Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Debordering Closure Results in Determinantal and Pfaffian Ideals

Anakin Dey, Zeyu Guo

TL;DR

This work advances the understanding of algebraic complexity for nonprincipal polynomial ideals, focusing on determinantal and Pfaffian ideals. By leveraging the isolation lemma together with straightening-law expansions and coefficient extraction, it converts border-approximate results into exact, small-depth circuits: for a nonzero $f$ in $I^{\det}_{n,m,r}$ (and similarly in $I^{\mathrm{pfaff}}_{2n,2r}$) of polynomial degree, a constant-depth $f$-oracle circuit computes a $t\times t$ determinant (resp. Pfaffian) exactly, with $t=\Theta(r^{1/3})$. This debordering relieves the need for approximation in these settings and yields exact reductions from $f$ to determinants/Pfaffians, with circuit sizes polynomial in $n,m,d$ (and in ABP parameters). The results generalize prior border-complexity findings and offer new avenues for hitting-set generation and PIT derandomization while highlighting open questions about small characteristic fields and extensions to broader nonprincipal ideals.

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 $t = Θ(r^{1/3})$ 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.

Debordering Closure Results in Determinantal and Pfaffian Ideals

TL;DR

This work advances the understanding of algebraic complexity for nonprincipal polynomial ideals, focusing on determinantal and Pfaffian ideals. By leveraging the isolation lemma together with straightening-law expansions and coefficient extraction, it converts border-approximate results into exact, small-depth circuits: for a nonzero in (and similarly in ) of polynomial degree, a constant-depth -oracle circuit computes a determinant (resp. Pfaffian) exactly, with . This debordering relieves the need for approximation in these settings and yields exact reductions from to determinants/Pfaffians, with circuit sizes polynomial in (and in ABP parameters). The results generalize prior border-complexity findings and offer new avenues for hitting-set generation and PIT derandomization while highlighting open questions about small characteristic fields and extensions to broader nonprincipal ideals.

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 generated by the minors of matrices. Over fields of characteristic zero or of sufficiently large characteristic, they showed that for any nonzero , the determinant of a matrix of variables with is approximately computed by a constant-depth, polynomial-size -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 has polynomial degree, the determinant is in fact exactly computed by a constant-depth, polynomial-size -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.

Paper Structure

This paper contains 19 sections, 46 theorems, 83 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Proof Overview
  4. Preliminaries
  5. Linear Algebra
  6. Bitableaux
  7. Determinantal Ideals
  8. Pfaffian Ideals
  9. Isolation Lemma
  10. Computing the Homogeneous Components
  11. Determinantal Ideals
  12. Isolating One Term
  13. Expressing Algebraic Branching Programs as Determinants
  14. Computing the Determinant of a Matrix
  15. Pfaffian Ideals
  16. ...and 4 more sections

Key Result

Theorem 1.3

Let $\mathbb{F}$ be a field of characteristic zero. Let $X = (x_{i,j})_{\substack{1 \leq i \leq n \\ 1 \leq j \leq m}}$ be an $n \times m$ matrix of variables over $\mathbb{F}$ and let $I^{\det}_{n,m,r} \subseteq \mathbb{F}[X] = \mathbb{F}[x_{i,j}]$ be the ideal generated by the $r \times r$ minors

Figures (4)

  • Figure 1: The Young diagrams for $\sigma = (5, 4, 2, 1)$ and $\widehat{\sigma} = (4, 3, 2, 2, 1)$.
  • Figure 2: Two examples of the order $\mathop{\mathrm{\leq_{\mathop{\mathrm{lex}}\nolimits}}}\nolimits$ on partitions. First, we have that $(5, 4, 2, 1) \mathop{\mathrm{\leq_{\mathop{\mathrm{lex}}\nolimits}}}\nolimits (5, 4, 3, 2, 1, 1)$ as $2 < 3$. Next, we have that $(5, 4, 2, 1) \mathop{\mathrm{\leq_{\mathop{\mathrm{lex}}\nolimits}}}\nolimits (5, 4, 2)$ as $(5, 4, 2)$ is a prefix of $(5, 4, 2, 1)$.
  • Figure 3: A Young tableau $T$ for $\sigma = (5, 4, 2, 1)$ as well as its conjugate $\widehat{T}$. Note that $T$ is semistandard and $\widehat{T}$ is conjugate semistandard.
  • Figure 4: The canonical and anti-canonical tableau for $\sigma = (5, 4, 2, 1)$ and $n = 7$.

Theorems & Definitions (88)

  • Definition 1.1
  • Conjecture 1.2: Grochow20
  • Remark
  • Theorem 1.3: AF21
  • Theorem 1.5: Informal version of \ref{['thrm:ABP_det_oracle']} and \ref{['cor:det_oracle_for_det']}
  • Theorem 1.6: Informal version of \ref{['thrm:ABP_pfaff_oracle']} and \ref{['cor:pfaff_oracle_for_pfaff']}
  • Definition 2.1: Oracle circuit
  • Definition 2.2: Algebraic branching program
  • Definition 2.3: Elementary matrix
  • Definition 2.4: Partition, Young diagram
  • ...and 78 more