Reciprocal polynomial
Template:Short description Template:More citations needed Template:MOS In algebra, given a polynomial
- <math>p(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n,</math>
with coefficients from an arbitrary field, its reciprocal polynomial or reflected polynomial,<ref name="concrete">*Template:Cite book</ref><ref name="Aigner">Template:Cite book</ref> denoted by Template:Math or Template:Math,<ref name="Aigner"/><ref name="concrete"/> is the polynomial<ref>Template:Harvnb</ref>
- <math>p^*(x) = a_n + a_{n-1}x + \cdots + a_0x^n = x^n p(x^{-1}).</math>
That is, the coefficients of Template:Math are the coefficients of Template:Math 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
- <math>p(z) = a_0 + a_1z + a_2z^2 + \cdots + a_nz^n,</math>
the conjugate reciprocal polynomial, denoted Template:Math, is defined by,
- <math>p^{\dagger}(z) = \overline{a_n} + \overline{a_{n-1}}z + \cdots + \overline{a_0}z^n = z^n\overline{p(\bar{z}^{-1})},</math>
where <math>\overline{a_i}</math> denotes the complex conjugate of <math>a_i</math>, and is also called the reciprocal polynomial when no confusion can arise.
A polynomial Template:Math is called self-reciprocal or palindromic if Template:Math. The coefficients of a self-reciprocal polynomial satisfy Template:Math for all Template:Math.
Properties
Reciprocal polynomials have several connections with their original polynomials, including:
- Template:Math
- Template:Math.<ref name="Aigner"/>
- For Template:Math is a root of a polynomial Template:Math if and only if Template:Math is a root of Template:Math or if <math> \alpha = 0 </math> and <math> p ^* </math> is of lower degree than <math> p </math>.<ref name="Pless 1990 loc=pg. 57">Template:Harvnb</ref>
- If Template:Math then Template:Math is irreducible if and only if Template:Math is irreducible.<ref name="Roman 1995 loc= pg. 37">Template:Harvnb</ref>
- Template:Math is primitive if and only if Template:Math is primitive.<ref name="Pless 1990 loc=pg. 57"/>
Other properties of reciprocal polynomials may be obtained, for instance:
- A self-reciprocal polynomial of odd degree is divisible by x+1, hence is not irreducible if its degree is > 1.
Template:Anchor 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
- <math> P(x) = \sum_{i=0}^n a_ix^i</math>
is a polynomial of degree Template:Math, then Template:Math is palindromic if Template:Math for Template:Math.
Similarly, a polynomial Template:Math of degree Template:Math is called antipalindromic if Template:Math for Template:Math. That is, a polynomial Template:Math is antipalindromic if Template:Math.
Examples
From the properties of the binomial coefficients, it follows that the polynomials Template:Math are palindromic for all positive integers Template:Math, while the polynomials Template:Math are palindromic when Template:Math is even and antipalindromic when Template:Math is odd.
Other examples of palindromic polynomials include cyclotomic polynomials and Eulerian polynomials.
Properties
- If Template:Math is a root of a polynomial that is either palindromic or antipalindromic, then Template:Sfrac is also a root and has the same multiplicity.<ref>Template:Harvnb for the palindromic case only</ref>
- The converse is true: If for each root Template:Math of polynomial, the value Template:Sfrac is also a root of the same multiplicity, then the polynomial is either palindromic or antipalindromic.
- For any polynomial Template:Math, the polynomial Template:Math is palindromic and the polynomial Template:Math is antipalindromic.
- It follows that any polynomial Template:Math can be written as the sum of a palindromic and an antipalindromic polynomial, since Template:Math.<ref>Template:Citation</ref>
- 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 Template:Math (it has –1 as a root) and its quotient by Template:Math is also palindromic.
- An antipalindromic polynomial over a field Template:Mvar with odd characteristic is a multiple of Template:Math (it has 1 as a root) and its quotient by Template:Math is palindromic.
- An antipalindromic polynomial of even degree is a multiple of Template:Math (it has −1 and 1 as roots) and its quotient by Template:Math is palindromic.
- If Template:Math is a palindromic polynomial of even degree 2Template:Mvar, then there is a polynomial Template:Math of degree Template:Math such that Template:Math.<ref>Template:Harvnb</ref>
- If Template:Math is a monic antipalindromic polynomial of even degree 2Template:Mvar over a field Template:Mvar of odd characteristic, then it can be written uniquely as Template:Math, where Template:Mvar is a monic polynomial of degree Template:Mvar with no constant term.<ref>Template:Citation</ref>
- If an antipalindromic polynomial Template:Math has even degree Template:Math over a field Template:Mvar of odd characteristic, then its "middle" coefficient (of power Template:Math) is 0 since Template:Math.
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.<ref>Template:Cite book</ref>
Conjugate reciprocal polynomialsTemplate:Anchor
A polynomial is conjugate reciprocal if <math>p(x) \equiv p^{\dagger}(x)</math> and self-inversive if <math>p(x) = \omega p^{\dagger}(x)</math> for a scale factor Template:Math on the unit circle.<ref name=SV08>Template:Cite book</ref>
If Template:Math is the minimal polynomial of Template:Math with Template:Math, and Template:Math has real coefficients, then Template:Math is self-reciprocal. This follows because
- <math>z_0^n\overline{p(1/\bar{z_0})} = z_0^n\overline{p(z_0)} = z_0^n\bar{0} = 0.</math>
So Template:Math is a root of the polynomial <math>z^n\overline{p(\bar{z}^{-1})}</math> which has degree Template:Math. But, the minimal polynomial is unique, hence
- <math>cp(z) = z^n\overline{p(\bar{z}^{-1})}</math>
for some constant Template:Math, i.e. <math>ca_i=\overline{a_{n-i}}=a_{n-i}</math>. Sum from Template:Math to Template:Math and note that 1 is not a root of Template:Math. We conclude that Template:Math.
A consequence is that the cyclotomic polynomials Template:Math are self-reciprocal for Template:Math. This is used in the special number field sieve to allow numbers of the form Template:Math and Template:Math to be factored taking advantage of the algebraic factors by using polynomials of degree 5, 6, 4 and 6 respectively – note that Template:Math (Euler's totient function) of the exponents are 10, 12, 8 and 12.Template:Citation needed
Per Cohn's theorem, a self-inversive polynomial has as many roots in the unit disk <math>\{z\in\mathbb{C}: |z| < 1\}</math> as the reciprocal polynomial of its derivative.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>
Application in coding theory
The reciprocal polynomial finds a use in the theory of cyclic error correcting codes. Suppose Template:Math can be factored into the product of two polynomials, say Template:Math. When Template:Math generates a cyclic code Template:Math, then the reciprocal polynomial Template:Math generates Template:Math, the orthogonal complement of Template:Math.<ref>Template:Harvnb</ref> Also, Template:Math is self-orthogonal (that is, Template:Math, if and only if Template:Math divides Template:Math.<ref>Template:Harvnb</ref>