Reciprocal polynomial

In algebra, given a polynomial
with coefficients from an arbitrary field, its reciprocal polynomial or reflected polynomial, denoted by or , is the polynomial
That is, the coefficients of are the coefficients of in reverse order. They 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 , is defined by,
where formula_5 denotes the complex conjugate of formula_6, and 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 for all . 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:
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.
Real coefficients.
A polynomial with real coefficients all of whose complex roots lie on the unit circle in the complex plane (that is, all the roots have modulus 1) is either palindromic or antipalindromic.
Conjugate reciprocal polynomials.
A polynomial is conjugate reciprocal if formula_8 and self-inversive if formula_9 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 formula_11 which has degree . But, the minimal polynomial is unique, hence 
for some constant , i.e. formula_13. 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 (Euler's totient function) of the exponents are 10, 12, 8 and 12.
Application in coding theory.
The reciprocal polynomial finds a use in the theory of cyclic 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 .