Testing low-degree polynomials over prime fields

  • Authors:

  • Charanjit S. Jutla;Anindya C. Patthak;Atri Rudra;David Zuckerman

  • Affiliations:

  • IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598;University of California, Riverside, California 92521;Department of Computer Science and Engineering, University at Buffalo, The State University of New York, Buffalo, New York 14260;Department of Computer Science, University of Texas at Austin, Austin, Texas 78712

  • Venue:
  • Random Structures & Algorithms
  • Year:
  • 2009

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Abstract

We present an efficient randomized algorithm to test if a given function f : 𝔽 pn → 𝔽p (where p is a prime) is a low-degree polynomial. This gives a local test for Generalized Reed-Muller codes over prime fields. For a given integer t and a given real ε 0, the algorithm queries f at O($ O({{1}\over{\epsilon}}+t.p^{{2t \over p-1}+1}) $) points to determine whether f can be described by a polynomial of degree at most t. If f is indeed a polynomial of degree at most t, our algorithm always accepts, and if f has a relative distance at least ε from every degree t polynomial, then our algorithm rejects f with probability at least $ {1\over 2} $. Our result is almost optimal since any such algorithm must query f on at least $ \Omega ( {1 \over \epsilon} + p^ {t+1 \over p-1})$ points. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009 A preliminary version of this paper appeared in 45th Symposium on Foundations of Computer Science, 2004. Most of this work was done while the author (Patthak) was at the University of Texas at Austin. This work was done while the author (Rudra) was at the University of Texas at Austin.