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On the sharpness of fewnomial bounds and the number of components of fewnomial hypersurfaces

  • Frédéric Bihan
  • J. Maurice Rojast
  • Frank Sottile
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 146)

Abstract

We prove the existence of systems of n polynomial equations in n variables with a total of n+k+1 distinct monomial terms possessing \( \left\lfloor {\tfrac{{n + k}} {k}} \right\rfloor ^k \) nondegenerate positive solutions. This shows that the recent upper bound of \( \tfrac{{e^2 + 3}} {4}2^{\left( \begin{subarray}{l} k \\ 2 \end{subarray} \right)} n^k \) for the number of nondegenerate positive solutions has the correct order for fixed k and large n. We also adapt a method of Perrucci to show that there are fewer than \( \tfrac{{e^2 + 3}} {4}2^{\left( {\begin{array}{*{20}c} {k + 1} \\ 2 \\ \end{array} } \right)} 2^n n^{k + 1} \) connected components in a smooth hypersurface in the positive orthant of ℝ N defined by a polynomial with n+k+1 monomials, where n is the dimension of the affine span of the exponent vectors. Our results hold for polynomials with real exponents.

Key words

Fewnomials connected component 

AMS(MOS) subject classifications

14P99 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Frédéric Bihan
    • 1
  • J. Maurice Rojast
    • 2
  • Frank Sottile
    • 2
  1. 1.Laboratoire de MathématiquesUniversité de SavoieLe Bourget-du-Lac CedexFrance
  2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA

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