Related papers: Exact & Numerical Tests of Generalised Root Identi…

Let HN denote the problem of determining whether a system of multivariate polynomials with integer coefficients has a complex root. It has long been known that HN in P implies P=NP and, thanks to recent work of Koiran, it is now known that…

We investigate the horizontal distribution of zeros of the derivative of the Riemann zeta function and compare this to the radial distribution of zeros of the derivative of the characteristic polynomial of a random unitary matrix. Both…

The main purpose of this paper is solve polynomial equations that are satisfied by (generalized) polynomials. More exactly, we deal with the following problem: let $\mathbb{F}$ be a field with $\mathrm{char}(\mathbb{F})=0$ and $P\in…

For every nonzero integer $m$ and every integer $n \ge 1$, the $n$\textsuperscript{th} harmonic number $H_n = 1 + \tfrac12 + \dots + \tfrac1n$ satisfies the identity \[ H_n \;=\; \frac{1}{m}\,\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k}\, \binom{m…

A new version of the Graeffe algorithm for finding all the roots of univariate complex polynomials is proposed. It is obtained from the classical algorithm by a process analogous to renormalization of dynamical systems. This iteration is…

We derive an integral expression $G(z)$ for the reciprocal gamma function, $1/\Gamma(z)=G(z)/\pi$, that is valid for all $z\in\mathbb{C}$, without the need for analytic continuation. The same integral avoids the singularities of the gamma…

Igusa's local zeta function $Z_{f,p}(s)$ is the generating function that counts the number of integral roots, $N_{k}(f)$, of $f(\mathbf x) \bmod p^k$, for all $k$. It is a famous result, in analytic number theory, that $Z_{f,p}$ is a…

A generalization of Arnold's strange duality to invertible polynomials in three variables by the first author and A.Takahashi includes the following relation. For some invertible polynomials $f$ the Saito dual of the reduced monodromy zeta…

We study the behavior of $r$-fold zeta-functions of Euler-Zagier type with identical arguments $\zeta_r(s,s,\ldots,s)$ on the real line. Our basic tool is an "infinite'' version of Newton's classical identities. We carry out numerical…

We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. Normal form algorithms provide an algebraic approach to solve this problem.…

An efficient randomized polynomial identity test for noncommutative polynomials given by noncommutative arithmetic circuits remains an open problem. The main bottleneck to applying known techniques is that a noncommutative circuit of size…

There has been an avalanche of recent research on multiple zeta values. We propose dividing identities for multiple zeta values into structural and specific types. Structural identities are valid for any generalized multiple zeta function,…

Satisfiability Modulo the Theory of Nonlinear Real Arithmetic, SMT(NRA) for short, concerns the satisfiability of polynomial formulas, which are quantifier-free Boolean combinations of polynomial equations and inequalities with integer…

Let $\Lambda\left(n\right)$ be the von Mangoldt function and $r_{G}\left(n\right) := \sum_{m_1 + m_2=n} \Lambda \left(m_1 \right) \Lambda\left(m_2 \right)$ be the counting function for the numbers that can be written as sum of two primes…

Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…

In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…

Let $(\mathbb{R}_{\alpha ,\beta ,\gamma }(z))_{m}(z)=z+\sum_{n=1}^{m}A_{n}z^{n+1}$ be the sequence of partial sums of the normalized Rabotnov functions $\mathbb{R}_{\alpha ,\beta ,\gamma }(z)=z+\sum_{n=1}^{\infty }A_{n}z^{n+1}$ where…

We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where "varieties" carry a PGL_n-action, regular…

Using techniques of algebraic and analytic number theory, we resolve a question on monoid rings posed by Kulosman, et. al., under the assumption of the Generalized Riemann Hypothesis (GRH). Specifically, we show that under an appropriate…

We introduce the zeta number, natural halting probability and natural complexity of a Turing machine and we relate them to Chaitin's Omega number, halting probability, and program-size complexity. A classification of Turing machines…