Low-degree test with polynomially small error

Abstract

A long line of work in Theoretical Computer Science shows that a function is close to a low-degree polynomial iff it is locally close to a low-degree polynomial. This is known as low-degree testing and is the core of the algebraic approach to construction of PCP. We obtain a low-degree test whose error, i.e., the probability it accepts a function that does not correspond to a low-degree polynomial, is polynomially smaller than existing low-degree tests. A key tool in our analysis is an analysis of the sampling properties of the incidence graph of degree-k curves and k′-tuples of points in a finite space \({\mathbb{F}^m}\). We show that the Sliding Scale Conjecture in PCP, namely the conjecture that there are PCP verifiers whose error is exponentially small in their randomness, would follow from a derandomization of our low-degree test.