This paper provides explicit bounds for Stillman's conjecture applicable to all degrees, improving understanding of projective dimensions of ideals generated by forms of bounded degree.

Contribution

It constructs explicit bounds for all degrees, extending prior non-constructive results, using a new bound D(k,d) and a recurrence relation.

Findings

Explicit bounds behave like power towers of height d^3/6+11d/6-4

Bound D(k,d) controls minimal prime generators over regular sequences

Recurrence relation derived from bounds refines projective dimension estimates

Abstract

In 2016 Ananyan and Hochster proved Stillman's conjecture by showing the existence of a uniform upper bound for the projective dimension of all homogeneous ideals, in polynomial rings over a field, generated by n forms of degree at most d. Explicit values of the bounds for forms of degrees 5 and higher are not yet known. The main result of this article is the construction of explicit such bounds, for all degrees d, which behave like power towers of height d^3/6+11d/6-4. This is done by establishing a bound D(k,d), which controls the number of generators of a minimal prime over an ideal of a regular sequence of k or fewer forms of degree d, and supplementing it into Ananyan and Hochster's proof in order to obtain a recurrence relation.