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

作　　者：MITTERAMSKOGLER Johann Josef WINKLER Franz 

机构地区：[1]Research Institute for Symbolic Computation(RISC),Johannes Kepler Universität Linz,A-4040 Linz,Austria

出　　处：《Journal of Systems Science & Complexity》2023年第4期1769-1788,共20页系统科学与复杂性学报（英文版）

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 field.Solutions 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 polynomial.As 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 field.These automorphisms are determined by linear rational functions,i.e.,Möbius transformations.Intrinsic 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 class.Furthermore,the desired linear rational function can be determined by solving a comparatively simple differential system over the ODE’s field of definition.These results hold for arbitrary differential fields of characteristic zero.

关 键 词：Algebraic curve algebraic differential equation general solution Möbius transformation rational parametrization 

分 类 号：O175[理学—数学]