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.
the conjugate reciprocal polynomial, p† given by,
where denotes the complex conjugate of , is also called the reciprocal polynomial when no confusion can arise.
A polynomial p is called self-reciprocal if .
The coefficients of a self-reciprocal polynomial satisfy ai = an−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.
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*.
- 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:
- If a polynomial is self-reciprocal and irreducible then it must have even degree.
Conjugate reciprocal polynomials
So z0 is a root of the polynomial which has degree n. But, the minimal polynomial is unique, hence
for some constant c, i.e. . 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 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 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. Also, C is self-orthogonal (that is, C ⊆ C⊥), if and only if p*(x) divides g(x).
- 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
|This article needs additional citations for verification. (June 2008)|
- Pless, Vera (1990), Introduction to the Theory of Error Correcting Codes (2nd ed.), New York: Wiley-Interscience, ISBN 0-471-61884-5