Bounds for D-finite closure properties

We provide bounds on the size of operators obtained by algorithms for executing D-finite closure properties. For operators of small order, we give bounds on the degree and on the height (bit-size). For higher order operators, we give degree bounds that are parameterized with respect to the order and reflect the phenomenon that higher order operators may have lower degrees (order-degree curves)

Efficient Algorithms for Mixed Creative Telescoping

Creative telescoping is a powerful computer algebra paradigm -initiated by Doron Zeilberger in the 90's- for dealing with definite integrals and sums with parameters. We address the mixed continuous-discrete case, and focus on the integration of bivariate hypergeometric-hyperexponential terms. We design a new creative telescoping algorithm operating on this class of inputs, based on a Hermite-like reduction procedure. The new algorithm has two nice features: it is efficient and it delivers, for a suitable representation of the input, a minimal-order telescoper. Its analysis reveals tight bounds on the sizes of the telescoper it produces.Comment: To be published in the proceedings of ISSAC'1

Reduction-Based Creative Telescoping for Algebraic Functions

Continuing a series of articles in the past few years on creative telescoping using reductions, we develop a new algorithm to construct minimal telescopers for algebraic functions. This algorithm is based on Trager's Hermite reduction and on polynomial reduction, which was originally designed for hyperexponential functions and extended to the algebraic case in this paper

Computing periods of rational integrals

A period of a rational integral is the result of integrating, with respect to one or several variables, a rational function over a closed path. This work focuses particularly on periods depending on a parameter: in this case the period under consideration satisfies a linear differential equation, the Picard-Fuchs equation. I give a reduction algorithm that extends the Griffiths-Dwork reduction and apply it to the computation of Picard-Fuchs equations. The resulting algorithm is elementary and has been successfully applied to problems that were previously out of reach.Comment: To appear in Math. comp. Supplementary material at http://pierre.lairez.fr/supp/periods