The Hilbert Function, Algebraic Extractors, and Recursive Fourier Sampling

In this paper, we apply tools from algebraic geometry to prove new results concerning extractors for algebraic sets, the recursive Fourier sampling problem, and VC dimension. We present a new construction of an extractor which works for algebraic sets defined by polynomials over GF(2) of substantially higher degree than the current state-of-the-art construction. We also exactly determine the GF(2)-polynomial degree of the recursive Fourier sampling problem and use this to provide new partial results towards a circuit lower bound for this problem. Finally, we answer a question concerning VC dimension, interpolation degree and the Hilbert function.