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Mathematische Zeitschrift

, Volume 253, Issue 2, pp 361–385 | Cite as

Polynomial systems with few real zeroes

  • Benoît Bertrand
  • Frédéric Bihan
  • Frank SottileEmail author
Article

Abstract

We study some systems of polynomials whose support lies in the convex hull of a circuit, giving a sharp upper bound for their numbers of real solutions. This upper bound is non-trivial in that it is smaller than either the Kouchnirenko or the Khovanskii bounds for these systems. When the support is exactly a circuit whose affine span is ℤ n , this bound is 2n+1, while the Khovanskii bound is exponential in n 2. The bound 2n+1 can be attained only for non-degenerate circuits. Our methods involve a mixture of combinatorics, geometry, and arithmetic.

Keywords

Convex Hull Real Solution Polynomial System Real Zero Affine Span 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Benoît Bertrand
    • 1
  • Frédéric Bihan
    • 2
  • Frank Sottile
    • 3
  1. 1.Section de mathematiquesUniversité de GenèveGenève 4Suisse
  2. 2.Laboratoire de MathématiquesUniversité de SavoieLe Bourget-du-Lac CedexFrance
  3. 3.Department of MathematicsTexas A&M UniversityCollege StationUSA

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