Bounds on the number of generators of prime ideals

TL;DR

This paper extends classical bounds on the number of quadrics in prime ideals to bounds on minimal generators in any degree, applicable over any field, and provides bounds on higher Betti numbers.

Contribution

It introduces explicit bounds on the minimal generators of prime ideals in any degree, removing the algebraically closed field assumption, and extends to bounds on higher Betti numbers.

Findings

Bound on generators in any degree depending only on degree and height

Removal of algebraically closed field assumption

Bounds on higher graded Betti numbers

Abstract

Let $S$ be a polynomial ring over any field $\Bbbk$, and let $P \subseteq S$ be a non-degenerate homogeneous prime ideal of height $h$. When $\Bbbk$ is algebraically closed, a classical result attributed to Castelnuovo establishes an upper bound on the number of linearly independent quadrics contained in $P$ which only depends on $h$. We significantly extend this result by proving that the number of minimal generators of $P$ in any degree $j$ can be bounded above by an explicit function that only depends on $j$ and $h$. In addition to providing a bound for generators in any degree $j$, not just for quadrics, our techniques allow us to drop the assumption that $\Bbbk$ is algebraically closed. By means of standard techniques, we also obtain analogous upper bounds on higher graded Betti numbers of any radical ideal.