 
Summary: Lower Bounds for Polynomial Calculus:
NonBinomial Case
Michael Alekhnovich # , Alexander A. Razborov +
September 27, 2000
Abstract
We generalize recent linear lower bounds for Polynomial Calculus
based on binomial ideals. We produce a general hardness criterion
(that we call immunity) which is satisfied by a random function and
prove linear lower bounds on the size of PC refutations for a wide class
of tautologies based on immune functions. As some applications of our
techniques, we introduce mod p Tseitin tautologies in the Boolean case
(e.g. in the presence of axioms x 2
i
= x i ), prove that they are hard for
PC over fields with characteristic di#erent from p, and generalize them
to Flow tautologies which are based on the MAJORITY function and
are proved to be hard over any field. We also show
the# n) lower
bound for random kCNF's over fields of characteristic 2.
1 Introduction
