Related papers: Operations for D-Algebraic Functions
We consider an ordinary nonlinear differential equation with generalized coefficients as an equation in differentials in algebra of new generalized functions. Then the solution of such equation will be a new generalized function. In the…
Telescopers for a function are linear differential (resp. difference) operators annihilated by the definite integral (resp. definite sum) of this function. They play a key role in Wilf-Zeilberger theory and algorithms for computing them…
We present an algorithm for computing a holonomic system for a definite integral of a holonomic function over a domain defined by polynomial inequalities. If the integrand satisfies a holonomic difference-differential system including…
In this paper, we introduce and study differential graded (DG for short) polynomial algebras. In brief, a DG polynomial algebra $\mathcal{A}$ is a connected cochain DG algebra such that its underlying graded algebra $\mathcal{A}^{\#}$ is a…
This paper is devoted to the comparison of the notions of regularity for algebraic connections and (holonomic) regularity for algebraic $\mathcal D$-modules.
Following and generalizing a construction by Kontsevich, we associate a zeta function to any matrix with entries in a ring of noncommutative Laurent polynomials with integer coefficients. We show that such a zeta function is an algebraic…
We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order…
A linear ordinary differential equation is called algebraic if all its solution are algebraic over its field of definition. In this paper we solve the problem of finding closed form solution to algebraic linear ordinary differential…
In this paper we study definable families of functions from an ordered abelian group into various naturally arising definable quotients. We show that for an ordered abelian group $G$ and definable family of convex subgroups…
Let C be an algebraically closed field and X a projective curve over C. Consider an ordinary linear differential equation, or a linear differ- ence equation, with coefficients in the field of rational functions of X, and assume that its…
A physically more adequate definition of a quaternionic holomorphic (H-holomorphic) function of one quaternionic variable compared to known ones and a quaternionic generalization of Cauchy-Riemann's equations are presented. At that a class…
The ideas behind the concept of algebraic ("integration-by-parts") algorithms for multiloop calculations are reviewed. For any topology and mass pattern, a finite iterative algebraic procedure is proved to exist which transforms the…
Within the framework of mappings between affine spaces, the notion of $n$-th polarization of a function will lead to an intrinsic characterization of polynomial functions. We prove that the characteristic features of derivations, such as…
In a recent article, the class of functions from the integers to the integers computable in polynomial time has been characterized using discrete ordinary differential equations (ODE), also known as finite differences. Doing so, we pointed…
We develop a functional calculus for $d$-tuples of non-commuting elements in a Banach algebra. The functions we apply are free analytic functions, that is nc functions that are bounded on certain polynomial polyhedra.
We show that any multiplicative bijection between the algebras of differentiable functions, defined on differentiable manifolds of positive dimension, is an algebra isomorphism, given by composition with a unique diffeomorphism.
We construct a complex entire function with arbitrary number of variables which has the following property: The infinite set consisting of all the values of all its partial derivatives of any orders at all algebraic points, including zero…
Let L be a bounded distributive lattice. We give several characterizations of those L^n --> L mappings that are polynomial functions, i.e., functions which can be obtained from projections and constant functions using binary joins and…
Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…
A well-known theorem factors a scalar coefficient differential operator given a linearly independent set of functions in its kernel. The goal of this paper is to generalize this useful result to other types of operators. In place of the…