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

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

is the polynomial[3]

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,

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

the conjugate reciprocal polynomial, $p †$ given by,

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 ${\overline {a_{i}}}$ denotes the complex conjugate of $a_{i}\,\!$ , 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 $a i = a n - i$ . 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:

$p (x) = x n p * (x -1)$ [2]
$α$ is a root of polynomial $p$ if and only if $α -1$ is a root of $p *$ .[4]
If $p (x) \neq x$ 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 self-reciprocal and irreducible then it must have even degree.[5]

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

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

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 $2 n$ , 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 $p(x)\equiv p^{\dagger }(x)$ and self-inversive if $p(x)=\omega p^{\dagger }(x)$ for a scale factor $ω$ on the unit circle.[10]

If $p (z)$ is the minimal polynomial of $z 0$ with $| z 0 | = 1, z 0 \neq 1$ , and $p (z)$ has real coefficients, then $p (z)$ is self-reciprocal. This follows because

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

So $z 0$ is a root of the polynomial $z^{n}{\overline {p({\bar {z}}^{-1})}}$ which has degree $n$ . But, the minimal polynomial is unique, hence

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

for some constant $c$ , i.e. $ca_{i}={\overline {a_{n-i}}}=a_{n-i}$ . 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 $Φ n$ are self-reciprocal for $n > 1$ . This is used in the special number field sieve to allow numbers of the form $x 11 \pm 1, x 13 \pm 1, x 15 \pm 1$ and $x 21 \pm 1$ 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 self-orthogonal (that is, $C \subseteq C ⊥)$ , if and only if $p *$ divides $g (x)$ .[12]

Notes

- Graham, Ronald; Knuth, Donald E.; Patashnik, Oren (1994). Concrete mathematics : a foundation for computer science (Second ed.). Reading, Mass: Addison-Wesley. p. 340. ISBN 978-0201558029.
Aigner, Martin (2007). A course in enumeration. Berlin New York: Springer. p. 94. ISBN 978-3540390329.
Roman 1995, pg.37
Pless 1990, pg. 57
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 : Sato-Tate Theorems for Finite Field Mellin Transformations, Princeton University Press, p. 146, ISBN 9780691153315
Markovsky, Ivan; Rao, Shodhan (2008), "Palindromic polynomials, time-reversible systems and conserved quantities" (PDF), Control and Automation: 125–130, doi:10.1109/MED.2008.4602018, ISBN 978-1-4244-2504-4
Sinclair, Christopher D.; Vaaler, Jeffrey D. (2008). "Self-inversive polynomials with all zeros on the unit circle". In McKee, James; Smyth, C. J. (eds.). 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 1334.11017.
Pless 1990, pg. 75, Theorem 48
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
Émile Durand (1961) Solutions numériques des équations algrébriques I, Masson et Cie: XV - polynômes dont les coefficients sont symétriques ou antisymétriques, p. 140-141.

External links

"The Fundamental Theorem for Palindromic Polynomials". MathPages.com.
Reciprocal Polynomial (on MathWorld)

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[concrete-1] Graham, Ronald; Knuth, Donald E.; Patashnik, Oren (1994). Concrete mathematics : a foundation for computer science (Second ed.). Reading, Mass: Addison-Wesley. p. 340. ISBN 978-0201558029.

[2] Graham, Ronald; Knuth, Donald E.; Patashnik, Oren (1994). Concrete mathematics : a foundation for computer science (Second ed.). Reading, Mass: Addison-Wesley. p. 340. ISBN 978-0201558029.

[Aigner-2] Aigner, Martin (2007). A course in enumeration. Berlin New York: Springer. p. 94. ISBN 978-3540390329.

[3] Roman 1995, pg.37

[Pless_1990_loc=pg._57-4] Pless 1990, pg. 57

[Roman_1995_loc=_pg._37-5] Roman 1995, pg. 37

[6] Pless 1990, pg. 57 for the palindromic case only

[7] Stein, Jonathan Y. (2000), Digital Signal Processing: A Computer Science Perspective, Wiley Interscience, p. 384, ISBN 9780471295464

[8] Katz, Nicholas M. (2012), Convolution and Equidistribution : Sato-Tate Theorems for Finite Field Mellin Transformations, Princeton University Press, p. 146, ISBN 9780691153315

[9] Markovsky, Ivan; Rao, Shodhan (2008), "Palindromic polynomials, time-reversible systems and conserved quantities" (PDF), Control and Automation: 125–130, doi:10.1109/MED.2008.4602018, ISBN 978-1-4244-2504-4

[SV08-10] Sinclair, Christopher D.; Vaaler, Jeffrey D. (2008). "Self-inversive polynomials with all zeros on the unit circle". In McKee, James; Smyth, C. J. (eds.). 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 1334.11017.

[11] Pless 1990, pg. 75, Theorem 48

[12] Pless 1990, pg. 77, Theorem 51

Reciprocal polynomial

Properties

Palindromic and antipalindromic polynomials

Examples

Properties

Real coefficients

Conjugate reciprocal polynomials

Application in coding theory

See also

Notes

References

External links