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The Ideal Proof System (IPS) of Grochow & Pitassi (FOCS 2014, J. ACM, 2018)is an algebraic proof system that uses algebraic circuits to refute thesolvability of unsatisfiable systems of polynomial equations. One potentialdrawback of IPS is that verifying an IPS proof is only known to be doable usingPolynomial Identity Testing (PIT), which is solvable by a randomized algorithm,but whose derandomization, even into NSUBEXP, is equivalent to strong lowerbounds. However, the circuits that are used in IPS proofs are not arbitrary,and it is conceivable that one could get around general PIT by leveraging somestructure in these circuits. This proposal may be even more tempting when IPSis used as a proof system for Boolean Unsatisfiability, where the equationsthemselves have additional structure.  Our main result is that, on the contrary, one cannot get around PIT as above:we show that IPS, even as a proof system for Boolean Unsatisfiability, can bep-simulated by a deterministically verifiable (Cook-Reckhow) proof system ifand only if PIT is in NP. We use our main result to propose a potentially newapproach to derandomizing PIT into NP.

Polynomial Identity Testing and the Ideal Proof System: PIT is in NP if and only if IPS can be p-simulated by a Cook-Reckhow proof system