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

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