Over the past ten years, many examples of natural polynomial families from combinatorics and number theory have emerged whose zeros for high degrees appear to converge to intriguing curves in the complex plane. One interesting collection of examples appears on the website [16] of Richard Stanley which includes chromatic polynomials of complete partite graphs, q-analogue of Catalan numbers, Bern...

Using a finite element approximation, an iterative optimization scheme is described to find the z zeros of Kniz) for fixed order n. Two computer programs have been implemented to find the complex zeros with a computational accuracy of either 13 or 27 significant digits. The optimization scheme described in the paper may also be readily applied to find real and complex zeros of an arbitrary func...

There is an intimate connection between the distribution of the nontrivial zeros of the Riemann zeta function ζ(s) and the distribution of prime numbers. Critical to many prime number problems is the horizontal distribution of zeros; here the Riemann Hypothesis (RH) asserts that the zeros all have real part 1 2 . There is also much interest in studying the distribution of the imaginary parts of...

The expected number of real zeros of polynomials a0+a1x+a2x+ · · · + an−1xn−1 with random coefficients is well studied. For n large and for the normal zero mean independent coefficients, irrespective of the distribution of coefficients, this expected number is known to be asymptotic to (2/π) logn. For the dependent cases studied so far it is shown that this asymptotic value remains O(logn). In ...

We prove a complex analytic analogue of the classical Rolle theorem asserting that the number of zeros of a real smooth function can exceed that of its derivative by at most 1. This result is used then to obtain upper bounds for the number of complex isolated zeros of : (1) functions defined by linear ordinary differential equations (in terms of the magnitude of the coefficients of the equation...