In algebra, the reciprocal polynomial, or reflected polynomial or, of a polynomial of degree with coefficients from an arbitrary field, such as is the polynomial Essentially, the coefficients are written in reverse order. They arise naturally in linear algebra as the characteristic polynomial of the inverse of a matrix. In the special case that the polynomial has complex coefficients, that is, the conjugate reciprocal polynomial, given by, where denotes the complex conjugate of, is also called the reciprocal polynomial when no confusion can arise. A polynomial is called self-reciprocal or palindromic if. The coefficients of a self-reciprocal polynomial satisfy. In the conjugate reciprocal case, the coefficients must be real to satisfy the condition.
Properties
Reciprocal polynomials have several connections with their original polynomials, including:
Other properties of reciprocal polynomials may be obtained, for instance:
If a polynomial is self-reciprocal and irreducible then it must have even degree.
Palindromic and antipalindromic polynomials
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, then is palindromic if for. Some authors use the terms palindromic and reciprocal interchangeably. Similarly,, a polynomial of degree, is called antipalindromic if for. That is, a polynomial is antipalindromic if.
Examples
From the properties of the binomial coefficients, it follows that the polynomials are palindromic for all positive integers, while the polynomials are palindromic when is even and antipalindromic when is odd. Other examples of palindromic polynomials include cyclotomic polynomials and Eulerian polynomials.
The converse is true: If a polynomial is such that if is a root then is also a root of the same multiplicity, then the polynomial is either palindromic or antipalindromic.
For any polynomial, the polynomial is palindromic and the polynomial is antipalindromic.
Any polynomial can be written as the sum of a palindromic and an antipalindromic polynomial.
The product of two palindromic or antipalindromic polynomials is palindromic.
The product of a palindromic polynomial and an antipalindromic polynomial is antipalindromic.
A palindromic polynomial of odd degree is a multiple of and its quotient by is also palindromic.
An antipalindromic polynomial is a multiple of and its quotient by is palindromic.
An antipalindromic polynomial of even degree is a multiple of and its quotient by is palindromic.
If is a palindromic polynomial of even degree, then there is a polynomial of degree such that .
If is a monic antipalindromic polynomial of even degree over a field with odd characteristic, then it can be written uniquely as, where is a monic polynomial of degree with no constant term.
If an antipalindromic polynomial has even degree, then its "middle" coefficient is 0 since.
Real coefficients
A polynomial with real coefficients all of whose complex roots lie on the unit circle in the complex plane is either palindromic or antipalindromic.
Conjugate reciprocal polynomials
A polynomial is conjugate reciprocal if and self-inversive if for a scale factor on the unit circle. If is the minimal polynomial of with, and has real coefficients, then is self-reciprocal. This follows because So is a root of the polynomial which has degree. But, the minimal polynomial is unique, hence for some constant, i.e.. Sum from to and note that 1 is not a root of. We conclude that. A consequence is that the cyclotomic polynomials are self-reciprocal for. This is used in the special number field sieve to allow numbers of the form and to be factored taking advantage of the algebraic factors by using polynomials of degree 5, 6, 4 and 6 respectively – note that of the exponents are 10, 12, 8 and 12.
The reciprocal polynomial finds a use in the theory ofcyclic error correcting codes. Suppose can be factored into the product of two polynomials, say. When generates a cyclic code, then the reciprocal polynomial generates, the orthogonal complement of. Also, is self-orthogonal (that is,, if and only if divides.
