Differentially Transcendental Functions

The aim of this paper 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 transcendency of solution of some functional equations. 1 Notation and preliminaries The theory DF0 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 DF0 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 DF0 has a model completion, and then defined DCF0 to be the model completion of DF0. Subsequently L. Blum found simple axioms of DFC0 without refereing to differential polynomials in more than one variable, see [23]. 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[x1,x2, . . . ,xn] denotes the set of (ordinary) algebraic polynomials over F in variables x1, x2, . . . , xn. 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, . . . ,DX) where f ∈ F(x,y1,y2, . . . ,yn). 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. Proposition 1.1 Let A ⊆ B ⊂ C be ordinary algebraic fields. Then a. td(B|A) ≤ td(C|A). b. td(C|A) = td(C|B) + td(B|A). Email address : [email protected] Email address : [email protected] Second author supported in part by the project MNTRS, Grant No. 1861.