On the zeros of univariate E-polynomials

Abstract

We consider two problems concerning real zeros of univariate E-polynomials. First, we prove an explicit upper bound for the absolute values of the zeroes of an E-polynomial defined by polynomials with inte-ger coefficients that improves the bounds known up to now. On the other hand, we extend the classical Budan—Fourier theorem for real polynomials to E-polynomials. This result gives, in particular, an upper bound for the number of real zeroes of an E-polynomial. We show this bound is sharp for particular families of these functions, which proves that a conjecture by D. Richardson is false.