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• W2893530259 abstract "In [2], 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 {it strength} condition), then any homogeneous basis for $V$ is an R$_eta$ sequence. The methods of cite{AH2} are not constructive. In this paper, we use related but different ideas that emphasize the notion of a {it 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 ${}^eta!A$ functions, and explicit recursions that determine the functions ${}^eta!B$ from the ${}^eta!A$ 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 $2^{n+1}(n - 2) + 4$." @default.
• W2893530259 created "2018-10-05" @default.
• W2893530259 creator A5033588315 @default.
• W2893530259 creator A5056285909 @default.
• W2893530259 date "2018-09-30" @default.
• W2893530259 modified "2023-09-27" @default.
• W2893530259 title "Strength conditions, small subalgebras, and Stillman bounds in degree $leq 4$" @default.
• W2893530259 cites W1519801683 @default.
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• W2893530259 hasPublicationYear "2018" @default.
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