IPS Lower Bounds for Formulas and Sum of ROABPs

TL;DR

This paper establishes new lower bounds for algebraic proof systems, specifically for variants of the Ideal Proof System (IPS), demonstrating limitations of certain algebraic refutations over different fields and polynomial classes.

Contribution

It provides the first nearly quadratic-size formula lower bounds and exponential-size sum-of-ROABPs lower bounds for IPS variants, advancing algebraic proof complexity understanding.

Findings

Nearly quadratic-size formula lower bounds for multilinear refutations.

Exponential-size sum-of-ROABPs lower bounds over characteristic zero fields.

Extensions of bounds to fields of positive characteristic with polynomial modifications.

Abstract

We give new lower bounds for the fragments of the Ideal Proof System (IPS) introduced by Grochow and Pitassi (JACM 2018). The Ideal Proof System is a central topic in algebraic proof complexity developed in the context of Nullstellensatz refutation (Beame, Impagliazzo, Krajicek, Pitassi, Pudlak, FOCS 1994) and simulates Extended Frege efficiently. Our main results are as follows. 1. mult-IPS_{Lin'}: We prove nearly quadratic-size formula lower bound for multilinear refutation (over the Boolean hypercube) of a variant of the subset-sum axiom polynomial. Extending this, we obtain a nearly matching qualitative statement for a constant degree target polynomial. 2. IPS_{Lin'}: Over the fields of characteristic zero, we prove exponential-size sum-of-ROABPs lower bound for the refutation of a variant of the subset-sum axiom polynomial. The result also extends over the fields of positive characteristics when the target polynomial is suitably modified. The modification is inspired by the recent results (Hakoniemi, Limaye, Tzameret, STOC 2024 and Behera, Limaye, Ramanathan, Srinivasan, ICALP 2025). The mult-IPS_{Lin'} lower bound result is obtained by combining the quadratic-size formula lower bound technique of Kalorkoti (SICOMP 1985) with some additional ideas. The proof technique of IPS_{Lin'} lower bound result is inspired by the recent lower bound result of Chatterjee, Kush, Saraf and Shpilka (CCC 2024).