In this thesis we study complex harmonic functions of the form f where f is the sum of a nonconstant analytic and a nonconstant anti-analytic function of one variable. The Fundamental Theorem of Algebra does not apply to such functions, so we ask how many zeros a complex harmonic function can have and where those zeros are located. This thesis focuses on the complex harmonic family of polynomials p_c where p_c is the sum of z+(c/2)z^2 and the conjugate of (c/(n-1))z^(n-1)+(1/n)z^n. We first establish properties of the critical curve, which separates orientation preserving and reversing regions. These properties are then used to show the sum of the orders of the zeros of p_c is -n. In turn, we use this to show p_c has n+2 zeros when 04 and n+4 zeros when c>4, n>5. The total number of zeros of p_c changes when zeros interact with the critical curve, so we investigate where zeros occur on the critical curve to understand how the number of zeros of p_c changes for c between 1 and 4.

This thesis studies complex harmonic polynomials of the form $f(z) = az^n + b\bar{z}^k+z$ where $n, k \in \mathbb{Z}$ with $n > k$ and $a, b > 0$. We show that the sum of the orders of the zeros of such functions is $n$ and investigate the locations of the zeros, including whether the zeros are in the sense-preserving or sense-reversing region and a set of conditions under which zeros have the same modulus. We also show that the number of zeros ranges from $n$ to $n+2k+2$ as long as certain criteria are met.

Cette thèse porte sur trois problèmes en combinatoire algébrique effective et algorithmique.Les premières parties proposent une approche alternative aux bases de Gröbner pour le calcul des invariants secondaires des groupes de permutations, par évaluation en des points choisis de manière appropriée. Cette méthode permet de tirer parti des symétries du problème pour confiner les calculs dans un quotient de petite dimension, et ainsi d'obtenir un meilleur contrôle de la complexité algorithmique, en particulier pour les groupes de grande taille. L'étude théorique est illustrée par de nombreux bancs d'essais utilisant une implantation fine des algorithmes. Un prérequis important est la génération efficace de vecteurs d'entiers modulo l'action d'un groupe de permutation, dont l'algorithmique fait l'objet d'une partie préliminaire.La quatrième partie cherche à déterminer, pour un certain quotient naturel d'une algèbre de Hecke affine, quelles spécialisations des paramètres aux racines de l'unité donne un comportement non générique.Finalement, la dernière partie présente une conjecture sur la structure d'une certaine $q$-déformation des polynômes harmoniques diagonaux en plusieurs paquets de variables pour la famille infinie de groupes de réflexions complexes.Tous ces chapitres s'appuient fortement sur l'exploration informatique, et font l'objet de multiples contributions au logiciel Sage. / This thesis concerns algorithmic approaches to three challenging problems in computational algebraic combinatorics.The firsts parts propose a Gröbner basis free approach for calculating the secondary invariants of a finite permutation group, proceeding by using evaluation at appropriately chosen points. This approach allows for exploiting the symmetries to confine the calculations into a smaller quotient space, which gives a tighter control on the algorithmic complexity, especially for large groups. The theoretical study is illustrated by extensive benchmarks using a fine implementation of algorithms. An important prerequisite is the generation of integer vectors modulo the action of a permutation group, whose algorithmic constitute a preliminary part of the thesis.The fourth part of this thesis is determining for a certain interesting quotient of an affine Hecke algebra exactly which root-of-unity specialization of its parameter lead to non-generic behavior.Finally, the last part presents a conjecture on the structure of certain q-deformed diagonal harmonics in many sets of variables for the infinite family of complex reflection groups.All chapters proceed widely by computer exploration, and most of established algorithms constitute contributions of the software Sage.