, Volume 17, Issue 3, pp 345–362 | Cite as

Polynomials with two values

  • Joachim von Zur Gathen
  • James R. Roche


This paper investigates the minimal degree of polynomialsfR[x] that take exactly two values on a given range of integers {0,...n}. We show that thegap, defined asn-deg(f), isO(n548). The maximal gap forn≤128 is 3. As an application, we obtain a bound on the Fourier degree of symmetric Boolean functions.

Mathematics Subject Classification (1991):

68R05 11B83 11Y50 11B39 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R. L. Graham, D. E. Knuth, andO. Patashnik:Concrete Mathematics, Addison-Wesley, New York, 1989.Google Scholar
  2. [2]
    C. J. Mozzochi: On the difference between consecutive primes,J. Number Theory,24 (1986), 181–187.Google Scholar
  3. [3]
    N. Nisan andM. Szegedy: On the degree of Boolean functions as real polynomials,STOC, 1992, Victoria, 462–474.Google Scholar
  4. [4]
    D. Singmaster: Repeated binomial coefficients and Fibonacci numbers,The Fibonacci Quarterly,13 (1975), 295–298.Google Scholar
  5. [5]
    B. M. Stewart:Theory of Numbers, 2nd edition, The Macmillan Co., New York, 1965.Google Scholar
  6. [6]
    C. A. Tovey: Multiple occurrences of binomial coefficients,The Fibonacci Quarterly,23 (1985), 356–358.Google Scholar

Copyright information

© János Bolyai mathematical Society 1997

Authors and Affiliations

  • Joachim von Zur Gathen
    • 1
  • James R. Roche
    • 2
  1. 1.FB Mathematik-InformatikUniversität-GH PaderbornPaderbornGermany
  2. 2.Center for Communications ResearchPrincetonUSA

Personalised recommendations