Polynomial systems supported on circuits and dessins d'enfants

Abstract

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