[Paper Review] Strength conditions, small subalgebras, and Stillman bounds in degree $\leq 4$

Tigran Ananyan, Melvin Hochster|arXiv (Cornell University)|Sep 30, 2018

Coding theory and cryptography28 references4 citations

TL;DR

This paper provides explicit, closed-form bounds for Stillman’s conjecture in degrees 2, 3, and 4 by introducing key functions $^\eta\!A(n,d)$ and $^\eta\!B(n,d)$, which control the generation of small subalgebras and regular sequences. It establishes that for $n$ quadrics, the projective dimension of $R/I$ is at most $2^{n+1}(n-2)+4$, yielding the best known bound for quadrics without restrictions on $n$. The results are valid over algebraically closed fields of characteristic not 2 or 3 for degree 4, and all characteristics for degrees 2 and 3.

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 ${}^η\!B(n,d)$ 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$_η$. A critical element in the proof is to show that there are functions ${}^η\!A(n,d)$ 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 ${}^η\!A(n,d)$ forms of strictly lower degree (we call this a {\it strength} condition), then any homogeneous basis for $V$ is an R$_η$ 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 ${}^η\!A(n,d)$ 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 ${}^η\!A$ functions, and explicit recursions that determine the functions ${}^η\!B$ from the ${}^η\!A$ functions. In degree 2, we obtain an explicit value for ${}^η\!B(n,2)$ 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 $2^{n+1}(n - 2) + 4$.

Motivation & Objective

• To provide explicit, constructive bounds for Stillman’s conjecture in degrees 2, 3, and 4, extending the non-constructive proof in [2].
• To define and compute the key functions $^\eta\!A(n,d)$ and $^\eta\!B(n,d)$ that control the generation of small subalgebras and regular sequences.
• To establish closed-form expressions for $^\eta\!A(n,d)$ in degrees 2, 3, and 4, and recursive formulas for $^\eta\!B(n,d)$.
• To improve the known Stillman bound for ideals generated by $n$ quadrics, giving the tightest known upper bound for projective dimension.

Proposed method

• Introduces the notion of $k$-collapse and $\kappa$-strong subspaces to define strength conditions that ensure homogeneous bases form regular sequences.
• Uses a new approach based on key functions to derive $^\eta\!A(n,d)$, differing from the non-constructive methods in [2], especially in degree 4.
• Derives closed-form expressions for $^\eta\!A(n,d)$ in degree 2, 3, and 4, with explicit formulas involving $A_3(2n+\eta)$ and a function $\mathfrak{K}_4$ for degree 4.
• Establishes recursive relations between $^\eta\!B(n,d)$ and $^\eta\!A(n,d)$, enabling computation of the size of small subalgebras generated by regular sequences.
• Applies the theory to compute the projective dimension bound for $n$ quadrics as $2^{n+1}(n-2)+4$, valid over any field.
• Restricts to algebraically closed fields of characteristic not 2 or 3 for degree 4, due to technical limitations in the method.

Experimental results

Research questions

• RQ1 What is the best possible explicit bound for $^\eta\!A(n,d)$ in degrees 2, 3, and 4, and can it be expressed in closed form?
• RQ2 Can the Stillman bound for ideals generated by $n$ forms of degree $d$ be made explicit and constructive, especially for $d=2$?
• RQ3 Is there a polynomial bound for $^\eta\!A(n,d)$ in terms of $n$ for fixed $d$, and if so, what is the degree of the polynomial?
• RQ4 Can the projective dimension of $R/I$ for $n$ quadrics be bounded more tightly than previous results?
• RQ5 What are the minimal possible values of $^\eta\!B(n,d)$, and is a polynomial bound possible for $^\eta\!B(n,d)$ in general?

Key findings

• For $n$ quadrics, the projective dimension of $R/I$ is at most $2^{n+1}(n-2)+4$, providing the best known bound for quadrics without restriction on $n$.
• In degree 2, the paper gives an explicit value for $^\eta\!B(n,2)$, yielding a tight bound on the size of small subalgebras generated by regular sequences.
• For degree 4 and fields of characteristic not 2 or 3, the paper provides a closed-form expression for $^\eta\!A(n,4)$ involving $\mathfrak{K}_4$ and $A_3(2n+\eta)$, with $A_3(2n+\eta) = 2(8n+4\eta-1)(2n+\eta-1)$.
• The paper establishes recursive formulas that determine $^\eta\!B(n,d)$ from $^\eta\!A(n,d)$, enabling algorithmic computation of Stillman bounds.
• The authors conjecture that $^\eta\!A(n,d)$ and $^\eta\!B(n,d)$ grow at most polynomially in $n$ for fixed $d$, with $^\eta\!A(n,d)$ bounded by $C_{d,\eta}n^{\lambda(d)}$ and $^\eta\!B(n,d)$ by $C_d n^d$.
• The paper shows that the ideal construction in [31] for $d=2$ achieves the bound $h(n-h+1)$, suggesting this may be sharp, and supports Conjecture 11.4 on the projective dimension of quadric ideals.