Polynomial systems supported on circuits and dessins d'enfants

• Autores: F. Bihan
• Localización: Journal of the London Mathematical Society, ISSN 0024-6107, Vol. 75, Nº 1, 2007, págs. 116-132
• Idioma: inglés
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• Resumen

  • We study polynomial systems in which equations have as common support a set of n + 2 points in n 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 , and the rank modulo 2 of . 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 n is n + 1, which is very small compared to the bound given by the Khovanskii fewnomial theorem