共被期刊论文引用了3次
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| 1 | The Algebro-Geometric Method for Solving Algebraic Differential Equations——A Survey显示文摘This paper presents the algebro-geometric method for computing explicit formula solutions for algebraic differential equations(ADEs). An algebraic differential equation is a polynomial relation between a function, some of its partial derivatives, and the variables in which the function is de?ned.Regarding all these quantities as unrelated variables, the polynomial relation leads to an algebraic relation de?ning a hypersurface on which the solution is to be found. A solution in a certain class of functions, such as rational or algebraic functions, determines a parametrization of the hypersurface in this class. So in the algebro-geometric method the author ?rst decides whether a given ADE can be parametrized with functions from a given class; and in the second step the author tries to transform a parametrization into one respecting also the differential conditions. This approach is relatively well understood for rational and algebraic solutions of single algebraic ordinary differential equations(AODEs). First steps are taken in a generalization to systems and to partial differential equations. | WINKLER Franz | 2019 | Journal of Systems Science & Complexity2019，32，1： | 3 |
| 2 | Rational Solutions of High-Order Algebraic Ordinary Differential Equations显示文摘This paper considers algebraic ordinary differential equations(AODEs)and study their polynomial and rational solutions.The authors first prove a sufficient condition for the existence of a bound on the degree of the possible polynomial solutions to an AODE.An AODE satisfying this condition is called noncritical.Then the authors prove that some common classes of low-order AODEs are noncritical.For rational solutions,the authors determine a class of AODEs,which are called maximally comparable,such that the possible poles of any rational solutions are recognizable from their coefficients.This generalizes the well-known fact that any pole of rational solutions to a linear ODE is contained in the set of zeros of its leading coefficient.Finally,the authors develop an algorithm to compute all rational solutions of certain maximally comparable AODEs,which is applicable to 78.54%of the AODEs in Kamke's collection of standard differential equations. | VO Thieu N. ZHANG Yi | 2020 | Journal of Systems Science & Complexity2020，33，3： | 0 |
| 3 | Descent of Ordinary Differential Equations with Rational General Solutions显示文摘Let F be an irreducible differential polynomial over k(t)with k being an algebraically closed field of characteristic zero.The authors prove that F=0 has rational general solutions if and only if the differential algebraic function field over k(t)associated to F is generated over k(t)by constants,i.e.,the variety defined by F descends to a variety over k.As a consequence,the authors prove that if F is of first order and has movable singularities then F has only finitely many rational solutions. | FENG Shuang FENG Ruyong | 2020 | Journal of Systems Science & Complexity2020，33，6： | 0 |
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