Linear Gaps Between Degrees for the Polynomial Calculus Modulo Distinct Primes Sam Buss1;2 Dima Grigoriev Department of Mathematics Computer Science and Engineering Univ. of Calif., San Diego Pennsylvania State University La Jolla, CA 92093-0112 University Park, PA 16802-6106
[email protected] [email protected] Russell Impagliazzo1;3 Toniann Pitassi1;4 Computer Science and Engineering Computer Science Univ. of Calif., San Diego University of Arizona La Jolla, CA 92093-0114 Tucson, AZ 85721-0077
[email protected] [email protected] Abstract e±cient search algorithms and in part by the desire to extend lower bounds on proposition proof complexity This paper gives nearly optimal lower bounds on the to stronger proof systems. minimum degree of polynomial calculus refutations of The Nullstellensatz proof system is a propositional Tseitin's graph tautologies and the mod p counting proof system based on Hilbert's Nullstellensatz and principles, p 2. The lower bounds apply to the was introduced in [1]. The polynomial calculus (PC) ¸ polynomial calculus over ¯elds or rings. These are is a stronger propositional proof system introduced the ¯rst linear lower bounds for polynomial calculus; ¯rst by [4]. (See [8] and [3] for subsequent, more moreover, they distinguish linearly between proofs over general treatments of algebraic proof systems.) In the ¯elds of characteristic q and r, q = r, and more polynomial calculus, one begins with an initial set of 6 generally distinguish linearly the rings Zq and Zr where polynomials and the goal is to prove that they cannot q and r do not have the identical prime factors.