Algorithmic regularity for polynomials and applications

In analogy with the regularity lemma of Szemerédi [Sze75], regularity lemmas for polynomials proved by Green and Tao [GT09] and by Kaufman and Lovett [KL08] show that one can modify a given collection of polynomials F = {P1, . . ., Pm} into a new collection F' so that the polynomials in F' are \"pseudorandom\". These lemmas have various applications, such as (special cases of) Reed-Muller testing and worst-case to average-case reductions for polynomials. However, the transformation from F to F' in these works is not algorithmic. We define new notions of regularity for polynomials which, while being qualitatively equivalent to the above, also allow for efficient algorithms. Using the algorithmic regularity lemmas, we obtain an algorithmic version of the inverse theorem for Gowers norm (for bounded degree polynomials) over fields of high characteristic, by Green and Tao [GT09]. As an application, we show that if a polynomial P of degree d is within (normalized) Hamming distance 1 − 1/|F| − ε of some unknown polynomial of degree k over a prime field F (for k < d < |F|), then there is an efficient algorithm for finding a degree-k polynomial Q, which is within distance 1 − 1/|F| − η of P, for some η depending on ε. This can be thought of as decoding the Reed-Muller code of order k beyond the list decoding radius, in the sense of finding one close codeword, when the received word P itself is a polynomial (of degree larger than k but smaller than |F|). We also show an algorithmic inverse theorem for polynomials over fields of small characteristic and somewhat simplify the original proof by Tao and Ziegler [TZ12]. We also obtain an algorithmic version of the worst-case to average-case reductions by Kaufman and Lovett [KL08]. They show that if a polynomial of degree d can be weakly approximated by a polynomial of lower degree, then it can be computed exactly using a collection of polynomials of degree at most d − 1. We give an efficient algorithm to find this collection. Finally, our algorithmic regularity lemma over low characteristics can be used to efficiently decompose low-degree polynomials over Fn (for any prime order field F) into polynomials P1, . . ., Pm: Fn → F that satisfy prescribed degree bounds and for which P(x) = Γ(P1(x), . . ., Pm(x)) for a given m and Γ.