Testing Low Complexity Affine-Invariant Properties

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 Fp 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 : Fp → [R] is testable, assuming that 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. ∗Center for Computational Intractability. Supported by NSF Grants CCF-0832797, 0830673, and 0528414. †Faculty of Computer Science. The research leading to these results has received funding from the ERC 7th Framework Programme grant number 202405. ‡School of Mathematics. Supported by NSF grant DMS-0835373. ar X iv :1 20 1. 03 30 v2 [ cs .C C ] 8 O ct 2 01 2