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 chi, either chi(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 chi(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 F_p. Our technical results are: 1. If P: K^n -> K is a polynomial of degree |K|^{-s} for some s > 0 and non-trivial additive character chi, then P is a function of O_{d, 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^n. Then, if P: K^n -> K is a polynomial of degree d that is sufficiently regular, then (P(x): x in 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^n -> 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^n -> K are testable with one-sided error

Improved Local Testing for Multiplicity Codes

Multiplicity codes are a generalization of Reed-Muller codes which include derivatives as well as the values of low degree polynomials, evaluated in every point in ?_p^m. Similarly to Reed-Muller codes, multiplicity codes have a local nature that allows for local correction and local testing. Recently, [Karliner et al., 2022] showed that the plane test, which tests the degree of the codeword on a random plane, is a good local tester for small enough degrees. In this work we simplify and extend the analysis of local testing for multiplicity codes, giving a more general and tight analysis. In particular, we show that multiplicity codes MRM_p(m, d, s) over prime fields with arbitrary d are locally testable by an appropriate k-flat test, which tests the degree of the codeword on a random k-dimensional affine subspace. The relationship between the degree parameter d and the required dimension k is shown to be nearly optimal, and improves on [Karliner et al., 2022] in the case of planes. Our analysis relies on a generalization of the technique of canonincal monomials introduced in [Haramaty et al., 2013]. Generalizing canonical monomials to the multiplicity case requires substantially different proofs which exploit the algebraic structure of multiplicity codes

The Multivariate Resultant is NP-hard in any Characteristic

The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n homogeneous equations in n variables is satisfiable (the resultant is a polynomial in the system's coefficients which vanishes if and only if the system is satisfiable). In this paper we present several NP-hardness results for testing whether a multivariate resultant vanishes, or equivalently for deciding whether a square system of homogeneous equations is satisfiable. Our main result is that testing the resultant for zero is NP-hard under deterministic reductions in any characteristic, for systems of low-degree polynomials with coefficients in the ground field (rather than in an extension). We also observe that in characteristic zero, this problem is in the Arthur-Merlin class AM if the generalized Riemann hypothesis holds true. In positive characteristic, the best upper bound remains PSPACE.Comment: 13 page