2007/07/20 by Julián Pfeifle, Julian Pfeifle, Pfeifle, Julian
Computer Science · Mathematics · #12D10 #52B35 #Advanced Combinatorial Mathematics #Basis (linear algebra) #Chromatic scale #Combinatorics #Combinatorics (math.CO) #Complex plane #Computer science #Degree (music) #Dual space #Duality (order theory) #FOS: Mathematics #Geometry #Mathematical analysis #Mathematics #Plane (geometry) #Point processes and geometric inequalities #Polynomial and algebraic computation #Pure mathematics #Space (punctuation) #Upper and lower bounds #Vector space #math.CO #msc:12D10 #msc:52B35
paper · pdf · doi:10.48550/arxiv.0707.3010
25 pages, 10 figures. Final version incorporating referees' comments, to appear in J. Comb. Theory, Ser. A
openalex publication_date 2007/07/20 · arxiv created 2009/11/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We bound the location of roots of polynomials that have nonnegative coefficients with respect to a fixed but arbitrary basis of the vector space of polynomials of degree at most d. For this, we interpret the basis polynomials as vector fields in the real plane, and at each point in the plane analyze the combinatorics of the Gale dual vector configuration. This approach permits us to incorporate arbitrary linear equations and inequalities among the coefficients in a unified manner to obtain more precise bounds on the location of roots. We apply our technique to bound the location of roots of Ehrhart and chromatic polynomials. Finally, we give an explanation for the clustering seen in plots of roots of random polynomials.