2014/12/15 by Abhishek Bhowmick, Shachar Lovett, Bhowmick, Abhishek +1
Computer Science · #68Qxx #Computational Complexity (cs.CC) #F.0 #FOS: Computer and information sciences #acm:68Qxx #cs.CC #msc:68Qxx
paper · pdf · doi:10.48550/arxiv.1412.4719
arxiv created 2014/12/15 · arxiv updated 2014/12/16
The problem of constructing explicit functions which cannot be approximated by low degree polynomials has been extensively studied in computational complexity, motivated by applications in circuit lower bounds, pseudo-randomness, constructions of Ramsey graphs and locally decodable codes. Still, most of the known lower bounds become trivial for polynomials of super-logarithmic degree. Here, we suggest a new barrier explaining this phenomenon. We show that many of the existing lower bound proof techniques extend to nonclassical polynomials, an extension of classical polynomials which arose in higher order Fourier analysis. Moreover, these techniques are tight for nonclassical polynomials of logarithmic degree.