Reciprocal polynomial

In algebra, the reciprocal polynomial p* of a polynomial p with coefficients from an arbitrary field, such as

\( p(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n, \,\! \)

is the polynomial[1]

\( p^*(x) = a_n + a_{n-1}x + \cdots + a_0x^n = x^n p(x^{-1}). \)

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 p has complex coefficients, that is,

\( p(z) = a_0 + a_1z + a_2z^2 + \cdots + a_nz^n, \,\! \)

the conjugate reciprocal polynomial, p† given by,

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

where \( \overline{a_i} \) denotes the complex conjugate of \( a_i \,\! \) , is also called the reciprocal polynomial when no confusion can arise.

A polynomial p is called self-reciprocal if \( p(x) \equiv p^{*}(x). \)

The coefficients of a self-reciprocal polynomial satisfy a_i = a_n−i, and in this case p is also called a palindromic polynomial. 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:

α is a root of polynomial p if and only if α−1 is a root of p*.[2]
If p(x) ≠ x then p is irreducible if and only if p* is irreducible.[3]
p is primitive if and only if p* is primitive.[2]

Other properties of reciprocal polynomials may be obtained, for instance:

If a polynomial is self-reciprocal and irreducible then it must have even degree.[3]

Conjugate reciprocal polynomials

A polynomial is conjugate reciprocal if \( p(x) \equiv p^{\dagger}(x) \) and self-inversive if \( p(x) = \omega p^{\dagger}(x) \) for a scale factor ω on the unit circle.[4]

If p(z) is the minimal polynomial of z0 with |z0| = 1, \( z_0\neq1 \), and p(z) has real coefficients, then p(z) is self-reciprocal. This follows because

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

So z0 is a root of the polynomial \( z^n\overline{p(\bar{z}^{-1})} which has degree n. But, the minimal polynomial is unique, hence

\( cp(z) = z^n\overline{p(\bar{z}^{-1})} \)

for some constant c, i.e. \( ca_i=\overline{a_{n-i}}=a_{n-i} \). Sum from i = 0 to n and note that 1 is not a root of p. We conclude that c = 1.

A consequence is that the cyclotomic polynomials \( \Phi_n \) are self-reciprocal for n > 1; this is used in the special number field sieve to allow numbers of the form \( x^{11} \pm 1, x^{13} \pm 1, x^{15} \pm 1 \) and \( x^{21} \pm 1 \) to be factored taking advantage of the algebraic factors by using polynomials of degree 5, 6, 4 and 6 respectively – note that \( \phi \) (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 xⁿ − 1 can be factored into the product of two polynomials, say xⁿ − 1 = g(x)p(x). When g(x) generates a cyclic code C, then the reciprocal polynomial p*(x) generates C^⊥, the orthogonal complement of C.^[5] Also, C is self-orthogonal (that is, C ⊆ C^⊥), if and only if p*(x) divides g(x).^[6]

Notes

Roman 1995, pg.37
Pless 1990, pg. 57
Roman 1995, pg. 37
Sinclair, Christopher D.; Vaaler, Jeffrey D. (2008). "Self-inversive polynomials with all zeros on the unit circle". In McKee, James; Smyth, C. J. Number theory and polynomials. Proceedings of the workshop, Bristol, UK, April 3–7, 2006. London Mathematical Society Lecture Note Series 352. Cambridge: Cambridge University Press. pp. 312–321. ISBN 978-0-521-71467-9. Zbl 06093092.
Pless 1990, pg. 75, Theorem 48
Pless 1990, pg. 77, Theorem 51

References

Pless, Vera (1990), Introduction to the Theory of Error Correcting Codes (2nd ed.), New York: Wiley-Interscience, ISBN 0-471-61884-5

Roman, Steven (1995), Field Theory, New York: Springer-Verlag, ISBN 0-387-94408-7

External links

Reciprocal Polynomial (on MathWorld)

Mathematics Encyclopedia