Algebraic independence of solutions of differential equations of the second order
In the paper the problem when a differential equation of special kind admits a ``first integral'' (in the frame of differential algebra) is discussed. The following theorem is proved. Theorem. let C be a field of constants. Let f,g,A,B\(\in {\mathbb{C}}\{y\}\), where B/A is a linear combination of logarithmic \({\mathbb{C}}\)-derivatives: \(B/A=\sum^{n}_{i=1}C_ i(\partial h_ i/h_ i)\), with \(C_ i\) in \({\mathbb{C}}\) linearly independent over \({\mathbb{Q}}\) and \(h_ i\) nonconstant rational functions in \({\mathbb{C}}(y)\), and let \(F=y'f+g\). Then over any differential field extension K of \({\mathbb{C}}\), the only prime ideal of order 1 which contains \(P=AF'-y'B\) is \(\{y'\}.\) It is also shown that there is a sufficiently large class of prime ideals of second order whose generic solutions have algebraic dependence of very special kinds. As consequence such ideal has an infinite countable set of generic solutions which is algebraically free over the field of constants.
scientific article; zbMATH DE number 4146078