2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/31741We show that constant-depth Frege systems with counting axioms modulo $m$ polynomially simulate Nullstellensatz refutations modulo $m$. Central to this is a new definition of reducibility from formulas to systems of polynomials with the property that, for most previously studied translations of formulas to systems of polynomials, a formula reduces to its translation. When combined with a previous result of the authors, this establishes the first size separation between Nullstellensatz and polynomial calculus refutations. We also obtain new, small refutations for certain CNFs by constant-depth Frege systems with counting axioms.17 pagesComputational ComplexityLogic in Computer ScienceF.4.1text