Real Algebraic and Analytic Geometry

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172. Frédéric Bihan:
Polynomial systems supported on circuits and dessins d'enfants.

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Submission: 2005, September 20.

Abstract:
We study real solutions to polynomial systems whose equations have as common support a set, called a circuit, of n+2 points in Z^n. We find a bound on the number of real solutions to such systems which depends on n and two characteristic numbers of the circuit. We prove that this bound is sharp by drawing so-called dessins d'enfant on the Riemann sphere. We also obtain that the maximal number of solutions with positive coordinates to such systems is n+1, which is very small comparatively to the bound given by the Khovanskii fewnomial theorem.

Mathematics Subject Classification (2000): 12D10, 14M25.

Keywords and Phrases: polynomial systems, real solutions, Fewnomials.

Full text, 19p.: pdf 260k.


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