Ruyong Feng

Rational general solutions of algebraic ordinary differential equations

We give a necessary and sufficient condition for an algebraic ODE to have a rational type general solution. For an autonomous first order ODE, we give an algorithm to compute a rational general solution if it exists. The algorithm is based on the relation between rational solutions of the first order ODE and rational parametrizations of the plane algebraic curve defined by the first order ODE and Padé approximants.

A polynomial time algorithm for finding rational general solutions of first order autonomous ODEs

Abstract We give a necessary and sufficient condition for an algebraic ODE to have a rational type general solution. For a first order autonomous ODE F = 0 , we give an exact degree bound for its rational solutions, based on the connection between rational solutions of F = 0 and rational parametrizations of the plane algebraic curve defined by F = 0 . For a first order autonomous ODE, we further give a polynomial time algorithm for computing a rational general solution if it exists based on the computation of Laurent series solutions and Pade approximants. Experimental results show that the algorithm is quite efficient.

On the existence of telescopers for mixed hypergeometric terms

We present a criterion for the existence of telescopers for mixed hypergeometric terms, which is based on additive and multiplicative decompositions. The criterion enables us to determine the termination of Zeilbergers algorithms for mixed hypergeometric inputs, and to verify that certain indefinite sums do not satisfy any polynomial differential equation.

Hrushovski's algorithm for computing the Galois group of a linear differential equation

Abstract We present a detailed and modified version of Hrushovskis algorithm that determines the Galois group of a linear differential equation. Moreover, we give explicit degree bounds for the defining polynomials of various linear algebraic groups that appear in the algorithm. These explicit bounds will play an important role to understand the complexity of the algorithm.

Liouvillian solutions of linear difference-differential equations

For a field k with an automorphism @s and a derivation @d, we introduce the notion of Liouvillian solutions of linear difference-differential systems {@s(Y)=AY,@d(Y)=BY} over k and characterize the existence of Liouvillian solutions in terms of the Galois group of the systems. In the forthcoming paper, we will propose an algorithm for deciding if linear difference-differential systems of prime order have Liouvillian solutions.

An algorithm to compute Liouvillian solutions of prime order linear difference-differential equations

A normal form is given for integrable linear difference-differential equations {@s(Y)=AY,@d(Y)=BY}, which is irreducible over C(x,t) and solvable in terms of Liouvillian solutions. We refine this normal form and devise an algorithm to compute all Liouvillian solutions of such kinds of systems of prime order.

Algebraic general solutions of algebraic ordinary differential equations

In this paper, we give a necessary and sufficient condition for an algebraic ODE to have an algebraic general solution. For a first order autonomous ODE, we give an optimal bound for the degree of its algebraic general solutions and a polynomial-time algorithm to compute an algebraic general solution if it exists. Here an algebraic ODE means that an ODE given by a differential polynomial.

On the structure of compatible rational functions

A finite number of rational functions are compatible if they satisfy the compatibility conditions of a first-order linear functional system involving differential, shift and q-shift operators. We present a theorem that describes the structure of compatible rational functions. The theorem enables us to decompose a solution of such a system as a product of a rational function, several symbolic powers, a hyperexponential function, a hypergeometric term, and a q-hypergeometric term. We outline an algorithm for computing this product, and present an application.

Parallel telescoping and parameterized Picard-Vessiot theory

Parallel telescoping is a natural generalization of differential creative-telescoping for single integrals to line integrals. It computes a linear ordinary differential operator L, called a parallel telescoper, for several multivariate functions, such that the application of L to the functions yields partial derivatives of a single function. We present a necessary and sufficient condition guaranteeing the existence of parallel telescopers for differentially finite functions, and develop an algorithm to compute minimal ones for compatible hyperexponential functions. Besides computing annihilators of parametric line integrals, we use the parallel telescoping for determining Galois groups of parameterized partial differential systems of first order.

Rational solutions of ordinary difference equations

In this paper, we generalize the results of Feng and Gao [Feng, R., Gao, X.S., 2006. A polynomial time algorithm to find rational general solutions of first order autonomous ODEs. J. Symbolic Comput., 41(7), 735-762] to the case of difference equations. We construct two classes of ordinary difference equations (O@DEs) whose solutions are exactly the univariate polynomial and rational functions respectively. On the basis of these O@DEs and the difference characteristic set method, we give a criterion for an O@DE with any order and nonconstant coefficients to have a rational type general solution. For the first-order autonomous (constant coefficient) O@DE, we give a polynomial time algorithm for finding the polynomial solutions and an algorithm for finding the rational solutions for a given degree.