Mathematische Zeitschrift

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

Polynomial systems with few real zeroes

  • Benoît Bertrand
  • Frédéric Bihan
  • Frank SottileEmail author


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.


Convex Hull Real Solution Polynomial System Real Zero Affine Span 
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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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