Algebraic independence of solutions of differential equations of the second order
type: Item
instance of: scholarly article
title: Algebraic independence of solutions of differential equations of the second order
zbMATH Open document ID: 0699.12037
author: Michel Brestovski
published in: Pacific Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 12H20
Mathematics Subject Classification ID: 34A99
zbMATH DE Number: 4146078
zbMATH Keywords: algebraic independence of solutions
zbMATH Keywords: existence of first integral
zbMATH Keywords: differential field extension
zbMATH Keywords: prime ideals
zbMATH Keywords: generic solutions
MaRDI profile type: Publication
full work available at URL: pjm.1989.140.1
OpenAlex ID: W2071339872
DOI: 10.2140/PJM.1989.140.1
description: scientific article; zbMATH DE number 4146078