Debordering Closure Results in Determinantal and Pfaffian Ideals

2025-11-20

AI Summary

This work addresses the central “debordering” problem in algebraic complexity: lifting border complexity results for determinant and Pfaffian ideals to exact computation. For input polynomials $ f $ satisfying a polynomial-degree constraint, we establish, for the first time, that both the determinant and the Pfaffian can be computed exactly by constant-depth, polynomial-size algebraic circuits with access to an $ f $-oracle. Our approach innovatively combines the Andrews–Forbes border approximation technique with the isolation lemma, and employs the straightening law to structurally analyze polynomial expansions within the ideal. This yields the first explicit, exact algebraic circuit constructions for determinant and Pfaffian ideals—overcoming the long-standing reliance on approximations. The result advances our understanding of the interplay between the structural properties of polynomial ideals and their algebraic computational power.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability

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• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 $r imes r$ minors of $n imes m$ 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 $t imes t$ 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.

Problem

• Debordering algebraic closure results for determinantal ideals
• Establishing exact computation in constant-depth f-oracle circuits
• Extending analogous debordering techniques to Pfaffian ideals

Innovation

• Debordered determinantal ideals via isolation lemma
• Applied straightening-law expansions for Pfaffians
• Exact computation with constant-depth oracle circuits