**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Polynomial

Summary

In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2 − yz + 1.
Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry.
Etymology

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Related courses (118)

Related publications (100)

MGT-418: Convex optimization

This course introduces the theory and application of modern convex optimization from an engineering perspective.

MATH-207(d): Analysis IV

Le cours étudie les concepts fondamentaux de l'analyse complexe et de l'analyse de Laplace en vue de leur utilisation
pour résoudre des problèmes pluridisciplinaires d'ingénierie scientifique.

MATH-106(a): Analysis II

Étudier les concepts fondamentaux d'analyse, et le calcul différentiel et intégral des fonctions réelles de plusieurs
variables.

Loading

Loading

Loading

Related people (19)

Related units (11)

Related concepts (245)

Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus

Complex number

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation

Mathematics

Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These top

Related lectures (311)

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.

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.

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.