On the Role of Algebra in the Efficient Verification of Proofs

This article extracts the elements of algebra that play a central role in the design of efficient probabilistic verifiers or "probabilistically checkable proof systems (PCPs)". The main algebraic elements are low-degree polynomials over finite fields. Their role can be broken up into three essential elements: 1. Their classical role in the design of error-correcting codes. 2. Their recently discovered property of being efficiently locally checkable. 3. The existence of characterizations via polynomials of fundamental complexity classes including NP, PSPACE and NEXPTIME