Polynomial systems supported on circuits and dessins d'enfants
Frédéric Bihan · Journal of the London Mathematical Society · 2007
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.