Explicit Stillman bounds for all degrees
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.
Keywords
Cite
@article{arxiv.2009.02826,
title = {Explicit Stillman bounds for all degrees},
author = {Giulio Caviglia and Yihui Liang},
journal= {arXiv preprint arXiv:2009.02826},
year = {2022}
}
Comments
A critical mistake was made at the end of section 4. We misinterpret the proof of Ananyan and Hochster's paper, the final bound grows even faster than a power tower thus would be meaningless to state