Few Product Gates But Many Zeros

  • Bernd Borchert
  • Pierre McKenzie
  • Klaus Reinhardt
Conference paper

DOI: 10.1007/978-3-642-03816-7_15

Part of the Lecture Notes in Computer Science book series (LNCS, volume 5734)
Cite this paper as:
Borchert B., McKenzie P., Reinhardt K. (2009) Few Product Gates But Many Zeros. In: Královič R., Niwiński D. (eds) Mathematical Foundations of Computer Science 2009. MFCS 2009. Lecture Notes in Computer Science, vol 5734. Springer, Berlin, Heidelberg

Abstract

A d-gem is a { + , − ,×}-circuit having very few ×-gates and computing from {x} ∪ ℤ a univariate polynomial of degree d having d distinct integer roots. We introduce d-gems because they could help factoring integers and because their existence for infinitely many d would blatantly disprove a variant of the Blum-Cucker-Shub-Smale conjecture. A natural step towards validating the conjecture would thus be to rule out d-gems for large d. Here we construct d-gems for several values of d up to 55. Our 2n-gems for n ≤ 4 are skew, that is, each { + , − }-gate adds an integer. We prove that skew 2n-gems if they exist require n { + , − }-gates, and that these for n ≥ 5 would imply new solutions to the Prouhet-Tarry-Escott problem in number theory. By contrast, skew d-gems over the real numbers are shown to exist for every d.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Bernd Borchert
    • 1
  • Pierre McKenzie
    • 2
  • Klaus Reinhardt
    • 1
  1. 1.WSIUniversität TübingenTübingenGermany
  2. 2.DIROUniv. de MontréalMontréalCanada

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