# Proof complexity in algebraic systems and bounded depth Frege systems with modular counting

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## Abstract

We prove a lower bound of the form*N*^{Ω(1)} on the degree of polynomials in a Nullstellensatz refutation of the*Count*_{ q } polynomials over ℤ_{ m }, where*q* is a prime not dividing*m*. In addition, we give an explicit construction of a degree*N*^{ Ω(1) } design for the*Count*_{ q } principle over ℤ_{ m }. As a corollary, using Beame*et al.* (1994) we obtain a lower bound of the form 2^{NΩ(1)} for the number of formulas in a constant-depth Frege proof of the modular counting principle*Count* _{ q } ^{ N } from instances of the counting principle*Count* _{ m } ^{ M } .

We discuss the polynomial calculus proof system and give a method of converting tree-like polynomial calculus derivations into low degree Nullstellensatz derivations.

Further we shwo that a lower bound for proofs in a bounded depth Frege system in the language with the modular counting connective*MOD*_{ p } follows from a lower bound on the degree of Nullstellensatz proofs with a constant number of levels of extension axioms, where the extension axioms comprise a formalization of the approximation method of Razborov (1987) and Smolensky (1987) (in fact, these two proof systems are basically equivalent).

## Keywords

Approximation Method Computational Mathematic Problem Complexity Algorithm Analysis Constant Number## Preview

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