Computer Algebra in the Service of Enumerative Combinatorics

Bostan, Alin

doi:10.1145/3452143.3465507

Cited by 4 publications

(3 citation statements)

References 70 publications

(84 reference statements)

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“…Diagonals of rational functions (or diagonals of algebraic functions) have been shown to emerge naturally [25] for n-fold integrals in physics (corresponding to solutions of linear differential operators of quite high order [26]); field theory; and enumerative combinatorics [27,28], and have been seen as "Periods" [29][30][31] of algebraic varieties (corresponding to the denominators of these rational functions). The fact that diagonals of rational or algebraic functions occur frequently in physics explains many unexpected mathematical properties encountered in physics that are far more obvious from a physics viewpoint.…”

Section: Diagonal Of Rational Functions Creative Telescoping Biration...mentioning

confidence: 99%

Plea for Diagonals and Telescopers of Rational Functions

Hassani,

Maillard,

Zenine

2024

Universe

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This paper is a plea for diagonals and telescopers of rational or algebraic functions using creative telescoping, using a computer algebra experimental mathematics learn-by-examples approach. We show that diagonals of rational functions (and this is also the case with diagonals of algebraic functions) are left-invariant when one performs an infinite set of birational transformations on the rational functions. These invariance results generalize to telescopers. We cast light on the almost systematic property of homomorphism to their adjoint of the telescopers of rational or algebraic functions. We shed some light on the reason why the telescopers, annihilating the diagonals of rational functions of the form P/Qk and 1/Q, are homomorphic. For telescopers with solutions (periods) corresponding to integration over non-vanishing cycles, we have a slight generalization of this result. We introduce some challenging examples of the generalization of diagonals of rational functions, like diagonals of transcendental functions, yielding simple F12 hypergeometric functions associated with elliptic curves, or the (differentially algebraic) lambda-extension of correlation of the Ising model.

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Section: Diagonal Of Rational Functions Creative Telescoping Biration...mentioning

confidence: 99%

Plea for Diagonals and Telescopers of Rational Functions

Hassani,

Maillard,

Zenine

2024

Universe

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“…In particular, it is proved that exactly 23 of the 79 models (roughly 29%) have a D-finite generating function and of those exactly 6 are algebraic. We refer to Bostan's habilitation thesis for an excellent summary on this topic [Bos17], see also [Bos21] for a short but clear-cut exposition.…”

Section: Importance In Practicementioning

confidence: 99%

The art of algorithmic guessing in gfun

Yurkevich

2022

Maple Trans.

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The technique of guessing can be very fruitful when dealing with sequences which arise in practice. This holds true especially when guessing is performed algorithmically and efficiently. The ideal tool for it exists as a package named gfun in the software Maple. In this text we explore and explain some of gfun's possibilities and illustrate them on two examples from recent mathematical research by the author and his collaborators.

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“…Combinatorial motivation. Recently, these computations have become relevant in the very active study of discrete walks in N d (see recent surveys [27,12] for numerous references). The relation between the Brownian motion and the heat equation can be exploited to derive the asymptotic number of walks in N d starting and ending at the origin and using n steps, all taken from a given finite set S ⊂ Z d [17].…”

Section: Introductionmentioning

confidence: 99%

Computation of Tight Enclosures for Laplacian Eigenvalues

Dahne,

Salvy

2020

Preprint

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Recently, there has been interest in high-precision approximations of the first eigenvalue of the Laplace-Beltrami operator on spherical triangles for combinatorial purposes. We compute improved and certified enclosures to these eigenvalues. This is achieved by applying the method of particular solutions in high precision, the enclosure being obtained by a combination of interval arithmetic and Taylor models. The index of the eigenvalue is certified by exploiting the monotonicity of the eigenvalue with respect to the domain. The classically troublesome case of singular corners is handled by combining expansions from all corners and an expansion from an interior point. In particular, this allows us to compute 80 digits of the fundamental eigenvalue for the 3D Kreweras model that has been the object of previous efforts.

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Computer Algebra in the Service of Enumerative Combinatorics

Cited by 4 publications

References 70 publications

Plea for Diagonals and Telescopers of Rational Functions

Plea for Diagonals and Telescopers of Rational Functions

The art of algorithmic guessing in gfun

Computation of Tight Enclosures for Laplacian Eigenvalues

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