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Journal of the London Mathematical Society Advance Access originally published online on January 18, 2007
Journal of the London Mathematical Society 2007 75(1):116-132; doi:10.1112/jlms/jdl013
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© 2007 London Mathematical Society

Polynomial systems supported on circuits and dessins d'enfants

Frédéric Bihan

Laboratoire de Mathématiques (LAMA)
Unite Mixte de Recherche 5127 CNRS
Université de Savoie
UFR SFA, Campus Scientifique
73376 Le Bourget-du-Lac cedex
France
frederic.bihan{at}univ-savoie.fr

We study polynomial systems in which equations have as common support a set C of n + 2 points in Zn called a circuit. We find a bound on the number of real solutions to such systems which depends on n, the dimension of the affine span of the minimal affinely dependent subset of C, and the rank modulo 2 of C. We prove that this bound is sharp by drawing the so-called dessins d’enfants on the Riemann sphere. We also obtain that the maximal number of solutions with positive coordinates to systems supported on circuits in Zn is n + 1, which is very small compared to the bound given by the Khovanskii fewnomial theorem.

2000 Mathematics Subject Classification 12D10, 14M25.

Received October 28, 2005; published online January 18, 2007.


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Home page
 Int Math Res NoticesHome page
D. J. Bates, F. Bihan, and F. Sottile
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Int Math Res Notices, January 1, 2007; 2007(rnm114): rnm114 - 7.
[Abstract] [PDF]


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