Explicit Stillman bounds for all degrees

TL;DR

This paper provides explicit bounds for Stillman's conjecture, including projective dimension and Castelnuovo-Mumford regularity, for ideals generated by forms of degree at most d, advancing understanding of uniform bounds in polynomial rings.

Contribution

It constructs explicit bounds for projective dimension and regularity for ideals generated by forms of degree at most d, extending prior existence results to explicit numerical bounds.

Findings

Explicit bounds for projective dimension of ideals generated by forms of degree d.

Explicit bounds for Castelnuovo-Mumford regularity in terms of multiplicity.

Extension of uniform bounds to degrees 5 and higher.

Abstract

In 2016 Ananyan and Hochster proved Stillman's conjecture by showing the existence of a uniform upper bound on the lengths of an $R_\eta$-sequence containing a fixed $n$ forms of degree at most $d$ in polynomial rings over a field. This result yields many other uniform bounds including bounds on the projective dimension of the ideals generated by $n$-forms of degree at most $d$. Explicit values of these bounds for forms of degrees $5$ and higher are not yet known. This article constructs such explicit bounds, one of which is an upper bound for the projective dimension of all homogeneous ideals, in polynomial rings over a field, generated by $n$ forms of degree at most $d$. In the settings of the Eisenbud-Goto conjecture, we derive an explicit bound of the Castelnuovo-Mumford regularity of a degenerate prime ideal $P$ in a polynomial ring $S$ in terms of the multiplicity of $S/P$.