In algebra, given a polynomial

:<math>p(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n,</math>

with coefficients from an arbitrary field, its reciprocal polynomial or reflected polynomial, denoted by or ,

:<math>p^*(x) = a_n + a_{n-1}x + \cdots + a_0x^n = x^n p(x^{-1}).</math>

That is, the coefficients of are the coefficients of 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

:<math>p(z) = a_0 + a_1z + a_2z^2 + \cdots + a_nz^n,</math>

the conjugate reciprocal polynomial, denoted , is defined by,

:<math>p^{\dagger}(z) = \overline{a_n} + \overline{a_{n-1z + \cdots + \overline{a_0}z^n = z^n\overline{p(\bar{z}^{-1})},</math>

where <math>\overline{a_i}</math> denotes the complex conjugate of <math>a_i</math>, 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 .

Properties

Reciprocal polynomials have several connections with their original polynomials, including:

  2. .
  3. If then is irreducible if and only if is irreducible.
  4. is primitive if and only if is primitive.
  • The converse is true: If for each root of polynomial, the value 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.
  • It follows that any polynomial can be written as the sum of a palindromic and an antipalindromic polynomial, since .
  • 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 (it has –1 as a root) and its quotient by is also palindromic.
  • An antipalindromic polynomial over a field with odd characteristic is a multiple of (it has 1 as a root) and its quotient by is palindromic.
  • An antipalindromic polynomial of even degree is a multiple of (it has −1 and 1 as roots) and its quotient by is palindromic.
  • If is a palindromic polynomial of even degree 2, then there is a polynomial of degree such that .
  • If is a monic antipalindromic polynomial of even degree 2 over a field of 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 over a field of odd characteristic, then its "middle" coefficient (of power ) is 0 since .

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 <math>p(x) \equiv p^{\dagger}(x)</math> and self-inversive if <math>p(x) = \omega p^{\dagger}(x)</math> for a scale factor on the unit circle.

If is the minimal polynomial of with , and has real coefficients, then is conjugate-reciprocal. This follows because

:<math>z_0^n\overline{p(1/\bar{z_0})} = z_0^n\overline{p(z_0)} = z_0^n\bar{0} = 0.</math>

So is a root of the polynomial <math>z^n\overline{p(\bar{z}^{-1})}</math> which has degree . But, the minimal polynomial is unique, hence

:<math>cp(z) = z^n\overline{p(\bar{z}^{-1})}</math>

for some constant , i.e. <math>ca_i=\overline{a_{n-i=a_{n-i}</math>. Sum from to and note that 1 is not a root of . We conclude that .

A consequence is that the cyclotomic polynomials are conjugate-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.

Per Cohn's theorem, a self-inversive polynomial has as many roots in the unit disk <math>\{z\in\mathbb{C}: |z| < 1\}</math> as the reciprocal polynomial of its derivative.

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 .

See also

  • Cohn's theorem

Notes

References