Nonclassical polynomials as a barrier to polynomial lower bounds

TL;DR

This paper introduces nonclassical polynomials as a new barrier in polynomial lower bounds, extending existing techniques and explaining the limitations of current proof methods for super-logarithmic degrees.

Contribution

It demonstrates that many lower bound techniques extend to nonclassical polynomials, providing a new barrier and tight bounds for logarithmic degree cases.

Findings

Existing lower bound techniques extend to nonclassical polynomials.

The techniques are tight for nonclassical polynomials of logarithmic degree.

Provides a new barrier explaining limitations for super-logarithmic degrees.

Abstract

The problem of constructing explicit functions which cannot be approximated by low degree polynomials has been extensively studied in computational complexity, motivated by applications in circuit lower bounds, pseudo-randomness, constructions of Ramsey graphs and locally decodable codes. Still, most of the known lower bounds become trivial for polynomials of super-logarithmic degree. Here, we suggest a new barrier explaining this phenomenon. We show that many of the existing lower bound proof techniques extend to nonclassical polynomials, an extension of classical polynomials which arose in higher order Fourier analysis. Moreover, these techniques are tight for nonclassical polynomials of logarithmic degree.