Continuous derivations
In many and very diverse problems of differential algebra a fundamental role is played implicitly or explicitly by a valuation related to the derivation. Let L be a field, v a rank one non-archimedean valuation of L trivial on the prime field of L. Then \((L,v)\) is called a valued field. Let K be a subfield of L and \(D: K\to L\) a derivation. Let \(S\subset K\). Then D is bounded below on S if there exists \(\gamma\) in the value group of v such that v(Dx/x)\(\geq \gamma\) for all \(x\in S\). It is shown (proposition 2.2) that if \(v\neq 0\), D is bounded below on K if and only if D is continuous on K when L is normed via \(\| x\| =e^{-v(x)}\), \(x\in L\). This paper is the first to investigate in a general way the properties of continuous derivations, thus coordinating many results scattered through the literature of the applications (most of which impose further restrictions on D). Many of the theorems generalize to all finite ranks. Let K/F be a finitely generated field extension. If K/F has transcendence degree one, a classic result assures that every F-derivation of K is continuous in the topology of every valuation of K/F. In the general case one asks first whether for a given valuation v of K over F, there exists a non-zero F-derivation of K continuous with respect to the topology of v; and second, whether, given an F-derivation D of K there exists a valuation v of K over F such that D is continuous in the corresponding topology. These questions, somewhat generalized, are answered in many cases. It is shown that the answer to the first question is affirmative in important cases but not always. Information is obtained (theorem 5.2 and corollary 5.3) on the dimension of the space of continuous derivations. The answer to the second question is affirmative if K/F is finitely generated and of characteristic 0, but there is a counter-example if K/F is not finitely generated. Pursuing a related but somewhat different question, theorems 4.1 and 7.1 b generalize earlier results of the author [Am. J. Math. 100, 245-261 (1978; Zbl 0405.13014)] and of the reviewer [in Contrib. Algebra, Collect. Papers dedicated E. Kolchin, 117-128 (1977; Zbl 0373.12013)] which have been applied to problems of differential specialization. The new proof is shorter and clearer than the earlier ones. From theorem 4.1 a criterion is obtained (theorem 7.2) for the important problem of deciding whether a derivation operates on a power series by term by term differentiation, and for generalizations of this problem.
scientific article; zbMATH DE number 4049751
scientific article