Characterization of circuits supporting polynomial systems with the\n maximal number of positive solutions

2016/03/06 by Boulos El Hilany, Hilany, Boulos El · 1 citation
Computer Science · Mathematics · #13P15 #14H57 #14P25 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Coding theory and cryptography #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1603.01813

openalex publication_date 2016/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A polynomial system with n equations in n variables supported on a set\n\W\⊂\ℝⁿ of n+2 points has at most n+1\nnon-degenerate positive solutions. Moreover, if this bound is reached, then\n\W is minimally affinely dependent, in other words, it is a circuit\nin \ℝⁿ. For any positive integer number n, we determine all\ncircuits \W\⊂\ℝⁿ which can support a polynomial system\nwith n+1 non-degenerate positive solutions. Restrictions on such circuits\n\W are obtained using Grothendieck's real dessins d'enfant, while\npolynomial systems with n+1 non-degenerate positive solutions are constructed\nusing Viro's combinatorial patchworking.\n

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