General Solutions of First-Order Algebraic ODEs in Simple Constant Extensions

作     者：MITTERAMSKOGLER Johann Josef WINKLER Franz MITTERAMSKOGLER Johann Josef;WINKLER Franz

作者机构：Research Institute for Symbolic Computation(RISC)Johannes Kepler Universität LinzA-4040 LinzAustria 

出 版 物：《系统科学与复杂性学报:英文版》 (Journal of Systems Science and Complexity)

年 卷 期：2023年第36卷第4期

页      面：1769-1788页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

主　　题：Algebraic curve algebraic differential equation general solution Möbius transformation rational parametrization 

摘      要：If a first-order algebraic ODE is defined over a certain differential field,then the most elementary solution class,in which one can hope to find a general solution,is given by the adjunction of a single arbitrary constant to this *** of this type give rise to a particular kind of generic point—a rational parametrization—of an algebraic curve which is associated in a natural way to the ODE’s defining *** for the opposite direction,we show that a suitable rational parametrization of the associated curve can be extended to a general solution of the ODE if and only if one can find a certain automorphism of the solution *** automorphisms are determined by linear rational functions,i.e.,Möbius *** properties of rational parametrizations,in combination with the particular shape of such automorphisms,lead to a number of necessary conditions on the existence of general solutions in this solution ***,the desired linear rational function can be determined by solving a comparatively simple differential system over the ODE’s field of *** results hold for arbitrary differential fields of characteristic zero.