Bounds on the number of generators of prime ideals

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 Castel\-nuovo 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.