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Transactions of the American Mathematical Society

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Strength conditions, small subalgebras, and Stillman bounds in degree $ \leq 4$

Authors: Tigran Ananyan and Melvin Hochster
Journal: Trans. Amer. Math. Soc. 373 (2020), 4757-4806
MSC (2010): Primary 13D05, 13F20
Published electronically: April 28, 2020
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Abstract: In an earlier work, the authors prove Stillman's conjecture in all characteristics and all degrees by showing that, independent of the algebraically closed field $ K$ or the number of variables, $ n$ forms of degree at most $ d$ in a polynomial ring $ R$ over $ K$ are contained in a polynomial subalgebra of $ R$ generated by a regular sequence consisting of at most $ {}^\eta \!B(n,d)$ forms of degree at most $ d$; we refer to these informally as ``small'' subalgebras. Moreover, these forms can be chosen so that the ideal generated by any subset defines a ring satisfying the Serre condition R$ _\eta $. A critical element in the proof is to show that there are functions $ {}^\eta \!A(n,d)$ with the following property: in a graded $ n$-dimensional $ K$-vector subspace $ V$ of $ R$ spanned by forms of degree at most $ d$, if no nonzero form in $ V$ is in an ideal generated by $ {}^\eta \!A(n,d)$ forms of strictly lower degree (we call this a strength condition), then any homogeneous basis for $ V$ is an R$ _\eta $ sequence. The methods of our earlier work are not constructive. In this paper, we use related but different ideas that emphasize the notion of a key function to obtain the functions $ {}^\eta \!A(n,d)$ in degrees 2, 3, and 4 (in degree 4 we must restrict to characteristic not 2, 3). We give bounds in closed form for the key functions and the $ {}^\eta \!{\underline {A}}$ functions, and explicit recursions that determine the functions $ {}^\eta \!B$ from the $ {}^\eta \!{\underline {A}}$ functions. In degree 2, we obtain an explicit value for $ {}^\eta \!B(n,2)$ that gives the best known bound in Stillman's conjecture for quadrics when there is no restriction on $ n$. In particular, for an ideal $ I$ generated by $ n$ quadrics, the projective dimension $ R/I$ is at most $ 2^{n+1}(n - 2) + 4$.

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Additional Information

Tigran Ananyan
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109–1043

Melvin Hochster
Affiliation: Altair Engineering, 1820 E. Big Beaver Road, Troy, Michigan 48083

Keywords: Polynomial ring, ideal, quartic form, cubic form, quadratic form, projective dimension, regular sequence
Received by editor(s): October 25, 2018
Received by editor(s) in revised form: October 6, 2019
Published electronically: April 28, 2020
Additional Notes: The second author was partially supported by grants from the National Science Foundation (DMS–0901145 and DMS–1401384)
Article copyright: © Copyright 2020 American Mathematical Society