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

Let f0 ; f1 ; : : : ; fk be n-variable polynomials over a finite prime field Fp . A proof of the ideal membership f0 2 hf1 ; : : : ; fk i in polynomial calculus is a sequence of polynomials h1 ; : : : ; h t such that h t = f0 , and such that every h i is either an f j , j 1, or obtained from h1 ; : : : ; h i\Gamma1 by one of the two inference rules: g1 and g2 entail any Fp-linear combination of g1 , g2 , and g entails g \Delta g 0 , for any polynomial g 0 . The degree of the proof is the maximum degree of h i 's. We give a condition on families ffN;0 ; : : : ; fN;k N gN!! of nN -variable polynomials of bounded degree implying that the minimum degree of polynomial calculus proofs of fN;0 from fN;1 ; : : : ; fN;k N cannot be bounded by an independent constant and, in fact, is\Omega\Gamma/31 (log(N))). In particular, we obtain an\Omega\Gamma/19 (log(N))) lower bound for the degrees of proofs of 1 (so called refutations) of the (N; m) - system (defined in [4]) formalizing ...

. We prove lower bounds for a proof system having exponential speed-up over both polynomial calculus and constant-depth Frege systems in DeMorgan language. Introduction An interesting open problem is to prove a lower bound for constant-depth subsystems of F (MOD p ) (cf. [5, Definition 12.6.1] or [3]), a system that combines boolean reasoning (DeMorgan language) with algebraic reasoning (counting modulo p). In this note we show that the lower bound for the degree of Nullstellensatz proofs for the onto pigeonhole principle PHP n+m n from [2], as well as the bound for the polynomial calculus proofs of the counting principles Count n q from [8], imply lower bounds for a weak subsystem of F (MOD p ) that has nevertheless an exponential speed-up over constant-depth Frege systems in DeMorgan language and over polynomial calculus. Namely, the proofs in the system, call it F c d (MOD p ) here, are F (MOD p ) - proofs that may use only formulas that can be obtained by substituting DeMorgan...