Explicit formula for the generating series of diagonal 3D rook paths

Let $a_n$ denote the number of ways in which a chess rook can move from a corner cell to the opposite corner cell of an $n \times n \times n$ three-dimensional chessboard, assuming that the piece moves closer to the goal cell at each step. We describe the computer-driven \emph{discovery and proof} of the fact that the generating series $G(x)= \sum_{n \geq 0} a_n x^n$ admits the following explicit expression in terms of a Gaussian hypergeometric function: G(x) = 1 + 6 \cdot \int_0^x \frac{\,\pFq21{1/3}{2/3}{2} {\frac{27 w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} \, dw.Comment: To appear in "S\'eminaire Lotharingien de Combinatoire

Analytic Combinatorics in Several Variables: Effective Asymptotics and Lattice Path Enumeration

The field of analytic combinatorics, which studies the asymptotic behaviour of sequences through analytic properties of their generating functions, has led to the development of deep and powerful tools with applications across mathematics and the natural sciences. In addition to the now classical univariate theory, recent work in the study of analytic combinatorics in several variables (ACSV) has shown how to derive asymptotics for the coefficients of certain D-finite functions represented by diagonals of multivariate rational functions. We give a pedagogical introduction to the methods of ACSV from a computer algebra viewpoint, developing rigorous algorithms and giving the first complexity results in this area under conditions which are broadly satisfied. Furthermore, we give several new applications of ACSV to the enumeration of lattice walks restricted to certain regions. In addition to proving several open conjectures on the asymptotics of such walks, a detailed study of lattice walk models with weighted steps is undertaken.Comment: PhD thesis, University of Waterloo and ENS Lyon - 259 page

Tropical Geometry in Singular

Das Ziel dieser Dissertation ist die Entwicklung und Implementation eines Algorithmus zur Berechnung von tropischen Varietäten über allgemeine bewertete Körper. Die Berechnung von tropischen Varietäten über Körper mit trivialer Bewertung ist ein hinreichend gelöstes Problem. Hierfür kombinieren die Autoren Bogart, Jensen, Speyer, Sturmfels und Thomas eindrucksvoll klassische Techniken der Computeralgebra mit konstruktiven Methoden der konvexer Geometrie. Haben wir allerdings einen Grundkörper mit nicht-trivialer Bewertung, wie zum Beispiel den Körper der $p$-adischen Zahlen $\mathbb{Q}_p$, dann stößt die konventionelle Gröbnerbasentheorie scheinbar an ihre Grenzen. Die zugrundeliegenden Monomordnungen sind nicht geeignet um Problemstellungen zu untersuchen, die von einer nicht-trivialen Bewertung auf den Koeffizienten abhängig sind. Dies führte zu einer Reihe von Arbeiten, welche die gängige Gröbnerbasentheorie modifizieren um die Bewertung des Grundkörpers einzubeziehen.$$\phantom{newline}$$ In dieser Arbeit präsentieren wir einen alternativen Ansatz und zeigen, wie sich die Bewertung mittels einer speziell eingeführten Variable emulieren lässt, so dass eine Modifikation der klassischen Werkzeuge nicht notwendig ist. Im Rahmen dessen wird Theorie der Standardbasen auf Potenzreihen über einen Koeffizientenring verallgemeinert. Hierbei wird besonders Wert darauf gelegt, dass alle Algorithmen bei polynomialen Eingabedaten mit ihren klassischen Pendants übereinstimmen, sodass für praktische Zwecke auf bereits etablierte Softwaresysteme zurückgegriffen werden kann. Darüber hinaus wird die Konstruktion des Gröbnerfächers sowie die Technik des Gröbnerwalks für leicht inhomogene Ideale eingeführt. Dies ist notwendig, da bei der Einführung der neuen Variable die Homogenität des Ausgangsideal gebrochen wird.$$\phantom{newline}$$ Alle Algorithmen wurden in Singular implementiert und sind als Teil der offiziellen Distribution erhältlich. Es ist die erste Implementation, welches in der Lage ist tropische Varietäten mit $p$-adischer Bewertung auszurechnen. Im Rahmen der Arbeit entstand ebenfalls ein Singular Paket für konvexe Geometrie, sowie eine Schnittstelle zu Polymake

Implementation and Evaluation of Algorithmic Skeletons: Parallelisation of Computer Algebra Algorithms

This thesis presents design and implementation approaches for the parallel algorithms of computer algebra. We use algorithmic skeletons and also further approaches, like data parallel arithmetic and actors. We have implemented skeletons for divide and conquer algorithms and some special parallel loops, that we call ‘repeated computation with a possibility of premature termination’. We introduce in this thesis a rational data parallel arithmetic. We focus on parallel symbolic computation algorithms, for these algorithms our arithmetic provides a generic parallelisation approach. The implementation is carried out in Eden, a parallel functional programming language based on Haskell. This choice enables us to encode both the skeletons and the programs in the same language. Moreover, it allows us to refrain from using two different languages—one for the implementation and one for the interface—for our implementation of computer algebra algorithms. Further, this thesis presents methods for evaluation and estimation of parallel execution times. We partition the parallel execution time into two components. One of them accounts for the quality of the parallelisation, we call it the ‘parallel penalty’. The other is the sequential execution time. For the estimation, we predict both components separately, using statistical methods. This enables very confident estimations, although using drastically less measurement points than other methods. We have applied both our evaluation and estimation approaches to the parallel programs presented in this thesis. We haven also used existing estimation methods. We developed divide and conquer skeletons for the implementation of fast parallel multiplication. We have implemented the Karatsuba algorithm, Strassen’s matrix multiplication algorithm and the fast Fourier transform. The latter was used to implement polynomial convolution that leads to a further fast multiplication algorithm. Specially for our implementation of Strassen algorithm we have designed and implemented a divide and conquer skeleton basing on actors. We have implemented the parallel fast Fourier transform, and not only did we use new divide and conquer skeletons, but also developed a map-and-transpose skeleton. It enables good parallelisation of the Fourier transform. The parallelisation of Karatsuba multiplication shows a very good performance. We have analysed the parallel penalty of our programs and compared it to the serial fraction—an approach, known from literature. We also performed execution time estimations of our divide and conquer programs. This thesis presents a parallel map+reduce skeleton scheme. It allows us to combine the usual parallel map skeletons, like parMap, farm, workpool, with a premature termination property. We use this to implement the so-called ‘parallel repeated computation’, a special form of a speculative parallel loop. We have implemented two probabilistic primality tests: the Rabin–Miller test and the Jacobi sum test. We parallelised both with our approach. We analysed the task distribution and stated the fitting configurations of the Jacobi sum test. We have shown formally that the Jacobi sum test can be implemented in parallel. Subsequently, we parallelised it, analysed the load balancing issues, and produced an optimisation. The latter enabled a good implementation, as verified using the parallel penalty. We have also estimated the performance of the tests for further input sizes and numbers of processing elements. Parallelisation of the Jacobi sum test and our generic parallelisation scheme for the repeated computation is our original contribution. The data parallel arithmetic was defined not only for integers, which is already known, but also for rationals. We handled the common factors of the numerator or denominator of the fraction with the modulus in a novel manner. This is required to obtain a true multiple-residue arithmetic, a novel result of our research. Using these mathematical advances, we have parallelised the determinant computation using the Gauß elimination. As always, we have performed task distribution analysis and estimation of the parallel execution time of our implementation. A similar computation in Maple emphasised the potential of our approach. Data parallel arithmetic enables parallelisation of entire classes of computer algebra algorithms. Summarising, this thesis presents and thoroughly evaluates new and existing design decisions for high-level parallelisations of computer algebra algorithms

Precision QCD corrections to gluon-initiated diphoton-plus-jet production at the LHC

In this thesis, we present recent advances at the precision frontier of higher-order quantum chromodynamics (QCD) calculations. We consider massless two-loop five-point amplitudes, with a particular focus on diphoton-plus-jet production through gluon fusion. We build a library of infrared functions up to at most next-to-next-to-leading order (NNLO) in QCD, which can be used to validate amplitudes and construct counterterms in subtraction schemes at NNLO. We review progress in the novel use of machine learning technology to optimise the evaluation of amplitudes in hadron collider simulations. We present the full-colour virtual QCD corrections to diphoton-plus-jet production through gluon fusion, discussing the new techniques developed to calculate these non-planar two-loop amplitudes. We use these amplitudes to compute the next-to-leading QCD corrections to the differential cross sections of diphoton-plus-jet production through gluon fusion at the Large Hadron Collider. We also present the leading-colour double-virtual corrections to hadronic trijet production. All derived amplitudes are made available in a public implementation that is ready for further phenomenological application

Modular polynomials on Hilbert surfaces

International audienceWe describe an evaluation/interpolation approach to compute modular polynomials on a Hilbert surface, which parametrizes abelian surfaces with maximal real multiplication. Under some heuristics we obtain a quasi-linear algorithm. The corresponding modular polynomials are much smaller than the ones on the Siegel threefold. We explain how to compute even smaller polynomials by using pullbacks of theta functions to the Hilbert surface

Techniques for high-multiplicity scattering amplitudes and applications to precision collider physics

In this thesis, we present state-of-the-art techniques for the computation of scattering amplitudes in Quantum Field Theories. Following an introduction to the topic, we describe a robust framework that enables the calculation of multi-scale two-loop amplitudes directly relevant to modern particle physics phenomenology at the Large Hadron Collider and beyond. We discuss in detail the use of finite fields to bypass the algebraic complexity of such computations, as well as the method of integration-by-parts relations and differential equations. We apply our framework to calculate the two-loop amplitudes contributing to three process: Higgs boson production in association with a bottom-quark pair, W boson production with a photon and a jet, as well as lepton-pair scattering with an off-shell and an on-shell photon. Finally, we draw our conclusions and discuss directions for future progress of amplitude computations