# Computing periods of rational integrals

@article{Lairez2016ComputingPO, title={Computing periods of rational integrals}, author={Pierre Lairez}, journal={Math. Comput.}, year={2016}, volume={85}, pages={1719-1752} }

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…

## 53 Citations

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An algorithm which takes as input a polynomial system defining S and an integer p and returns the n-dimensional volume of S at absolute precision 2^-p and improves upon the previous exponential bounds obtained by Monte-Carlo or moment-based methods.

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This lecture notes address the more general case when the coefficients are polynomials, and focuses on left ideals, or D-ideals, which represent holonomic functions in several variables by the linear differential equations they satisfy.

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This work shows how to obtain dominant asymptotics for the diagonal coefficient sequence of multivariate rational functions under some genericity assumptions using symbolic-numeric techniques and is the first completely automatic treatment and complexity analysis for the asymPTotic enumeration of rational functions in an arbitrary number of variables.

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