Algorithmic regularity for polynomials and applications

In analogy with the regularity lemma of Szemeredi, regularity lemmas for polynomials shown by Green and Tao (Contrib. Discrete Math. 2009) and by Kaufman and Lovett (FOCS 2008) modify a given collection of polynomials calF = {P_1,...,P_m} to a new collection calF so that the polynomials in calF 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 calF to calF is not algorithmic for either regularity lemma. We define new notions of regularity for polynomials, which are analogous to the above, but which allow for an efficient algorithm to compute the pseudorandom collection calF. In particular, when the field is of high characteristic, in polynomial time, we can refine calF into calF where every nonzero linear combination of polynomials in calF has desirably small Gowers norm. Using the algorithmic regularity lemmas, we show that if a polynomial P of degree d is within (normalized) Hamming distance 1-1/|F| -eps 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| -eta of P, for some eta depending on eps. This can be thought of as decoding the Reed-Muller code of order k beyond the list decoding radius (finding one close codeword), when the received word P itself is a polynomial of degree d (with k < d < |F|). We also obtain an algorithmic version of the worst-case to average-case reductions by Kaufman and Lovett. 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 (randomized) algorithm to find this collection.