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 the conjugate reciprocal polynomial, denoted p†, is defined by, where
, and is also called the reciprocal polynomial when no confusion can arise.
The coefficients of a self-reciprocal polynomial satisfy ai = an−i for all i.
From the properties of the binomial coefficients, it follows that the polynomials P(x) = (x + 1)n are palindromic for all positive integers n, while the polynomials Q(x) = (x – 1)n are palindromic when n is even and antipalindromic when n is odd.
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.
for a scale factor ω on the unit circle.
[11] If p(z) is the minimal polynomial of z0 with |z0| = 1, z0 ≠ 1, and p(z) has real coefficients, then p(z) is self-reciprocal.
which has degree n. But, the minimal polynomial is unique, hence for some constant c, i.e.
A consequence is that the cyclotomic polynomials Φn are self-reciprocal for n > 1.
This is used in the special number field sieve to allow numbers of the form x11 ± 1, x13 ± 1, x15 ± 1 and x21 ± 1 to be factored taking advantage of the algebraic factors by using polynomials of degree 5, 6, 4 and 6 respectively – note that φ (Euler's totient function) of the exponents are 10, 12, 8 and 12.
[citation needed] Per Cohn's theorem, a self-inversive polynomial has as many roots in the unit disk
[12][13] The reciprocal polynomial finds a use in the theory of cyclic error correcting codes.
When g(x) generates a cyclic code C, then the reciprocal polynomial p∗ generates C⊥, the orthogonal complement of C.[14] Also, C is self-orthogonal (that is, C ⊆ C⊥), if and only if p∗ divides g(x).