Bounds on the Total Coefficient Size of Nullstellensatz Proofs of the Pigeonhole Principle

Authors Aaron Potechin , Aaron Zhang



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Aaron Potechin
  • University of Chicago, IL, USA
Aaron Zhang
  • The Voleon Group, Berkeley, CA, USA

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Aaron Potechin and Aaron Zhang. Bounds on the Total Coefficient Size of Nullstellensatz Proofs of the Pigeonhole Principle. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 117:1-117:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.117

Abstract

We show that the minimum total coefficient size of a Nullstellensatz proof of the pigeonhole principle on n+1 pigeons and n holes is 2^{Θ(n)}. We also investigate the ordering principle and construct an explicit Nullstellensatz proof for the ordering principle on n elements with total coefficient size 2ⁿ - n.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
Keywords
  • Proof complexity
  • Nullstellensatz
  • pigeonhole principle
  • coefficient size

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