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Polynomial systems supported on circuits and dessins d'enfants
Author(s) -
Bihan Frédéric
Publication year - 2007
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdl013
Subject(s) - modulo , mathematics , dimension (graph theory) , polynomial , riemann hypothesis , rank (graph theory) , affine transformation , upper and lower bounds , set (abstract data type) , electronic circuit , span (engineering) , combinatorics , discrete mathematics , pure mathematics , mathematical analysis , computer science , physics , civil engineering , quantum mechanics , engineering , programming language
We study polynomial systems in which equations have as common support a set of n + 2 points in ℤ 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 ℤ n is n + 1, which is very small compared to the bound given by the Khovanskii fewnomial theorem.