Optimality of size-degree tradeoffs for polynomial calculus

There are methods to turn short refutations in <i>polynomial calculus</i> (Pc) and <i>polynomial calculus with resolution</i> (Pcr) into refutations of low degree. Bonet and Galesi [1999, 2003] asked if such size-degree tradeoffs for Pc [Clegg et al. 1996; Impagliazzo et al. 1999] and Pcr [Alekhnovich et al. 2004] are optimal. We answer this question by showing a polynomial encoding of the <i>graph ordering principle</i> on <i>m</i> variables which requires Pc and Pcr refutations of degree Ω(&sqrt; <i>m</i>). Tradeoff optimality follows from our result and from the short refutations of the graph ordering principle in Bonet and Galesi [1999, 2001]. We then introduce the algebraic proof system Pcr<i>_k</i> which combines together polynomial calculus and <i>k-DNF resolution</i> (Res<i>_k</i>). We show a size hierarchy theorem for Pcr<i>_k</i>: Pcr<i>_k</i> is exponentially separated from Pcr<i>_k+1</i>. This follows from the previous degree lower bound and from techniques developed for Res<i>_k</i>. Finally we show that random formulas in conjunctive normal form (3-CNF) are hard to refute in Pcr<i>_k</i>.