A Degree--Size Relation for Resolution over Polynomials

2026-09-30

AI Summary

This study addresses the relationship between linear degree in Polynomial Calculus (PC) and exponential size in polynomial resolution for constant-width CNF formulas. The authors demonstrate that PC linear degree implies exponential size in polynomial resolution over the same field. Methodologically, they propose a Razborov-Smolensky approximation construction without introducing extension variables, utilizing least common multiple techniques to build inference-preserving approximations while measuring errors via multiplicative mapping rank. These contributions yield exponential lower bounds for systems such as Res(PC_r/F_p), improve lower-order resolution bounds, and derive super-polynomial lower bounds for AC^0[p]-Frege systems, thereby advancing algebraic proof complexity theory.

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.

Problem

• Polynomial Calculus
• Resolution over polynomials
• Proof complexity
• Degree-size relation
• Lower bounds

Innovation

• Polynomial Calculus
• Resolution over Polynomials
• Razborov-Smolensky Approximation
• Proof Complexity
• Common-Multiplier