2005/09/09 by Frédéric Bihan, F. Bihan, Bihan, F.
Computer Science · Mathematics · #12D10 #14M25 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AG #msc:12D10 #msc:14M25
paper · pdf · doi:10.48550/arxiv.math/0509219
19 pages, 5 figures, Section 3.1 revised, minor changes in other sections
openalex publication_date 2005/09/09 · arxiv created 2005/09/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study polynomial systems whose 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 so-called dessins d'enfant 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 comparatively to the bound given by the Khovanskii fewnomial theorem.