The paper presents an $L^{r}-$ analogue of an inequality regarding the $s^{th}$ derivative of a polynomial having zeros outside a circle of arbitrary radius but greater or equal to one. Our result provides improvements and generalizations of some well-known polynomial inequalities.

We study a subtle inequity in the distribution of differences between imaginary parts of zeros of the Riemann zeta function. We establish a precise measure which explains the phenomenon, first observed by R. P. Marco, that the location of each Riemann zero is encoded in the distribution of large Riemann zeros. We also extend these results to zeros of more general L-functions. In particular, we ...

If p(z) is univariate polynomial with complex coefficients having all its zeros inside the closed unit disk, then the Gauss-Lucas theorem states that all zeros of p′(z) lie in the same disk. We study the following question: what is the maximum distance from the arithmetic mean of all zeros of p(z) to a nearest zero of p′(z)? We obtain bounds for this distance depending on degree. We also show t...

admits a meromorphic continuation to the entire complex plane, with the unique and simple pole of residue 1 at s = 1. In the half-plane {s : Rs ≤ 0}, the Riemann zeta function has simple zeros at −2,−4,−6, . . ., and only at these points which are called trivial zeros. There exist also non-trivial zeros in the band {s : 0 < Rs < 1}. We refer for these basic facts for instance to [Bl] (Propositi...

The root count developed by Bernshtein, Kushnirenko and Khovanskii only counts the number of isolated zeros of a polynomial system in the algebraic torus (C∗)n. In this paper, we modify this bound slightly so that it counts the number of isolated zeros in Cn. Our bound is, apparently, significantly sharper than the recent root counts found by Rojas and in many cases easier to compute. As a cons...

One of the most famous problems in mathematics is the Riemann hypothesis: that the nontrivial zeros of the Riemann zeta function lie on a line in the complex plane. One way to prove the hypothesis would be to identify the zeros as eigenvalues of a Hermitian operator, many of whose properties can be derived through the analogy to quantum chaos. Using this, we construct a set of quantum graphs th...

We present an efficient and numerically reliable approach to compute the zeros of a periodic system. The zeros are defined in terms of the transfer-function matrix corresponding to an equivalent lifted statespace representation as constant system. The proposed method performs locally row compressions of the associated system pencil to extract a low order pencil which contains the zeros (both fi...

The concept of invariant zeros in a linear time-invariant system with state delay is considered. In the state-space framework, invariant zeros are treated as triples: complex number, nonzero state-zero direction, input-zero direction. Such a treatment is strictly related to the output-zeroing problem and in that spirit the zeros can be easily interpreted. The problem of zeroing the system outpu...

The question how the classical definition of the Smith zeros of an LTI continuous-time singular control system S(E, A, B, C, D) can be generalized and related to state-space methods is discussed. The zeros are defined as those complex numbers for which there exists a zero direction with a nonzero state-zero direction. Such a definition allows an infinite number of zeros (then the system is call...

We propose a computationally efficient and numerically reliable algorithm to compute the finite zeros of a linear discrete-time periodic system. The zeros are defined in terms of the transfer-function matrix corresponding to an equivalent lifted time-invariant state-space system. The proposed method relies on structure preserving manipulations of the associated system pencil to extract successi...