Strength conditions, small subalgebras, and Stillman bounds in degree $\leq 4$
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- by Tigran Ananyan and Melvin Hochster PDF
- Trans. Amer. Math. Soc. 373 (2020), 4757-4806 Request permission
Corrigendum: Trans. Amer. Math. Soc. 374 (2021), 8307-8308.
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 $2^{n+1}(n - 2) + 4$.References
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Additional Information
- Tigran Ananyan
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109–1043
- MR Author ID: 902622
- Email: hochster@umich.edu
- Melvin Hochster
- Affiliation: Altair Engineering, 1820 E. Big Beaver Road, Troy, Michigan 48083
- MR Author ID: 86705
- ORCID: 0000-0002-9158-6486
- Email: antigran@gmail.com
- 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)
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 4757-4806
- MSC (2010): Primary 13D05, 13F20
- DOI: https://doi.org/10.1090/tran/8060
- MathSciNet review: 4127862