Gale duality bounds for roots of polynomials with nonnegative coefficients

Details Gale duality bounds for roots of polynomials with nonnegative coefficients Gale duality bounds for roots of polynomials with nonnegative coefficients

2007-07-20

arxiv.org/abs/0707.3010v2

Mathematics

Combinatorics

25 pages, 10 figures. Final version incorporating referees' comments, to appear in J. Comb. Theory, Ser. A

Scientific paper

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.

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