E-Algebraic Functions over Fields of Positive Characteristic—An Analogue of Differentially Algebraic Functions (original) (raw)

Linear Differential Equations Over Arbitrary Algebraically Closed Fields

2014

Let K be an arbitrary algebraically closed field of characteristic zero and let K[[x]] be the ring of integral formal power series; let Ω be the K-subalgebra of K[[x]] generated by x and the subset TK = {exp(λx) : λ ∈ K}. In this note we supply some easy and elegant proofs for some classical results on the preimage of elements of the form xQ(x) exp(rx) through a linear differential operator with coefficients in K. We also make some theoretical considerations on the structure of the space of all solutions for a linear ODE defined over K[[x]]. 1. Some introductory remarks Let K be an algebraically closed field of characteristic zero [LS] and let x be a variable over K (simply an element x not belonging to K). LetK[[x]] be the ring of all formal integral power series f = a0+a1x+..., ai ∈ K, i = 0, 1, .... Here d dx : K[[x]] → K[[x]], df dx = a1 + 2a2x + ... + nanx n−1 + ..., is the usual differential operator defined on K[[x]]. For y ∈ K[[x]], we also denote y = dy dxn =   d dx ◦ .....

Differentially Transcendental Functions

arXiv: General Mathematics, 2004

The aim of this article is to exhibit a method for proving that certain analytic functions are not solutions of algebraic differential equations. The method is based on model-theoretic properties of differential fields and properties of certain known transcendental differential functions, as of Γ(x). Furthermore, it also determines differential transcendence of solution of some functional equations. 1 Notation and preliminaries The theory DF 0 of differential fields of characteristic 0 is the theory of fields with additional two axioms that relate to the derivative D: D(x + y) = Dx + Dy, D(xy) = xDy + yDx. Thus, a model of DF 0 is a differential field K = (K, +, •, D, 0, 1) where (K, +, •, 0, 1) is a field and D is a differential operator satisfying the above axioms. A. Robinson proved that DF 0 has a model completion, and then defined DCF 0 to be the model completion of DF 0. Subsequently, L. Blum found simple axioms of DFC 0 without refereing to differential polynomials in more than one variable, see [22]. In the following, if not otherwise stated, F, K, L,. .. will denote differential fields, F, L, K,. .. their domains while F * , K * , L * ,. .. will denote their field parts, i.e. F * =(F, +, •, 0, 1). Thus, F * [x 1 , x 2 ,. .. , x n ] denotes the set of (ordinary) algebraic polynomials over F * in variables x 1 , x 2 ,. .. , x n. The symbol L{X} denotes the ring of differential polynomials over L in the variable X. Hence, if f ∈ L{X} then for some n ∈ N, N = {0, 1, 2,. . .}, f = f (X, DX,. .. , D n X) where f ∈ F * (x, y 1 , y 2 ,. .. , y n). Suppose L ⊆ K. The symbol td(K|L) denotes the transcendental degree of K * over L *. The basic properties of td are described in the following proposition.

Galois theory of differential fields of positive characteristic

Pacific Journal of Mathematics, 1989

The strongly normal extensions of a differential field K of positive characteristic are defined. On the set G of all differential isomorphisms of a strongly normal extension N of K, a structure of an algebraic group is induced. Correspondences between subgroups of G and intermediate differential fields of N and K are studied.

Families of Commuting Formal Power Series and Formal Functional Equations

Annales Mathematicae Silesianae

In this paper we describe families of commuting invertible formal power series in one indeterminate over ℂ, using the method of formal functional equations. We give a characterization of such families where the set of multipliers (first coefficients) σ of its members F (x) = σx + . . . is infinite, in particular of such families which are maximal with respect to inclusion, so called families of type I. The description of these families is based on Aczél–Jabotinsky differential equations, iteration groups, and on some results on normal forms of invertible series with respect to conjugation.

Fr\'echet-valued formal power series

arXiv (Cornell University), 2019

Let A be a non-projectively-pluripolar set in a Fréchet space E. We give sufficient conditions to ensure the convergence on some zero-neighbourhood in E of a (sequence of) formal power series of Fréchet-valued continuous homogeneous polynomials provided that the convergence holds at a zero-neighbourhood of each complex line ℓa := Ca for every a ∈ A.