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

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 $2^{n+1}(n - 2) + 4$.