Polynôme réciproque

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.