Worst Case to Average Case Reductions for Polynomials
Abstract
A degree$d$ polynomial $p$ in $n$ variables over a field $\F$ is {\em equidistributed} if it takes on each of its $\F$ values close to equally often, and {\em biased} otherwise. We say that $p$ has a {\em low rank} if it can be expressed as a bounded combination of polynomials of lower degree. Green and Tao [gt07] have shown that bias imply low rank over large fields (i.e. for the case $d < \F$). They have also conjectured that bias imply low rank over general fields. In this work we affirmatively answer their conjecture. Using this result we obtain a general worst case to average case reductions for polynomials. That is, we show that a polynomial that can be {\em approximated} by few polynomials of bounded degree, can be {\em computed} by few polynomials of bounded degree. We derive some relations between our results to the construction of pseudorandom generators, and to the question of testing concise representations.
 Publication:

arXiv eprints
 Pub Date:
 June 2008
 DOI:
 10.48550/arXiv.0806.4535
 arXiv:
 arXiv:0806.4535
 Bibcode:
 2008arXiv0806.4535K
 Keywords:

 Mathematics  Combinatorics