<p>The main focus of this dissertation is to find and explain the periodic points of certain algebraic functions that are related to some modular functions, which themselves can be represented by continued fractions. Some of these continued fractions are first explored by Srinivasa Ramanujan in early 20th century. Later on, much work has been done in terms of studying the continued fractions, and proving several relations, identities, and giving different representations for them.</p> <p><br></p> <p>The layout of this report is as follows. Chapter 1 has all the basic background knowledge and ingredients about algebraic number theory, class field theory, Ramanujan’s theta functions, etc. In Chapter 2, we look at the Ramanujan-Göllnitz-Gordon continued fraction that we call v(τ) and evaluate it at certain arguments in the field K = Q(√−d), with −d ≡ 1 (mod 8), in which the ideal (2) = ℘₂℘′₂  is a product of two prime ideals. We prove several identities related to itself and with other modular functions. Some of these are new, while some of them are known but with different proofs. These values of v(τ) are shown to generate the inertia field of ℘₂ or ℘′₂ in an extended ring class field over the field K. The conjugates over Q of these same values, together with 0, −1 ± √2, are shown to form the exact set of periodic points of a fixed algebraic function ˆF(x), independent of d. These are analogues of similar results for the Rogers-Ramanujan continued fraction. See [1] and [2]. This joint work with my advisor Dr. Morton, is submitted for publication to the New York Journal.</p> <p><br> In Chapters 3 and 4, we take a similar approach in studying two more continued fractions c(τ) and u(τ), the first of which is more commonly known as the Ramanujan’s cubic continued fraction. We show what fields a value of this continued fraction generates over Q, and we describe how the periodic points for described functions arise as values of these continued fractions. Then in the last chapter, we summarise all these results, give some possible directions for future research as well as mentioning some conjectures.</p>

Dans cette thèse, nous étudions des algorithmes pour un problème de recherche de relations à une ou plusieurs variables. Il généralise celui de calculer une solution à un système d’équations linéaires modulaires sur un anneau de polynômes, et inclut par exemple le calcul d’approximants de Hermite-Padé ou d’interpolants bivariés. Plutôt qu’une seule solution, nous nous attacherons à calculer un ensemble de générateurs possédant de bonnes propriétés. Précisément, l’entrée de notre problème consiste en un module de dimension finie spécifié par l’action des variables sur ses éléments, et en un certain nombre d’éléments de ce module ; il s’agit de calculer une base de Gröbner du modules des relations entre ces éléments. En termes d’algèbre linéaire, l’entrée décrit une matrice avec une structure de type Krylov, et il s’agit de calculer sous forme compacte une base du noyau de cette matrice. Nous proposons plusieurs algorithmes en fonction de la forme des matrices de multiplication qui représentent l’action des variables. Dans le cas d’une matrice de Jordan,nous accélérons le calcul d’interpolants multivariés sous certaines contraintes de degré ; nos résultats pour une forme de Frobenius permettent d’accélérer le calcul de formes normales de matrices polynomiales univariées. Enfin, dans le cas de plusieurs matrices denses, nous accélérons le changement d’ordre pour des bases de Gröbner d’idéaux multivariés zéro-dimensionnels. / In this thesis, we study algorithms for a problem of finding relations in one or several variables. It generalizes that of computing a solution to a system of linear modular equations over a polynomial ring, including in particular the computation of Hermite- Padéapproximants and bivariate interpolants. Rather than a single solution, we aim at computing generators of the solution set which have good properties. Precisely, the input of our problem consists of a finite-dimensional module given by the action of the variables on its elements, and of some elements of this module; the goal is to compute a Gröbner basis of the module of syzygies between these elements. In terms of linear algebra, the input describes a matrix with a type of Krylov structure, and the goal is to compute a compact representation of a basis of the nullspace of this matrix. We propose several algorithms in accordance with the structure of the multiplication matrices which specify the action of the variables. In the case of a Jordan matrix, we accelerate the computation of multivariate interpolants under degree constraints; our result for a Frobenius matrix leads to a faster algorithm for computing normal forms of univariate polynomial matrices. In the case of several dense matrices, we accelerate the change of monomial order for Gröbner bases of multivariate zero-dimensional ideals.