Reciprocal polynomial
In algebra, the reciprocal polynomial p* of a polynomial p with coefficients from an arbitrary field, such as
is the polynomial[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,
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.
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 and self-inversive if
for a scale factor ω on the unit circle.[4]
If p(z) is the minimal polynomial of z0 with |z0| = 1, , and p(z) has real coefficients, then p(z) is self-reciprocal. This follows because
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.[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
- ↑ 2.0 2.1 Pless 1990, pg. 57
- ↑ 3.0 3.1 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