Algebraic \(D\)-groups and differential Galois theory
The author gives another exposition of his new differential Galois theory. As he thinks, a previous presentation [Ill. J. Math. 42, No. 4, 678--699 (1998; Zbl 0916.03028)] was too model-theoretic and so somewhat obscure to differential algebraists. This new representation is based on some generalization of \(G\)-primitive extensions of \textit{E. R. Kolchin} [Differential algebra and algebraic groups. Pure and Applied Mathematics, 54. New York-London: Academic Press (1973; Zbl 0264.12102)], where the algebraic group \(G\) is substituted by a differential algebraic one, and the \(G\)-primitive element is defined through some generalization of a logarithmic derivative. The author gives a concrete example of this extension of a new type which does not concern the strongly normal one in this paper. Let \(K=\mathbb C(e^{ct}: c\in \mathbb C)\) and \(L=K(t,e^{t^2})\) then \(L\) is a differential Galois extension of \(K\) for the equation \(\frac{d}{dt}(y^{-1}\frac{d}{dt}(y))=2\). The Galois group consists of elements of the form \(\lambda*e^{\mu t}\), where \(\lambda\in\mathbb C^{*}\) and \(\mu\in\mathbb C\). Let us remark, that to assign the group to the extension \(\mathbb C(t,e^{t^2})\) instead of \(L\) would meet more to the spirit of ideas of Kolchin. Other approaches to construct a differential Galois theory for nonlinear differential equations can be found in \textit{B. Malgrange} [Chin. Ann. Math., Ser. B 23, No. 2, 219--226 (2002; Zbl 1009.12005)] and \textit{J. F. Pommaret} [Differential Galois theory. Mathematics and Its Applications, Vol. 15. New York: Gordon and Breach (1983; Zbl 0539.12013)].
scientific article; zbMATH DE number 2138449