Related papers: Hilbert's tenth problem for complex meromorphic fu…

In this note we give a recursive formula for the derivatives of isotropic positive definite functions on the Hilbert sphere. We then use it to prove a conjecture stated by Tr\"ubner and Ziegel, which says that for a positive definite…

A meromorphic inner function is a bounded holomorphic function in the upper half-plane which is unimodular on the real line and extends to a meromorphic function in the whole complex plane. The argument of a meromorphic inner function on…

We develop a theory of Valuation Hilbert Modules and prove a version of Beurling's theorem for these. Then we apply our version of Beurling's theorem to obtain complete descriptions of the closed invariant subspaces of a number of Hilbert…

We consider holomorphic functions on the unit disc whose images are contained in a strip of the complex plane. Under an additional condition, such functions are constants. We also consider appropriate operator valued versions. Applications…

While routinely used in other areas of dynamics, image sets are ill-defined objects in general non-invertible measurable dynamics. We propose a way of consistently working with image sets of null-preserving (and hence, in particular, of…

We give an elementary characterization of rational functions among meromorphic functions in the complex plane.

We give an upper bound for the number of functionally independent meromorphic first integrals that a discrete dynamical system generated by an analytic map $f$ can have in a neighborhood of one of its fixed points. This bound is obtained in…

In this article, I give an iterative closed form formula for the Hilbert-Kunz function for any binomial hypersurface in general, over any feild of arbitrary positive characteristic. I prove that the Hilbert-Kunz multiplicity associated to…

We investigate the existence of non-trivial holomorphic and meromorphic solutions of Fermat functional equations over an open Riemann surface $S$. When $S$ is hyperbolic, we prove that any $k$-term Fermat functional equation always exists…

With the help of the notion of weighted sharing of sets, this paper dealt with the question posed by \emph{Yi} \cite{Yi-SC-1994} regarding the uniqueness of meromorphic functions concerning three set sharing. A result has been proved which…

We introduce a first-order theory of finite full binary trees and then identify decidable and undecidable fragments of this theory. We show that the analogue of Hilbert`s 10th Problem is undecidable by constructing a many-to-one reduction…

A new mathematical notation is proposed for the iteration of functions. It facilitates the application of the iteration of functions in mathematical and logical expressions, definitions of sets, and formulations of algorithms. Illustrations…

A functional Menger $\cap$-algebra is a set of n-place functions containing n projections and closed under the so-called Menger's compositions of n-place functions and the set-theoretic intersection of functions. We give the abstract…

Sufficient and necessary conditions on the spectral measure of a self-adjoint operator $A$, acting in a Hilbert space, are obtained, under which for any continuous scalar function the operator function $\phi(A+\gamma B)$ is holomorphic with…

A new concept of meromorphic $\Sigma$-factorization, for H\"{o}lder continuous functions defined on a contour $\Gamma$ that is the pullback of $\dot{\mathbb{R}}$ (or the unit circle) in a Riemann surface $\Sigma$ of genus 1, is introduced…

We prove the existence of ternary forms admitting apolar sets of points of cardinality equal to the Waring rank, but having different Hilbert function and different regularity. This is done exploiting liaison theory and Cayley-Bacharach…

In this paper we prove a number of results concerning uniqueness of a meromorphic function as well as its derivative sharing one or two sets. In particular, we deal with the specific question raised in [18], [19], [20] and ultimately…

Let k be a subfield of a p-adic field of odd residue characteristic, and let L be the function field of a variety of dimension n >= 1 over k. Then Hilbert's Tenth Problem for L is undecidable. In particular, Hilbert's Tenth Problem for…

It remains an open problem to classify the Hilbert functions of double points in $\mathbb{P}^2$. Given a valid Hilbert function $H$ of a zero-dimensional scheme in $\mathbb{P}^2$, we show how to construct a set of fat points $Z \subseteq…

We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of calculating the complex dimension of an algebraic…