Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?
Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur Graph Search.
Concept
Polynôme réciproque
Résumé
In algebra, given a polynomial
with coefficients from an arbitrary field, its reciprocal polynomial or reflected polynomial, denoted by p∗ or pR, is the polynomial
That is, the coefficients of p∗ are the coefficients of p in reverse order. Reciprocal polynomials arise naturally in linear algebra as the characteristic polynomial of the inverse of a matrix.
In the special case where the field is the complex numbers, when
the conjugate reciprocal polynomial, denoted p†, is defined by,
where denotes the complex conjugate of , and is also called the reciprocal polynomial when no confusion can arise.
A polynomial p is called self-reciprocal or palindromic if p(x) = p∗(x).
The coefficients of a self-reciprocal polynomial satisfy ai = an−i for all i.
Reciprocal polynomials have several connections with their original polynomials, including:
deg p = deg p∗ if is not 0.
p(x) = xnp∗(x−1).
α is a root of a polynomial p if and only if α−1 is a root of p∗.
If p(x) ≠ x then p is irreducible if and only if p∗ is irreducible.
p is primitive if and only if p∗ is primitive.
Other properties of reciprocal polynomials may be obtained, for instance:
A self-reciprocal polynomial of odd degree is divisible by x+1, hence is not irreducible if its degree is > 1.
A self-reciprocal polynomial is also called palindromic because its coefficients, when the polynomial is written in the order of ascending or descending powers, form a palindrome. That is, if
is a polynomial of degree n, then P is palindromic if ai = an−i for i = 0, 1, ..., n.
Similarly, a polynomial P of degree n is called antipalindromic if ai = −an−i for i = 0, 1, ..., n. That is, a polynomial P is antipalindromic if P(x) = –P∗(x).
From the properties of the binomial coefficients, it follows that the polynomials P(x) = (x + 1)n are palindromic for all positive integers n, while the polynomials Q(x) = (x – 1)n are palindromic when n is even and antipalindromic when n is odd.
Other examples of palindromic polynomials include cyclotomic polynomials and Eulerian polynomials.
Source officielle
À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.
Cours associés (1)
MATH-489: Number theory II.c - Cryptography
The goal of the course is to introduce basic notions from public key cryptography (PKC) as well as basic number-theoretic methods and algorithms for cryptanalysis of protocols and schemes based on PKC
Séances de cours associées (14)
Algèbre linéaire: représentation matricielle des applications linéaires
Explore la représentation matricielle des applications linéaires, en mettant l'accent sur les valeurs propres et les bases.
Diagonalisation orthogonale
Explore la diagonalisation orthogonale des matrices symétriques en utilisant des bases orthonormées et la méthode de Gram-Schmidt.
Polynôme caractéristique : sélections et équations
Discute des polynômes caractéristiques, des sélections de racines réelles et des équations de tiques.