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

TL;DR

This paper establishes exponential lower bounds on the total coefficient size of Nullstellensatz proofs for the pigeonhole principle and provides tight bounds for the ordering principle, advancing understanding of proof complexity.

Contribution

It proves exponential lower bounds for the pigeonhole principle and tight bounds for the ordering principle regarding Nullstellensatz proof sizes.

Findings

Nullstellensatz proofs of the pigeonhole principle require size 2^{Ω(n)}

Existence of Nullstellensatz proofs for the ordering principle with size 2^n - n

Advances understanding of proof complexity in algebraic proof systems

Abstract

In this paper, we investigate the total coefficient size of Nullstellensatz proofs. We show that Nullstellensatz proofs of the pigeonhole principle on $n$ pigeons require total coefficient size $2^{\Omega(n)}$ and that there exist Nullstellensatz proofs of the ordering principle on $n$ elements with total coefficient size $2^n - n$.