Mathematical Research Letters

Volume 19 (2012)

Number 1

Ideals Generated by Quadratic Polynomials

Pages: 233 – 244



Tigran Ananyan (Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109-1043, USA)

Melvin Hochster (Altair Engineering, 1820 Big Beaver Rd, Troy, MI 48083, USA)


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 $2n^{2n}$.

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