In algebra, given a polynomial with coefficients from an arbitrary field, its reciprocal polynomial or reflected polynomial, denoted by p∗ or pR , is the polynomial That is, the coefficients of p∗ are the coefficients of p 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 the conjugate reciprocal polynomial, denoted p† , is defined by, where denotes the complex conjugate of a_i, and is also called the reciprocal polynomial when no confusion can arise. A polynomial p is called self-reciprocal or palindromic if p(x) = p∗(x) . The coefficients of a self-reciprocal polynomial satisfy ai = an−i for all i .

Properties

Reciprocal polynomials have several connections with their original polynomials, including: deg p = deg p∗ if a_0 is not 0. p(x) = xnp∗(x−1) . α is a root of a polynomial p if and only if α−1 is a root of p∗ . 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:

Palindromic and antipalindromic polynomials

A self-reciprocal polynomial is also called palindromic because its coefficients, when the polynomial is written in the order of ascending or descending powers, form a palindrome. That is, if is a polynomial of degree n , then P is palindromic if ai = an−i for i = 0, 1, ..., n . Similarly, a polynomial P of degree n is called antipalindromic if ai = −an−i for i = 0, 1, ..., n . That is, a polynomial P is antipalindromic if P(x) = –P∗(x) .

Examples

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. Other examples of palindromic polynomials include cyclotomic polynomials and Eulerian polynomials.

Properties

a is a root of a polynomial that is either palindromic or antipalindromic, then 1⁄a is also a root and has the same multiplicity. a of polynomial, the value 1⁄a is also a root of the same multiplicity, then the polynomial is either palindromic or antipalindromic. q , the polynomial q + q∗ is palindromic and the polynomial q − q∗ is antipalindromic. q can be written as the sum of a palindromic and an antipalindromic polynomial, since q = (q + q∗)/2 + (q − q∗)/2 . x + 1 (it has –1 as a root) and its quotient by x + 1 is also palindromic. x – 1 (it has 1 as a root) and its quotient by x – 1 is palindromic. x2 – 1 (it has −1 and 1 as roots) and its quotient by x2 – 1 is palindromic. p(x) is a palindromic polynomial of even degree 2d, then there is a polynomial q of degree d such that p(x) = xdq(x + 1⁄x) . p(x) is a monic antipalindromic polynomial of even degree 2d over a field k of odd characteristic, then it can be written uniquely as p(x) = xd(Q(x) − Q(1⁄x)) , where Q is a monic polynomial of degree d with no constant term. P has even degree 2n over a field k of odd characteristic, then its "middle" coefficient (of power n ) is 0 since an = −a2n – n .

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 and self-inversive if for a scale factor ω on the unit circle. If p(z) is the minimal polynomial of z0 with = 1, 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 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. Per Cohn's theorem, a self-inversive polynomial has as many roots in the unit disk 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 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∗ generates C⊥ , the orthogonal complement of C . Also, C is self-orthogonal (that is, C ⊆ C⊥) , if and only if p∗ divides g(x) .