Polynôme

Résumé

thumb|Courbe représentative d'une fonction cubique. En mathématiques, un polynôme est une expression formée uniquement de produits et de sommes de constantes et d'indéterminées, habituellement notées X, Y, Z… Ces objets sont largement utilisés en pratique, ne serait-ce que parce qu'ils donnent localement une valeur approchée de toute fonction dérivable (voir l'article Développement limité) et permettent de représenter des formes lisses (voir l'article Courbe de Bézier, décrivant un cas particulier de fonction polynomiale). Un polynôme, en algèbre générale, à une indéterminée sur un anneau unitaire est une expression de la forme : : a_0 + a_1 X^1 + a_2 X^2 + \dots + a_n X^n , où X est un symbole appelé « indéterminée du polynôme », supposé être distinct de tout élément de l'anneau, les coefficients a{{ind|i}} sont dans l'anneau et n est un entier naturel. Si, en mathématiques appliquées, en analyse et en algèbre linéaire, il est fréquent de con

Source officielle

https://fr.wikipedia.org/wiki/Polyn%C3%B4me

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Sur la méthode de Buser-Silhol pour l'uniformisation des surfaces de Riemann hyperelliptiques

Grégoire Aubry

The Uniformization Theorem due to Koebe and Poincaré implies that every compact Riemann surface of genus greater or equal to 2 can be endowed with a metric of constant curvature – 1. On the other hand, a compact Riemann surface is a complex algebraic curve and is therefore described by a polynomial equation with complex coefficients. The uniformization problem is then to link explicitly these two descriptions. In [BS05b], Peter Buser and Robert Silhol develop a new uniformization method for compact Riemann surfaces of genus two. Given such a surface S, the method describes a polynomial equation of an algebraic curve conformally equivalent to S. However, in this method appear a complex number τ BS and a function f BS which is holomorphic on the unit disk, both being characterized by some functional equations. This means that τ BS, f BS are given implicitly. P. Buser and R. Silhol then approximate them numerically by a complex number τ and a polynomial p using the approximation method developped in [BS05a]. In cases where the equation of the algebraic curve is known, they notice that these approximations are very good. In this thesis we prove a convergence theorem for the approximation method of P. Buser and R. Silhol, and we propose an adaptation of their method that allows to solve some of the numerical problems to which it is prone. Moreover, we generalize this uniformization method to hyperelliptic Riemann surfaces of genus greater than 2, and we give some examples of numerical uniformization in genus 3.

EPFL2010

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L'algèbre al-Badi d'al-Karagi

Christophe Hebeisen

This work contains the study of the algebra called al-Badī‘ fī al-ḥisāb (literally : "the Wonderful on calculation"), written by the Persian mathematician Abu Bakr Muḥammad ibn al-Ḥusain al-Karaǧi (previously known as al-Karẖī, native from Karaǧ, Persia. Written c. 1010 in Bagdad, this work takes an important place in history of mathematics in general. Of particular interest are the first known appearance of a theory on root extracting of algebraic polynomials, and the beginning of a tendency to get rid of illustrating formulas and the resolutions of equations with help of geometric figures, which makes it a pure algebraic text. This work of high level adresses to a public with advanced mathematic knowledge. This algebra is, by will of the author, written in three main parts (books), containing part of Euclid's Elements (book I), a theory on root extracting of algebraic polynomials (book II), and a collection of problems on indeterminate analysis (book III). Some chapters are written hastily, while others go into the details. We provide a complete translation of the Badī‘, based on the transcription of the manuscript 36,1 of the Vatican library Barberini Orientale by Adel Anbouba (edited in Beyrouth in 1964), as well as a glossary. This translation comes with a mathematical commentary, and includes a list of significant words used by the author. We will also relate this algebra with other prior and later works containing the same problems.

EPFL2009

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We use the method of interlacing families of polynomials to derive a simple proof of Bourgain and Tzafriri's Restricted Invertibility Principle, and then to sharpen the result in two ways. We show that the stable rank can be replaced by the Schatten 4-norm stable rank and that tighter bounds hold when the number of columns in the matrix under consideration does not greatly exceed its number of rows. Our bounds are derived from an analysis of the smallest zeros of Jacobi and associated Laguerre polynomials.

HEBREW UNIV MAGNES PRESS2021

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