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We bound the location of roots of polynomials that have nonnegativecoefficients with respect to a fixed but arbitrary basis of the vector space ofpolynomials of degree at most $d$. For this, we interpret the basis polynomialsas vector fields in the real plane, and at each point in the plane analyze thecombinatorics of the Gale dual vector configuration. This approach permits usto incorporate arbitrary linear equations and inequalities among thecoefficients in a unified manner to obtain more precise bounds on the locationof roots. We apply our technique to bound the location of roots of Ehrhart andchromatic polynomials. Finally, we give an explanation for the clustering seenin plots of roots of random polynomials.

Gale duality bounds for roots of polynomials with nonnegative coefficients