Testing Low Complexity Affine-Invariant Properties

TL;DR

This paper proves that all low complexity affine-invariant properties of functions over finite fields are testable with a constant number of queries, using algebraic and Fourier analysis techniques.

Contribution

It establishes a general testability result for affine-invariant properties with low complexity, extending prior specific cases like Reed-Muller codes.

Findings

All low complexity affine-invariant properties are testable with constant queries.

Reed-Muller codes over finite fields are testable without detailed polynomial algebra.

Develops algebraic analogs of graph-theoretic techniques using higher-order Fourier analysis.

Abstract

Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affine-invariant property of multivariate functions over finite fields is testable with a constant number of queries. This immediately reproves, for instance, that the Reed-Muller code over F_p of degree d < p is testable, with an argument that uses no detailed algebraic information about polynomials except that low degree is preserved by composition with affine maps. The complexity of an affine-invariant property P refers to the maximum complexity, as defined by Green and Tao (Ann. Math. 2008), of the sets of linear forms used to characterize P. A more precise statement of our main result is that for any fixed prime p >=2 and fixed integer R >= 2, any affine-invariant property P of functions f: F_p^n -> [R] is testable, assuming the complexity of the property is less than p. Our proof involves developing analogs of graph-theoretic techniques in an algebraic setting, using tools from higher-order Fourier analysis.