The main results of this article are asymptotic formulas for the variance of the number of zeros of a Gaussian random polynomial of degree N in an open set U ⊂ C as the degree N → ∞, and more generally for the zeros of random holomorphic sections of high powers of any positive line bundle over any Riemann surface. The formulas were conjectured in special cases by Forrester and Honner. In higher...

Of central importance in number theory is the distribution of the complex zeros of f(s), all of which are in the critical strip 0 < a < 1 and are symmetrically located about the real axis and about the critical line a = 1/2. Riemann conjectured in 1859 that all of these zeros are on the critical line; this conjecture, which is still unproved, is known as the Riemann Hypothesis. The number of ze...

The characteristic equation of a system of delay diierential equations (DDEs) is a nonlinear equation with innnitely many zeros. The stability of a steady state solution of such a DDE system is determined by the number of zeros of this equation with positive real part. We present a numerical algorithm to compute the rightmost, i.e. stability determining, zeros of the characteristic equation. Th...

In this paper we consider the numerical problems faced by a blind dereverberation algorithm based on a multi-channel linear prediction. One hypothesis frequently incorporated in multimicrophone dereverberation algorithms is that channels do not share common zeros. However, it is known that real room transfer functions have a large number of zeros close to the unit circle on the z-plane, and thu...

We consider the number of zeros of the integral I(h) = ∮ Γh ω of real polynomial form ω of degree not greater than n over a family of vanishing cycles on curves Γh : y 2 + 3x − x = h, where the integral is considered as a function of the parameter h. We prove that the number of zeros of I(h), for 0 < h < 2, is bounded above by 2[n−1 2 ] + 1.

which is valid for any complex s, it follows that ζ(s) has zeros at s = −2,−4, . . . . These zeros are called the “trivial” zeros of ζ(s), to distinguish them from the complex zeros of ζ(s). The zeta-function has also an infinity of complex zeros. It is well-known that all complex zeros of ζ(s) lie in the so-called “critical strip” 0 < σ = R s < 1, and if N(T ) denotes the number of zeros ρ = β...