Linear gaps between degrees for the polynomial calculus modulo distinct primes

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 MOD_pⁿ of constant-degree polynomials expressing the negation of the counting principle. The Tseitin mod p principles, TS_n(p), are translations of the MOD_pⁿ into the Fourier basis. The present paper gives linear lower bounds on the degree of polynomial calculus refutations of MOD_pⁿ over p fields of characteristic q ≠ p and over rings Z_q 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 F_p, our results imply that the MOD_pⁿ polynomials have a linear gap between proof complexity for the polynomial calculus over F_p and over F_q. We also obtain a linear gap for the polynomial calculus over rings Z_p and Z_q where p, q do not have identical prime factors