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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(More)
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(More)
For a field k with an automorphism σ and a derivation δ, we introduce the notion of liouvillian solutions of linear difference-differential systems {σ(Y) = AY, δ(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(More)
A normal form is given for integrable linear difference-differential equations {σ(Y) = AY, δ(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 kind of systems of prime order.
We present two criteria for the existence of telescopers for bivariate hyperexponential-hypergeometric functions. One is for the existence of telescopers with respect to the continuous variable, the other for telescopers with respect to the discrete one. Our criteria are based on a standard representations of bivariate hyperexponential-hypergeometric(More)
Most work on finding elementary function solutions for differential equations focussed on linear equations [4, 2, 6, 1, 3]. In this paper, we try to find polynomial solutions to non-linear differential equations. Instead of finding arbitrary polynomial solutions, we will find the polynomial general solutions. For example, the general solution for(More)
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 <i>q</i>-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(More)