Algorithms in Intersection Theory in the Plane

This thesis presents an algorithm to find the local structure of intersections of plane curves. More precisely, we address the question of describing the scheme of the quotient ring of a bivariate zero-dimensional ideal $I\subseteq \mathbb K[x,y]$, \textit{i.e.} finding the points (maximal ideals of $\mathbb K[x,y]/I$) and describing the regular functions on those points. A natural way to address this problem is via Gr\"obner bases as they reduce the problem of finding the points to a problem of factorisation, and the sheaf of rings of regular functions can be studied with those bases through the division algorithm and localisation. Let $I\subseteq \mathbb K[x,y]$ be an ideal generated by $\mathcal F$, a subset of $\mathbb A[x,y]$ with $\mathbb A\hookrightarrow\mathbb K$ and $\mathbb K$ a field. We present an algorithm that features a quadratic convergence to find a Gr\"obner basis of $I$ or its primary component at the origin. We introduce an $\mathfrak m$-adic Newton iteration to lift the lexicographic Gr\"obner basis of any finite intersection of zero-dimensional primary components of $I$ if $\mathfrak m\subseteq \mathbb A$ is a \textit{good} maximal ideal. It relies on a structural result about the syzygies in such a basis due to Conca \textit{\&} Valla (2008), from which arises an explicit map between ideals in a stratum (or Gr\"obner cell) and points in the associated moduli space. We also qualify what makes a maximal ideal $\mathfrak m$ suitable for our filtration. When the field $\mathbb K$ is \textit{large enough}, endowed with an Archimedean or ultrametric valuation, and admits a fraction reconstruction algorithm, we use this result to give a complete $\mathfrak m$-adic algorithm to recover $\mathcal G$, the Gr\"obner basis of $I$. We observe that previous results of Lazard that use Hermite normal forms to compute Gr\"obner bases for ideals with two generators can be generalised to a set of $n$ generators. We use this result to obtain a bound on the height of the coefficients of $\mathcal G$ and to control the probability of choosing a \textit{good} maximal ideal $\mathfrak m\subseteq\mathbb A$ to build the $\mathfrak m$-adic expansion of $\mathcal G$. Inspired by Pardue (1994), we also give a constructive proof to characterise a Zariski open set of $\mathrm{GL}_2(\mathbb K)$ (with action on $\mathbb K[x,y]$) that changes coordinates in such a way as to ensure the initial term ideal of a zero-dimensional $I$ becomes Borel-fixed when $|\mathbb K|$ is sufficiently large. This sharpens our analysis to obtain, when $\mathbb A=\mathbb Z$ or $\mathbb A=k[t]$, a complexity less than cubic in terms of the dimension of $\mathbb Q[x,y]/\langle \mathcal G\rangle$ and softly linear in the height of the coefficients of $\mathcal G$. We adapt the resulting method and present the analysis to find the $\langle x,y\rangle$-primary component of $I$. We also discuss the transition towards other primary components via linear mappings, called \emph{untangling} and \emph{tangling}, introduced by van der Hoeven and Lecerf (2017). The two maps form one isomorphism to find points with an isomorphic local structure and, at the origin, bind them. We give a slightly faster tangling algorithm and discuss new applications of these techniques. We show how to extend these ideas to bivariate settings and give a bound on the arithmetic complexity for certain algebras

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

Modular and Analytical Methods for Solving Kinematics and Dynamics of Series-Parallel Hybrid Robots

While serial robots are known for their versatility in applications, larger workspace, simpler modeling and control, they have certain disadvantages like limited precision, lower stiffness and poor dynamic characteristics in general. A parallel robot can offer higher stiffness, speed, accuracy and payload capacity, at the downside of a reduced workspace and a more complex geometry that needs careful analysis and control. To bring the best of the two worlds, parallel submechanism modules can be connected in series to achieve a series-parallel hybrid robot with better dynamic characteristics and larger workspace. Such a design philosophy is being used in several robots not only at DFKI (for e.g., Mantis, Charlie, Recupera Exoskeleton, RH5 humanoid etc.) but also around the world, for e.g. Lola (TUM), Valkyrie (NASA), THOR (Virginia Tech.) etc.These robots inherit the complexity of both serial and parallel architectures. Hence, solving their kinematics and dynamics is challenging because they are subjected to additional geometric loop closure constraints. Most approaches in multi-body dynamics adopt numerical resolution of these constraints for the sake of generality but may suffer from inaccuracy and performance issues. They also do not exploit the modularity in robot design. Further, closed loop systems can have variable mobility, different assembly modes and can impose redundant constraints on the equations of motion which deteriorates the quality of many multi-body dynamics solvers. Very often only a local view to the system behavior is possible. Hence, it is interesting for geometers or kinematics researchers, to study the analytical solutions to geometric problems associated with a specific type of parallel mechanism and their importance over numerical solutions is irrefutable. Techniques such as screw theory, computational algebraic geometry, elimination and continuation methods are popular in this domain. But this domain specific knowledge is often underrepresented in the design of model based kinematics and dynamics software frameworks. The contributions of this thesis are two-fold. Firstly, a rigorous and comprehensive kinematic analysis is performed for the novel parallel mechanisms invented recently at DFKI-RIC such as RH5 ankle mechanism and Active Ankle using approaches from computational algebraic geometry and screw theory. Secondly, the general idea of a modular software framework called Hybrid Robot Dynamics (HyRoDyn) is presented which can be used to solve the geometry, kinematics and dynamics of series-parallel hybrid robotic systems with the help of a software database which stores the analytical solutions for parallel submechanism modules in a configurable and unit testable manner. HyRoDyn approach is suitable for both high fidelity simulations and real-time control of complex series-parallel hybrid robots. The results from this thesis has been applied to two robotic systems namely Recupera-Reha exoskeleton and RH5 humanoid. The aim of this software tool is to assist both designers and control engineers in developing complex robotic systems of the future. Efficient kinematic and dynamic modeling can lead to more compliant behavior, better whole body control, walking and manipulating capabilities etc. which are highly desired in the present day and future robotic applications

Métodos numérico-simbólicos para calcular soluciones liouvillianas de ecuaciones diferenciales lineales

El objetivo de esta tesis es dar un algoritmo para decidir si un sistema explicitable de ecuaciones diferenciales kJiferenciales de orden superior sobre las funciones racionales complejas, dado simbólicamente,admite !Soluciones liouvillianas no nulas, calculando una (de laforma dada por un teorema de Singer) en caso !afirmativo. mediante métodos numérico-simbólicos del tipo Introducido por van der Hoeven.donde el uso de álculo numérico no compromete la corrección simbólica. Para ello se Introduce untipo de grupos algebraicos lineales, los grupos euriméricos, y se calcula el cierre eurimérico del grupo de Galois diferencial,mediante una modificación del algoritmo de Derksen y van der Hoeven, dado por los generadores de Ramis.Departamento de Algebra, Análisis Matemático, Geometría y Topologí

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics

Cutting Feynman Amplitudes: from Adaptive Integrand Decomposition to Differential Equations on Maximal Cut

In this thesis we discuss, within the framework of the Standard Model (SM) of particle physics, advanced methods for the computation of scattering amplitudes at higher-order in perturbation theory. We offer a new insight into the role played by the unitarity of scattering amplitudes in the theoretical understanding and in the computational simplification of multi-loop calculations, at both the algebraic and the analytical level. On the algebraic side, generalized unitarity can be used, within the integrand reduction method, to express the integrand associated to a multi-loop amplitude as a sum of fundamental, irreducible contributions, yielding to a decomposition of the amplitude as a linear combination of master integrals. In this framework, we propose an adaptive formulation of the integrand decomposition algorithm, which systematically adjusts to the kinematics of the individual integrands the dimensionality of the momentum space, where unitarity cuts are performed. This new formulation makes the integrand decomposition method, which in the past played a key role in streamlining one-loop computations, an efficient tool also at multi-loop level. We provide evidence of the generality of the proposed method by determining a universal parametrization of the integrand basis for two-loop amplitudes in arbitrary kinematics and we illustrate its technical feasibility in the first automated implementation of the analytic integrand decomposition at one- and two-loop level. On the analytic side, we discuss the role of maximal-unitarity for the solution of differential equations for dimensionally regulated Feynman integrals. The determination of the analytic expression of the master integrals as a Laurent expansion in the dimensional regulating parameter requires the knowledge of the solutions of the homogeneous part of their differential equations at d=4. In all cases where Feynman integrals fulfil genuine first-order differential equations with a linear dependence on d, the corresponding homogeneous solutions can be determined through the Magnus exponential expansion. In this work we apply the latter to two-loop corrections to several SM processes such as the Higgs decay to weak vector bosons, H → WW, triple gauge couplings ZWW and γ∗WW and to the elastic scattering μe → μe in quantum electrodynamics. In some cases, the inadequacy of the Magnus method hints at the presence of master integrals that obey higher-order differential equations, for which no general theory exists. In this thesis we show that maximal-cuts of Feynman integrals solve, by construction, such homogeneous equations regardless of their order and complexity. Hence, whenever a Feynman integral obeys an irreducible higher-order differential equation, the computation of its maximal-cut along independent contours provides a closed integral representation of the full set of independent homogeneous solutions. We apply this strategy to the two-loop elliptic integrals that appear in heavy-quark mediated corrections to gg → gg and gg → gH as well as to the three-loop massive banana graph, which constitute the first example of Feynman integral that obeys a third-order differential equation. In the light of the results presented in this thesis, generalized unitarity emerges as a powerful tool not only for handling the algebraic complexity of perturbative calculations but also for investigating the nature of new classes of mathematical functions encountered in particle physics

Applications of Convex Analysis to Signomial and Polynomial Nonnegativity Problems

Here is a question that is easy to state, but often hard to answer: Is this function nonnegative on this set? When faced with such a question, one often makes appeals to known inequalities. One crafts arguments that are sufficient to establish the nonnegativity of the function, rather than determining the function's precise range of values. This thesis studies sufficient conditions for nonnegativity of signomials and polynomials. Conceptually, signomials may be viewed as generalized polynomials that feature arbitrary real exponents, but with variables restricted to the positive orthant. Our methods leverage efficient algorithms for a type of convex optimization known as relative entropy programming (REP). By virtue of this integration with REP, our methods can help answer questions like the following: Is there some function, in this particular space of functions, that is nonnegative on this set? The ability to answer such questions is extremely useful in applied mathematics. Alternative approaches in this same vein (e.g., methods for polynomials based on semidefinite programming) have been used successfully as convex relaxation frameworks for nonconvex optimization, as mechanisms for analyzing dynamical systems, and even as tools for solving nonlinear partial differential equations. This thesis builds from the sums of arithmetic-geometric exponentials or SAGE approach to signomial nonnegativity. The term "exponential" appears in the SAGE acronym because SAGE parameterizes signomials in terms of exponential functions. Our first round of contributions concern the original SAGE approach. We employ basic techniques in convex analysis and convex geometry to derive structural results for spaces of SAGE signomials and exactness results for SAGE-based REP relaxations of nonconvex signomial optimization problems. We frame our analysis primarily in terms of the coefficients of a signomial's basis expansion rather than in terms of signomials themselves. The effect of this framing is that our results for signomials readily transfer to polynomials. In particular, we are led to define a new concept of SAGE polynomials. For sparse polynomials, this method offers an exponential efficiency improvement relative to certificates of nonnegativity obtained through semidefinite programming. We go on to create the conditional SAGE methodology for exploiting convex substructure in constrained signomial nonnegativity problems. The basic insight here is that since the standard relative entropy representation of SAGE signomials is obtained by a suitable application of convex duality, we are free to add additional convex constraints into the duality argument. In the course of explaining this idea we provide some illustrative examples in signomial optimization and analysis of chemical dynamics. The majority of this thesis is dedicated to exploring fundamental questions surrounding conditional SAGE signomials. We approach these questions through analysis frameworks of sublinear circuits and signomial rings. These sublinear circuits generalize simplicial circuits of affine-linear matroids, and lead to rich modes of analysis for sets that are simultaneously convex in the usual sense and convex under a logarithmic transformation. The concept of signomial rings lets us develop a powerful signomial Positivstellensatz and an elementary signomial moment theory. The Positivstellensatz provides for an effective hierarchy of REP relaxations for approaching the value of a nonconvex signomial minimization problem from below, as well as a first-of-its-kind hierarchy for approaching the same value from above. In parallel with our mathematical work, we have developed the sageopt python package. Sageopt drives all the examples and experiments used throughout this thesis, and has been used by engineers to solve high-degree polynomial optimization problems at scales unattainable by alternative methods. We conclude this thesis with an explanation of how our theoretical results affected sageopt's design.</p