Contents Online
Mathematical Research Letters
Volume 19 (2012)
Number 1
Ideals Generated by Quadratic Polynomials
Pages: 233 – 244
DOI: https://dx.doi.org/10.4310/MRL.2012.v19.n1.a18
Authors
Abstract
Let $R$ be a polynomial ring in $N$ variables over an arbitrary field $K$ and let $I$ bean ideal of $R$ generated by $n$ polynomials of degree at most 2. We show that there is a boundon the projective dimension of $R/I$ that depends only on $n$, and not on $N$. The proof dependson showing that if $K$ is infinite and $n$ is a positive integer, there exists a positive integer $C(n)$,independent of $N$, such that any $n$ forms of degree at most 2 in $R$ are contained in a subringof $R$ generated over $K$ by at most $t \leq C(n)$ forms $G_1, \, \ldots, \, G_t$ of degree 1 or 2such that $G_1, \, \ldots, \, G_t$ is a regular sequence in $R$. $C(n)$ is asymptotic to $2n^{2n}$.
Published 2 May 2012