Ideals Generated by Quadratic Polynomials

Abstract

Let R be a polynomial ring in N variables over an arbitrary field K and let I be an ideal of R generated by n polynomials of degree at most 2. We show that there is a bound on the projective dimension of R/I that depends only on n, and not on N . The proof depends on 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 subring of R generated over K by at most t ≤ C(n) forms G1, . . . , Gt of degree 1 or 2 such that G1, . . . , Gt is a regular sequence in R. C(n) is asymptotic to 2n2n .

Bounding Projective Dimension

“…, β n ) are proven to be linearly presented. Towards this end one uses Lemma 3.14, the proof of which can be found in [1]. From here one deduces that there were exactly h nonzero g i which, together with the leading, front, primary and secondary coefficient variables in the standard form of I, are the generators of A.…”

“…Since the techniques of Ananyan and Hochster can be applied when the minimal generators are nonhomogeneous, we reserve the use of the term quadratic form for a homogeneous polynomial of degree two and we call a possibly not homogeneous polynomial of degree two a quadratic polynomial. We then illustrate the techniques of [1] for the case of ideals generated by three homogeneous quadratics.…”

“…We remark that question (1) is still wide open. We hope this paper serves as a convenient survey of these results and spurs future work in this area.…”

“…We now sketch how the construction of standard form associated to an ideal I is applied to finding bounds for the projective dimension of R/I in [1]. The main idea is that the use of standard forms allows one to find a suitable polynomial algebra A generated by linear and quadratic forms that contains the ideal I while having a number of generators that can be bounded in terms of m, n, h. The leading, front and primary coefficient variables are all included as generators of A.…”

“…In [1], Ananyan and Hochster propose a method of analyzing the projective dimension of ideals generated by polynomials of degree at most two, which need not be homogeneous. We review their idea of using a specific standard form as well as the derived recursive bound on projective dimension.…”