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Concept
Reciprocal polynomial
Summary
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.
Official source
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