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arXiv · 2610.00837

A Degree--Size Relation for Resolution over Polynomials

Shuo Pang
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Abstract

For every constant-width CNF, we show that linear degree in polynomial calculus (PC) implies exponential size in resolution over constant-degree polynomials, over the same prime field. Applications include exponential lower bounds for CNFs in $\operatorname{Res}(\operatorname{PC}_r/\mathbb{F}_p)$ and hence in $\operatorname{Res}(\oplus_p)$, separations between different moduli, improved lower bounds for $\operatorname{Res}(k)$ up to $k=\varepsilon\log n$, proof-search consequences, and an implication of super-polynomial $AC^0[p]$-Frege bounds from very strong PC degree lower bounds. The proof uses the common-multiplier idea isolated from Braun [arXiv:2609.23015] to construct a Razborov--Smolensky approximation that preserves inferences, without introducing extension variables. The approximation errors are measured by ranks of the multiplication maps induced by the error-witness polynomials, modulo bounded-degree PC consequences.

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Shuo Pang. 2026-09-30. A Degree--Size Relation for Resolution over Polynomials. https://arxiv.org/abs/2610.00837

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