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

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

Computing Periods of Hypersurfaces

We give an algorithm to compute the periods of smooth projective hypersurfaces of any dimension. This is an improvement over existing algorithms which could only compute the periods of plane curves. Our algorithm reduces the evaluation of period integrals to an initial value problem for ordinary differential equations of Picard-Fuchs type. In this way, the periods can be computed to extreme-precision in order to study their arithmetic properties. The initial conditions are obtained by an exact determination of the cohomology pairing on Fermat hypersurfaces with respect to a natural basis.Comment: 33 pages; Final version. Fixed typos, minor expository changes. Changed code repository lin

Algorithms for minimal Picard-Fuchs operators of Feynman integrals

In even space-time dimensions the multi-loop Feynman integrals are integrals of rational function in projective space. By using an algorithm that extends the Griffiths--Dwork reduction for the case of projective hypersurfaces with singularities, we derive Fuchsian linear differential equations, the Picard--Fuchs equations, with respect to kinematic parameters for a large class of massive multi-loop Feynman integrals. With this approach we obtain the differential operator for Feynman integrals to high multiplicities and high loop orders. Using recent factorisation algorithms we give the minimal order differential operator in most of the cases studied in this paper. Amongst our results are that the order of Picard--Fuchs operator for the generic massive two-point $n-1$-loop sunset integral in two-dimensions is $2^{n}-\binom{n+1}{\left\lfloor \frac{n+1}{2}\right\rfloor }$ supporting the conjecture that the sunset Feynman integrals are relative periods of Calabi--Yau of dimensions $n-2$. We have checked this explicitly till six loops. As well, we obtain a particular Picard--Fuchs operator of order 11 for the massive five-point tardigrade non-planar two-loop integral in four dimensions for generic mass and kinematic configurations, suggesting that it arises from $K3$ surface with Picard number 11. We determine as well Picard--Fuchs operators of two-loop graphs with various multiplicities in four dimensions, finding Fuchsian differential operators with either Liouvillian or elliptic solutions.Comment: 57 pages. Results for differential operators are on the repository : https://github.com/pierrevanhove/PicardFuchs#readm

A Baxter class of a different kind, and other bijective results using tableau sequences ending with a row shape

Tableau sequences of bounded height have been central to the analysis of k-noncrossing set partitions and matchings. We show here that familes of sequences that end with a row shape are particularly compelling and lead to some interesting connections. First, we prove that hesitating tableaux of height at most two ending with a row shape are counted by Baxter numbers. This permits us to define three new Baxter classes which, remarkably, do not obviously possess the antipodal symmetry of other known Baxter classes. We then conjecture that oscillating tableau of height bounded by k ending in a row are in bijection with Young tableaux of bounded height 2k. We prove this conjecture for k at most eight by a generating function analysis. Many of our proofs are analytic in nature, so there are intriguing combinatorial bijections to be found.Comment: 10 pages, extended abstrac

Constructing minimal telescopers for rational functions in three discrete variables

We present a new algorithm for constructing minimal telescopers for rational functions in three discrete variables. This is the first discrete reduction-based algorithm that goes beyond the bivariate case. The termination of the algorithm is guaranteed by a known existence criterion of telescopers. Our approach has the important feature that it avoids the potentially costly computation of certificates. Computational experiments are also provided so as to illustrate the efficiency of our approach

Reduction-Based Creative Telescoping for Definite Summation of D-finite Functions

Creative telescoping is an algorithmic method initiated by Zeilberger to compute definite sums by synthesizing summands that telescope, called certificates. We describe a creative telescoping algorithm that computes telescopers for definite sums of D-finite functions as well as the associated certificates in a compact form. The algorithm relies on a discrete analogue of the generalized Hermite reduction, or equivalently, a generalization of the Abramov-Petkov\v sek reduction. We provide a Maple implementation with good timings on a variety of examples.Comment: 15 page

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