The method of differentiating under the integral sign
Journal of Symbolic Computation
ISSAC '91 Proceedings of the 1991 international symposium on Symbolic and algebraic computation
On polynomial solutions of linear operator equations
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
Fast algorithms for Taylor shifts and certain difference equations
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Modern computer algebra
About the polynomial solutions of homogeneous linear differential equations depending on parameters
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
On rational solutions of systems of linear differential equations
Journal of Symbolic Computation - Special issue on differential algebra and differential equations
Liouvillian solutions of linear differential equations of order three and higher
Journal of Symbolic Computation - Special issue on differential algebra and differential equations
An extension of Zeilberger's fast algorithm to general holonomic functions
Discrete Mathematics
HAKMEM
Modular Algorithms In Symbolic Summation And Symbolic Integration (Lecture Notes in Computer Science)
D-finiteness: algorithms and applications
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Low complexity algorithms for linear recurrences
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Solving difference equations whose coefficients are not transcendental
Theoretical Computer Science
Fast algorithms for differential equations in positive characteristic
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
NumGfun: a package for numerical and analytic computation with D-finite functions
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
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We investigate polynomial solutions of homogeneous linear differential equations with coefficients that are polynomials with integer coefficients. The problems we consider are the existence of nonzero polynomial solutions, the determination of the dimension of the vector space of polynomial solutions, the computation of a basis of this space. Previous algorithms have a bit complexity that is at least quadratic in the largest integer valuation N of formal Laurent series solutions at infinity, even for merely detecting the existence of nonzero polynomial solutions. We give a deterministic algorithm that computes a compact representation of a basis of polynomial solutions in O(Nlog3N) bit operations. We also give a probabilistic algorithm that computes the dimension of the space of polynomial solutions in O(√Nlog2N) bit operations. In general, the integer N is not polynomially bounded in the bit size of the input differential equation. We isolate a class of equations for which detecting nonzero polynomial solutions can be performed in polynomial complexity. We discuss implementation issues and possible extensions.