Related papers: Separately regular and separately Nash functions

We characterize pairs of rational functions $A$, $B$ such that $A$ is semiconjugate to $B$, and $B$ is semiconjugate to $A$.

The purpose of this paper is to introduce the notion of Nash functions in the context of slice regular functions of one quaternionic or octonionic variable. We begin with a detailed analysis of the possible definitions of Nash slice regular…

This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, and here we extend it to…

We investigate semiconjugate rational functions, that is rational functions $A,$ $B$ related by the functional equation $A\circ X=X\circ B$, where $X$ is a rational function of degree at least two. We show that if $A$ and $B$ is a pair of…

Expanding upon recent work, a new class of $A$-functions is introduced that can be viewed as an appropriate generalization of the class of regular $A$-functions, the class of structured $A$-functions, and the class of perfect $A$-functions.…

The paper gives some criteria for partial sums of rational number sequences to be not rational functions and to be not algebraic functions. As an application, we study partial sums of some famous rational number sequences in mathematical…

We establish a general result on the existence of partially defined semiconjugacies between rational functions acting on the Riemann sphere. The semiconjugacies are defined on the complements to at most one-dimensional sets. They are…

We formulate some problems and conjectures about semigroups of rational functions under composition. The considered problems arise in different contexts, but most of them are united by a certain relationship to the concept of amenability.

Let X be an affine real algebraic set . We investigate on the theory of algebraically constructible functions on X and the description of the semi-algebraic subsets of X when we replace the polynomial functions on X by some rational…

Let $W$ be a subset of the set of real points of a real algebraic variety $X$. We investigate which functions $f: W \to \mathbb R$ are the restrictions of rational functions on $X$. We introduce two new notions: ${\it curve-rational \,…

In this paper we extend the notions of Schwartz functions, tempered functions and generalized Schwartz functions to Nash (i.e. smooth semi-algebraic) manifolds. We reprove for this case classically known properties of Schwartz functions on…

Differentially-algebraic (D-algebraic) functions are solutions of polynomial equations in the function, its derivatives, and the independent variables. We revisit closure properties of these functions by providing constructive proofs. We…

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

We define and study rational discrete analytic functions and prove the existence of a coisometric realization for discrete analytic Schur multipliers.

In this paper, an upper semismooth function is defined to be a lower semicontinuous function whose radial subderivative satisfies a mild directional upper semicontinuity property. Examples of upper semismooth functions are the proper lower…

The analysis of solutions to algebraic equations is further simplified. A couple of functions and their analytic continuation or root findings are required.

We present criteria for deciding whether a bivariate rational function in two variables can be written as a sum of two (q-)differences of bivariate rational functions. Using these criteria, we show how certain double sums can be evaluated,…

A function is differentially algebraic (or simply D-algebraic) if there is a polynomial relationship between some of its derivatives and the indeterminate variable. Many functions in the sciences, such as Mathieu functions, the Weierstrass…

This note studies, and partially solves, 3 elementary questions about continuous rational functions on real (and p-adic) algebraic varieties: Can one restrict such a function to a subvariety? Can one extend such a function from a…

The derivatives of Nash functions are Nash functions which are derived algebraically from their minimal polynomial equations. In this paper we show that, for any non-Nash analytic function, it is impossible to derive its derivatives…