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Reciprocal polynomial
Polynomial obtained from another polynomial by reversing its coefficients

In algebra, given a polynomial

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

with coefficients from an arbitrary field, its reciprocal polynomial or reflected polynomial, denoted by p∗ or pR, is the polynomial

p ∗ ( x ) = a n + a n − 1 x + ⋯ + a 0 x n = x n p ( x − 1 ) . {\displaystyle p^{*}(x)=a_{n}+a_{n-1}x+\cdots +a_{0}x^{n}=x^{n}p(x^{-1}).}

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

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

the conjugate reciprocal polynomial, denoted p†, is defined by,

p † ( z ) = a n ¯ + a n − 1 ¯ z + ⋯ + a 0 ¯ z n = z n p ( z ¯ − 1 ) ¯ , {\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 a i ¯ {\displaystyle {\overline {a_{i}}}} denotes the complex conjugate of a i {\displaystyle 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.

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Properties

Reciprocal polynomials have several connections with their original polynomials, including:

  1. deg p = deg p∗ if a 0 {\displaystyle a_{0}} is not 0.
  2. p(x) = xnp∗(x−1).6
  3. α is a root of a polynomial p if and only if α−1 is a root of p∗.7
  4. If x ∤ p(x) then p is irreducible if and only if p∗ is irreducible.8
  5. p is primitive if and only if p∗ is primitive.9

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.

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

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

  • 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.10
  • The converse is true: If for each root a of polynomial, the value ⁠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.
  • It follows that any polynomial q can be written as the sum of a palindromic and an antipalindromic polynomial, since q = (q + q∗)/2 + (q − q∗)/2.11
  • 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 over a field k with odd characteristic 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 x2 – 1 (it has −1 and 1 as roots) and its quotient by x2 – 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) = xdq(x + ⁠1/x⁠).12
  • If 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.13
  • If an antipalindromic polynomial 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.14

Conjugate reciprocal polynomials

A polynomial is conjugate reciprocal if p ( x ) ≡ p † ( x ) {\displaystyle p(x)\equiv p^{\dagger }(x)} and self-inversive if p ( x ) = ω p † ( x ) {\displaystyle p(x)=\omega p^{\dagger }(x)} for a scale factor ω on the unit circle.15

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

z 0 n p ( 1 / z 0 ¯ ) ¯ = z 0 n p ( z 0 ) ¯ = z 0 n 0 ¯ = 0. {\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 z n p ( z ¯ − 1 ) ¯ {\displaystyle z^{n}{\overline {p({\bar {z}}^{-1})}}} which has degree n. But, the minimal polynomial is unique, hence

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

for some constant c, i.e. c a i = a n − i ¯ = a n − i {\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.

Per Cohn's theorem, a self-inversive polynomial has as many roots in the unit disk { z ∈ C : | z | < 1 } {\displaystyle \{z\in \mathbb {C} :|z|<1\}} as the reciprocal polynomial of its derivative.1617

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.18 Also, C is self-orthogonal (that is, C ⊆ C⊥), if and only if p∗ divides g(x).19

See also

Notes

  • 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
  • Durand, Émile (1961), "Masson et Cie: XV - polynômes dont les coefficients sont symétriques ou antisymétriques", Solutions numériques des équations algrébriques, vol. I, pp. 140–141

References

  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. 978-0201558029 ↩

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

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

  4. *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. 978-0201558029 ↩

  5. Roman 1995, pg.37 - Roman, Steven (1995), Field Theory, New York: Springer-Verlag, ISBN 0-387-94408-7 ↩

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

  7. Pless 1990, pg. 57 - Pless, Vera (1990), Introduction to the Theory of Error Correcting Codes (2nd ed.), New York: Wiley-Interscience, ISBN 0-471-61884-5 ↩

  8. Roman 1995, pg. 37 - Roman, Steven (1995), Field Theory, New York: Springer-Verlag, ISBN 0-387-94408-7 ↩

  9. Pless 1990, pg. 57 - Pless, Vera (1990), Introduction to the Theory of Error Correcting Codes (2nd ed.), New York: Wiley-Interscience, ISBN 0-471-61884-5 ↩

  10. Pless 1990, pg. 57 for the palindromic case only - Pless, Vera (1990), Introduction to the Theory of Error Correcting Codes (2nd ed.), New York: Wiley-Interscience, ISBN 0-471-61884-5 ↩

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

  12. Durand 1961 - Durand, Émile (1961), "Masson et Cie: XV - polynômes dont les coefficients sont symétriques ou antisymétriques", Solutions numériques des équations algrébriques, vol. I, pp. 140–141 ↩

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

  14. Markovsky, Ivan; Rao, Shodhan (2008). "Palindromic polynomials, time-reversible systems, and conserved quantities". 2008 16th Mediterranean Conference on Control and Automation (PDF). IEEE. pp. 125–130. doi:10.1109/MED.2008.4602018. ISBN 978-1-4244-2504-4. S2CID 14122451. 978-1-4244-2504-4 ↩

  15. 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. Vol. 352. Cambridge: Cambridge University Press. pp. 312–321. ISBN 978-0-521-71467-9. Zbl 1334.11017. 978-0-521-71467-9 ↩

  16. Ancochea, Germán (1953). "Zeros of self-inversive polynomials". Proceedings of the American Mathematical Society. 4 (6): 900–902. doi:10.1090/S0002-9939-1953-0058748-8. ISSN 0002-9939. https://www.ams.org/proc/1953-004-06/S0002-9939-1953-0058748-8/ ↩

  17. Bonsall, F. F.; Marden, Morris (1952). "Zeros of self-inversive polynomials". Proceedings of the American Mathematical Society. 3 (3): 471–475. doi:10.1090/S0002-9939-1952-0047828-8. ISSN 0002-9939. https://www.ams.org/proc/1952-003-03/S0002-9939-1952-0047828-8/ ↩

  18. Pless 1990, pg. 75, Theorem 48 - Pless, Vera (1990), Introduction to the Theory of Error Correcting Codes (2nd ed.), New York: Wiley-Interscience, ISBN 0-471-61884-5 ↩

  19. Pless 1990, pg. 77, Theorem 51 - Pless, Vera (1990), Introduction to the Theory of Error Correcting Codes (2nd ed.), New York: Wiley-Interscience, ISBN 0-471-61884-5 ↩