Debordering Closure Results in Determinantal and Pfaffian Ideals

TL;DR

This paper proves that for polynomial degree polynomials in determinantal and Pfaffian ideals, the determinant and Pfaffian are exactly computable by small, constant-depth algebraic circuits with oracle access, improving previous approximate results.

Contribution

It deborders previous border-approximate results by showing exact computation of determinants and Pfaffians in polynomial degree ideals using small algebraic circuits.

Findings

Determinant of polynomial degree in determinantal ideals is exactly computed by small circuits.

Pfaffian of polynomial degree in Pfaffian ideals is exactly computed by small circuits.

Results use the isolation lemma and straightening-law analysis.

Abstract

One important question in algebraic complexity is understanding the complexity of polynomial ideals (Grochow, Bulletin of EATCS 131, 2020). Andrews and Forbes (STOC 2022) studied the determinantal ideals $I^{\det}_{n,m,r}$ generated by the $r\times r$ minors of $n\times m$ matrices. Over fields of characteristic zero or of sufficiently large characteristic, they showed that for any nonzero $f \in I^{\det}_{n,m,r}$, the determinant of a $t \times t$ matrix of variables with $t = \Theta(r^{1/3})$ is approximately computed by a constant-depth, polynomial-size $f$-oracle algebraic circuit, in the sense that the determinant lies in the border of such circuits. An analogous result was also obtained for Pfaffians in the same paper. In this work, we deborder the result of Andrews and Forbes by showing that when $f$ has polynomial degree, the determinant is in fact exactly computed by a constant-depth, polynomial-size $f$-oracle algebraic circuit. We further establish an analogous result for Pfaffian ideals. Our results are established using the isolation lemma, combined with a careful analysis of straightening-law expansions of polynomials in determinantal and Pfaffian ideals.