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Linear gaps between degrees for the polynomial calculus modulo distinct primes

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4 Author(s)
Buss, S. ; Dept. of Math., California Univ., San Diego, La Jolla, CA, USA ; Grigoriev, D. ; Impagliazzo, R. ; Pitassi, T.

Two important algebraic proof systems are the Nullstellensatz system and the polynomial calculus (also called the Grobner system). The Nullstellensatz system is a propositional proof system based on Hilbert's Nullstellensatz, and the polynomial calculus (PC) is a proof system which allows derivations of polynomials, over some field. The complexity of a proof in these systems is measured in terms of the degree of the polynomials used in the proof. The mod p counting principle can be formulated as a set MODpn of constant-degree polynomials expressing the negation of the counting principle. The Tseitin mod p principles, TSn(p), are translations of the MODpn into the Fourier basis. The present paper gives linear lower bounds on the degree of polynomial calculus refutations of MODpn over p fields of characteristic q ≠ p and over rings Zq with q,p relatively prime. These are the first linear lower bounds for the polynomial calculus. As it is well-known to be easy to give constant degree polynomial calculus (and even Nullstellensatz) refutations of the MOD pn polynomials over Fp, our results imply that the MODpn polynomials have a linear gap between proof complexity for the polynomial calculus over Fp and over Fq. We also obtain a linear gap for the polynomial calculus over rings Zp and Zq where p, q do not have identical prime factors

Published in:

Computational Complexity, 1999. Proceedings. Fourteenth Annual IEEE Conference on

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