Lower bounds for the polynomial calculus

Abstract.

We show that polynomial calculus proofs (sometimes also called Groebner proofs) of the pigeonhole principle $P H P^m_n$ must have degree at least $(n/2)+1$ over any field. This is the first non-trivial lower bound on the degree of polynomial calculus proofs obtained without using unproved complexity assumptions. We also show that for some modifications of $P H P^m_n$, expressible by polynomials of at most logarithmic degree, our bound can be improved to linear in the number of variables. Finally, we show that for any Boolean function $f_n$ in n variables, every polynomial calculus proof of the statement “$f_n$ cannot be computed by any circuit of size t,” must have degree $\Omega(t/n)$. Loosely speaking, this means that low degree polynomial calculus proofs do not prove ${\bf NP}\not\subseteq {\bf P}/poly$.