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Debordering Closure Results in Determinantal and Pfaffian Ideals

Authors Anakin Dey , Zeyu Guo

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LIPIcs.ITCS.2026.49.pdf
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ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Algebraic circuit complexity
  • Isolation lemma
  • Debordering

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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× r minors of n× m matrices. Over fields of characteristic zero or of sufficiently large characteristic, they showed that for any nonzero f ∈ I^{det}_{n,m,r}, the determinant of a t × 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.

Cite As Get BibTex

Anakin Dey and Zeyu Guo. Debordering Closure Results in Determinantal and Pfaffian Ideals. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 49:1-49:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.49

Author Details

Anakin Dey
  • Department of Mathematics, The Ohio State University, Columbus, OH, USA
Zeyu Guo
  • Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA

Funding

  • Guo, Zeyu: Supported by NSF CAREER award CCF-2440926.

Acknowledgements

We thank David Anderson, Robert Andrews, and Srikanth Srinivasan for helpful discussions. We also thank the anonymous reviewers of ITCS 2026 for their careful reading and insightful comments.

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