# Reciprocal polynomial

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

${\displaystyle p(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n},\,\!}$

is the polynomial[1]

${\displaystyle p^{*}(x)=a_{n}+a_{n-1}x+\cdots +a_{0}x^{n}=x^{n}p(x^{-1}).}$

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,

${\displaystyle p(z)=a_{0}+a_{1}z+a_{2}z^{2}+\cdots +a_{n}z^{n},\,\!}$

the conjugate reciprocal polynomial, p given by,

${\displaystyle p^{\dagger }(z)={\overline {a_{n}}}+{\overline {a_{n-1}}}z+\cdots +{\overline {a_{0}}}z^{n}=z^{n}{\overline {p({\bar {z}}^{-1})}},}$

where ${\displaystyle {\overline {a_{i}}}}$ denotes the complex conjugate of ${\displaystyle a_{i}\,\!}$, is also called the reciprocal polynomial when no confusion can arise.

A polynomial p is called self-reciprocal if p(x) = p(x).

The coefficients of a self-reciprocal polynomial satisfy ai = ani, 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:

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

## Palindromic and antipalindromic polynomials

A self-reciprocal polynomial is called palindromic because its coefficients, when the polynomial is written in ascending or descending order, form a palindrome. That is, if

${\displaystyle P(x)=\sum _{i=0}^{n}a_{i}x^{i}}$

is a polynomial of degree n, then P is palindromic if ai = ani for i = 0, 1, ..., n. Some authors use the terms "palindromic" and "reciprocal" interchangeably.

Similarly, P, a polynomial of degree n, is called antipalindromic if ai = −ani for i = 0, 1, ... n. That is, a polynomial P is antipalindromic if P(x) = – P(x).

Due to the properties of the binomial coefficients 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. Also, cyclotomic polynomials are palindromic.

### General properties

1. If 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.[4]
2. The converse is true: If a polynomial is such that if a is a root then 1/a is also a root of the same multiplicity, then the polynomial is either palindromic or antipalindromic.
3. For any polynomial q, the polynomial q + q is palindromic and the polynomial qq is antipalindromic.
4. Any polynomial q can be written as the sum of a palindromic and an antipalindromic polynomial.[5]
5. The product of two palindromic or antipalindromic polynomials is palindromic.
6. The product of a palindromic polynomial and an antipalindromic polynomial is antipalindromic.
7. A palindromic polynomial of odd degree is a multiple of x + 1 (it has -1 as a root) and its quotient by x + 1 is also palindromic.
8. An antipalindromic polynomial is a multiple of x – 1 (it has 1 as a root) and its quotient by x – 1 is palindromic.
9. An antipalindromic polynomial of even degree is a multiple of x2 – 1 (it has -1 and 1 as a roots) and its quotient by x2 – 1 is palindromic.
10. If 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).
11. If p(x) is a monic antipalindromic polynomial of even degree 2d over a field k with 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.[6]

It follows from the definition that if P is of even degree n (so it has an odd number of terms), then it can only be antipalindromic when the "middle" term is 0, i.e., am = −an – m, where n = 2m.

### Real coefficients

A polynomial with real coefficients all of whose complex roots lie on the unit circle in the complex plane (all the roots are unimodular) is either palindromic or antipalindromic.[7]

## Conjugate reciprocal polynomials

A polynomial is conjugate reciprocal if ${\displaystyle p(x)\equiv p^{\dagger }(x)}$ and self-inversive if ${\displaystyle p(x)=\omega p^{\dagger }(x)}$ for a scale factor ω on the unit circle.[8]

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. This follows because

${\displaystyle z_{0}^{n}{\overline {p(1/{\bar {z_{0}}})}}=z_{0}^{n}{\overline {p(z_{0})}}=z_{0}^{n}{\bar {0}}=0.}$

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

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

for some constant c, i.e. ${\displaystyle ca_{i}={\overline {a_{n-i}}}=a_{n-i}}$. 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 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.

## 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.[9] Also, C is self-orthogonal (that is, CC), if and only if p divides g(x).[10]

## Notes

1. ^ Roman 1995, pg.37
2. ^ a b Pless 1990, pg. 57
3. ^ a b Roman 1995, pg. 37
4. ^ Pless 1990, pg. 57 for the palindromic case only
5. ^ Stein, Jonathan Y. (2000), Digital Signal Processing: A Computer Science Perspective, Wiley Interscience, p. 384, ISBN 9780471295464
6. ^ Katz, Nicholas M. (2012), Convolution and Equidistribution : Sato-Tate Theorems for Finite Field Mellin Transformations, Princeton University Press, p. 146, ISBN 9780691153315
7. ^ Markovsky, Ivan; Rao, Shodhan (2008), "Palindromic polynomials, time-reversible systems and conserved quantities", Control and Automation, doi:10.1109/MED.2008.4602018
8. ^ 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.
9. ^ Pless 1990, pg. 75, Theorem 48
10. ^ 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