On Higher-Order Fourier Analysis over Non-Prime Fields

The celebrated Weil bound for character sums says that for any low-degree polynomial P and any additive character χ, either χ(P ) is a constant function or it is distributed close to uniform. The goal of higher-order Fourier analysis is to understand the connection between the algebraic and analytic properties of polynomials (and functions, generally) at a more detailed level. For instance, what is the tradeoff between the equidistribution of χ(P ) and its “structure"? Previously, most of the work in this area was over fields of prime order. We extend the tools of higher-order Fourier analysis to analyze functions over general finite fields. Let K be a field extension of a prime finite field Fp. Our technical results are: 1. If P : K → K is a polynomial of degree 6 d, and E[χ(P (x))] > |K|−s for some s > 0 and nontrivial additive character χ, then P is a function of Od,s(1) many non-classical polynomials of weight degree < d. The definition of non-classical polynomials over non-prime fields is one of the contributions of this work. 2. Suppose K and F are of bounded order, and let H be an affine subspace of K. Then, if P : K → K is a polynomial of degree d that is sufficiently regular, then (P (x) : x ∈ H) is distributed almost as uniformly as possible subject to constraints imposed by the degree of P . Such a theorem was previously known for H an affine subspace over a prime field. The tools of higher-order Fourier analysis have found use in different areas of computer science, including list decoding, algorithmic decomposition and testing. Using our new results, we revisit some of these areas. (i) For any fixed finite field K, we show that the list decoding radius of the generalized Reed Muller code over K equals the minimum distance of the code. (ii) For any fixed finite field K, we give a polynomial time algorithm to decide whether a given polynomial P : K → K can be decomposed as a particular composition of lesser degree polynomials. (iii) For any fixed finite field K, we prove that all locally characterized affine-invariant properties of functions f : K → K are testable with one-sided error. 1998 ACM Subject Classification G.2 Discrete Mathematics, D.2.5 Testing and Debugging