Lower Bounds on Hilbert's Nullstellensatz and Propositional Proofs

Abstract

The so-called weak form of Hilbert's Nullstellensatz says that a system of algebraic equations over a field, Q i ( x ¯ ) = 0 , does not have a solution in the algebraic closure if and only if 1 is in theideal generated by the polynomials Q i ( x ¯ ) . We shall prove a lower bound on the degrees of polynomials P i ( x ¯ ) such that ∑ i P i ( x ¯ ) Q i ( x ¯ ) = 1 . This result has the following application. The modular counting principle states that no finite set whose cardinality is not divisible by q can be partitioned into q-element classes. For each fixed cardinality N, this principle can be expressed as a propositional formula Count Count q N ( x e , … ) with underlying variables xe, where e ranges over q-element subsets of N. Ajtai [4] proved recently that, whenever p,q are two different primes, the propositional formulas Count Count q q n + 1 do not have polynomial size, constant-depth Frege proofs from substitution instances of Count Count p m , where m ≢ 0 (mod p). We give a new proof of this theorem based on the lower bound for Hilbert's Nullstellensatz. Furthermore our technique enables us to extend the independence results for counting principles to composite numbers p and q. This improved lower bound together with new upper bounds yield an exact characterization of when Countq can be proved efficiently from Countp, for all values of p and q