Reciprocal polynomial
In algebra, the reciprocal polynomial, or reflected polynomial^{[1]}^{[2]} p^{∗} or p^{R},^{[2]}^{[1]} of a polynomial p of degree n with coefficients from an arbitrary field, such as
is the polynomial^{[3]}
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 selfreciprocal or palindromic if p(x) = p^{∗}(x). The coefficients of a selfreciprocal polynomial satisfy a_{i} = a_{n−i}. In the conjugate reciprocal case, the coefficients must be real to satisfy the condition.
Contents
Properties
Reciprocal polynomials have several connections with their original polynomials, including:
 p(x)=x^{n}p^{*}(x^{−1})^{[2]}
 α is a root of polynomial p if and only if α^{−1} is a root of p^{∗}.^{[4]}
 If p(x) ≠ x then p is irreducible if and only if p^{∗} is irreducible.^{[5]}
 p is primitive if and only if p^{∗} is primitive.^{[4]}
Other properties of reciprocal polynomials may be obtained, for instance:
 If a polynomial is selfreciprocal and irreducible then it must have even degree.^{[5]}
Palindromic and antipalindromic polynomials
A selfreciprocal 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 a_{i} = a_{n − i} 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 a_{i} = −a_{n − 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
 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.^{[6]}
 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.
 For any polynomial q, the polynomial q + q^{∗} is palindromic and the polynomial q − q^{∗} is antipalindromic.
 Any polynomial q can be written as the sum of a palindromic and an antipalindromic polynomial.^{[7]}
 The product of two palindromic or antipalindromic polynomials is palindromic.
 The product of a palindromic polynomial and an antipalindromic polynomial is antipalindromic.
 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.
 An antipalindromic polynomial is a multiple of x – 1 (it has 1 as a root) and its quotient by x – 1 is palindromic.
 An antipalindromic polynomial of even degree is a multiple of x^{2} – 1 (it has 1 and 1 as a roots) and its quotient by x^{2} – 1 is palindromic.
 If p(x) is a palindromic polynomial of even degree 2d, then there is a polynomial q of degree d such that p(x) = x^{d}q(x + 1/x) (Durand 1961).
 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) = x^{d} (Q(x) − Q(1/x)), where Q is a monic polynomial of degree d with no constant term.^{[8]}
 If an antipalindromic polynomial P has even degree 2n, then its "middle" coefficient (of power n) is 0 since a_{n} = −a_{2n – n}.
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.^{[9]}
Conjugate reciprocal polynomials
A polynomial is conjugate reciprocal if and selfinversive if for a scale factor ω on the unit circle.^{[10]}
If p(z) is the minimal polynomial of z_{0} with z_{0} = 1, z_{0} ≠ 1, and p(z) has real coefficients, then p(z) is selfreciprocal. This follows because
So z_{0} 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 Φ_{n} are selfreciprocal for n > 1. This is used in the special number field sieve to allow numbers of the form x^{11} ± 1, x^{13} ± 1, x^{15} ± 1 and x^{21} ± 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 x^{n} − 1 can be factored into the product of two polynomials, say x^{n} − 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.^{[11]} Also, C is selforthogonal (that is, C ⊆ C^{⊥}), if and only if p^{∗} divides g(x).^{[12]}
Notes
 ^ ^{a} ^{b} *Graham, Ronald; Knuth, Donald E.; Patashnik, Oren (1994). Concrete mathematics : a foundation for computer science. Reading, Mass: AddisonWesley. p. 339. ISBN 9780201558029.
 ^ ^{a} ^{b} ^{c} Aigner, Martin (2007). A course in enumeration. Berlin New York: Springer. p. 94. ISBN 9783540390329.
 ^ Roman 1995, pg.37
 ^ ^{a} ^{b} Pless 1990, pg. 57
 ^ ^{a} ^{b} Roman 1995, pg. 37
 ^ Pless 1990, pg. 57 for the palindromic case only
 ^ Stein, Jonathan Y. (2000), Digital Signal Processing: A Computer Science Perspective, Wiley Interscience, p. 384, ISBN 9780471295464
 ^ Katz, Nicholas M. (2012), Convolution and Equidistribution : SatoTate Theorems for Finite Field Mellin Transformations, Princeton University Press, p. 146, ISBN 9780691153315
 ^ Markovsky, Ivan; Rao, Shodhan (2008), "Palindromic polynomials, timereversible systems and conserved quantities", Control and Automation, doi:10.1109/MED.2008.4602018
 ^ Sinclair, Christopher D.; Vaaler, Jeffrey D. (2008). "Selfinversive 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 9780521714679. Zbl 1334.11017.
 ^ Pless 1990, pg. 75, Theorem 48
 ^ Pless 1990, pg. 77, Theorem 51
References
This article needs additional citations for verification. (June 2008) (Learn how and when to remove this template message)

 Pless, Vera (1990), Introduction to the Theory of Error Correcting Codes (2nd ed.), New York: WileyInterscience, ISBN 0471618845
 Roman, Steven (1995), Field Theory, New York: SpringerVerlag, ISBN 0387944087
 Émile Durand (1961) Solutions numériques des équations algrébriques I, Masson et Cie: XV  polynômes dont les coefficients sont symétriques ou antisymétriques, p. 140141.
External links
 "The Fundamental Theorem for Palindromic Polynomials". MathPages.com.
 Reciprocal Polynomial (on MathWorld)