Lower Bounds for Polynomial Calculus with Extension Variables over Finite Fields.

For every prime p > 0, every n > 0 and κ = O(log n), we show the existence of an unsatisfiable system of polynomial equations over O(n log n) variables of degree O(log n) such that any Polynomial Calculus refutation over Fp with M extension variables, each depending on at most κ original variables requires size exp ( Ω(n2/(κ22κ(M+ n log(n)))) ) . ∗Supported by the Simons Foundation and NSF grant CCF-1909634. †Supported by the Simons Foundation and NSF grant CCF-1909634. ‡This material is based upon work supported by the National Science Foundation Grant No. CCF1900460, and by the IAS School of Mathematics. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.