Valuations and the Ritt problem
The author generalizes his previous result [see Am. J. Math. 108, 505-523 (1986)] concerned with the Ritt problem. He proves by means of methods of valuation theory the following result: Main theorem. Let F be an ordinary differential field of characteristic 0. Let \(\alpha\) be differentially algebraic over F and specializing differentially over F to \(\beta\) with tr deg F\(<\alpha >/F-tr \deg F<\beta >/F\geq 2\). Then there exists \(\alpha^*\) satisfying (1) \(\alpha\) specializes differentially over F to \(\alpha^*;\) (2) there is a rank one valuation \(\mu\) of \(F<\alpha^*>/F\) which is also a differential valuation and which is such that the associated place maps \(\alpha^*\) to \(\beta\), and \(\mu\) is of rational rank less than tr deg F\(<\alpha >/F-tr \deg F<\beta >/F\).
scientific article; zbMATH DE number 3952900