On the Asymptotic Nullstellensatz and Polynomial Calculus Proof Complexity

We show that the asymptotic complexity of uniformly generated (expressible in First-Order (FO) logic) propositional tautologies for the Nullstellensatz proof system (NS) as well as for Polynomial Calculus, (PC) has four distinct types of asymptotic behavior over fields of finite characteristic. More precisely, based on some highly non-trivial work by Krajicek, we show that for each prime p there exists a function l(n) \in \Omega(\log(n)) for NS and l(n) \in \Omega(\log(\log(n)) for PC, such that the propositional translation of any FO formula (that  fails in all finite models), has degree proof complexity over fields of characteristic p, that behave in 4 mutually distinct ways:(i) The degree complexity is bound by a constant.(ii) The degree complexity is at least l(n) for all values of n.(iii)  The degree complexity is at least l(n) except in a finite number of regular subsequences of inifinite size, where the degree is constant.(iv) The degree complexity fluctuates in a very particular way with the degree complexity taking different constant values on an infinite number of regular subsequences each of infinite size.We leave it as an open question whether the classification remains valid for l(n) \in n^\Omega(1) or even for l(n) \in \Omega(n).  Finally, we show that for any non-empty proper subset A \subseteq \(i), (ii), (iii), (iv)\ the decision problem of whether a given input FO formula \psi has type belonging to A -  is undecidable.