Reciprocal polynomial

In mathematics , a reciprocal polynomial is a polynomial whose coefficients are symmetric in a suitable sense:

A polynomial

${\displaystyle c_{n}X^{n}+c_{n-1}X^{n-1}+\ldots +c_{1}X+c_{0}}$

of degree is called reciprocal if for k = 0, ..., n holds (the sequence of coefficients is thus mirror-symmetric). ${\displaystyle n}$ ${\displaystyle c_{k}=c_{n-k}\,}$

This is the case if and only if

${\displaystyle p(X)=X^{n}\cdot p\!\left({\frac {1}{X}}\right)}$.

Sometimes the polynomial is called the reciprocal polynomial of . ${\displaystyle p^{*}(X)=X^{n}\cdot p\!\left({\frac {1}{X}}\right)}$ ${\displaystyle p(X)}$

In this case, polynomials that meet the symmetry condition are called self-reciprocal - this is the usual way of speaking in the English-language specialist literature.

Reciprocal polynomials are often used over finite fields .

Examples

1. The circle division polynomials are reciprocal.
2. Alexander polynomials of knots (see knot theory ) are reciprocal. For an Alexander polynomial of the form (after scaling with ) the substitution leads to the Conway polynomial . ${\displaystyle c_{0}+c_{1}t+\cdots +c_{0}t^{n}}$ ${\displaystyle t^{-n/2}}$ ${\displaystyle z^{2}=t+{\frac {1}{t}}-2}$

properties

For example, reciprocal polynomials have the following properties:

• If there is a zero of a reciprocal polynomial, then there is also a zero. ${\displaystyle x}$ ${\displaystyle 1/x}$
• It follows from this: if the degree of a reciprocal polynomial is odd, then there is a zero. Then by divisible. The quotient (see polynomial division ) is again a reciprocal polynomial. ${\displaystyle p}$ ${\displaystyle -1}$ ${\displaystyle p(X)}$ ${\displaystyle X+1}$
• If the degree of a reciprocal polynomial is even, it can be written as ${\displaystyle n=2k}$ ${\displaystyle p}$ ${\displaystyle p(X)}$

${\displaystyle p(X)=X^{k}q\!\left(X+{\frac {1}{X}}\right)}$

with a uniquely determined polynomial of degree . So the zeros of are exactly the solutions of ${\displaystyle q}$ ${\displaystyle k=n/2}$ ${\displaystyle p}$ ${\displaystyle x}$

${\displaystyle x+{\frac {1}{x}}=z}$

for the zeros of . ${\displaystyle z}$ ${\displaystyle q}$

variants

version 1

The symmetry condition can be modified as follows: Polynomials

${\displaystyle c_{n}X^{n}+c_{n-1}X^{n-1}+\ldots +c_{1}X+c_{0}}$

of the degree for which ${\displaystyle n}$

${\displaystyle c_{k}=-c_{n-k}\,}$ for k = 0, ..., n

holds, have similar properties as reciprocal polynomials:

• They are exactly the polynomials of degree that satisfy. ${\displaystyle p}$ ${\displaystyle n}$ ${\displaystyle p(X)=-X^{n}\cdot p\!\left({\frac {1}{X}}\right)}$
• If there is a zero, so too . Every such polynomial has the root 1. ${\displaystyle x}$ ${\displaystyle 1/x}$

Variant 2

We assume that the basic body used does not have characteristic 2.

One can use polynomials

${\displaystyle c_{n}X^{n}+c_{n-1}X^{n-1}+\ldots +c_{1}X+c_{0}}$

from the degree consider their coefficient ${\displaystyle n}$

${\displaystyle c_{k}=(-1)^{k}c_{n-k}\,}$ for k = 0, ..., n

fulfill. Non-trivial polynomials of this kind are only possible for even . ${\displaystyle n}$

They have the following properties:

• They are characterized by

${\displaystyle p(X)=X^{n}\cdot p\!\left(-{\frac {1}{X}}\right).}$

• If there is a zero, so too ${\displaystyle x}$ ${\displaystyle -1/x.}$
• If it is not divisible by 4, then and are zeros. Such a polynomial is divisible by; the quotient is again a polynomial of the same kind, the degree of which is divisible by 4. ${\displaystyle n}$ ${\displaystyle \mathrm {i} }$ ${\displaystyle -\mathrm {i} }$ ${\displaystyle X^{2}+1}$
• If it is divisible by 4, such a polynomial can be written as having a uniquely determined polynomial of degree . So the zeros of are the solutions of the equations for the zeros of . ${\displaystyle n=4k}$ ${\displaystyle p}$ ${\displaystyle p(X)=X^{2k}\cdot q\!\left(X-{\frac {1}{X}}\right)}$ ${\displaystyle q}$ ${\displaystyle 2k=n/2}$ ${\displaystyle p}$ ${\displaystyle x}$ ${\displaystyle x-{\frac {1}{x}}=z}$ ${\displaystyle z}$ ${\displaystyle q}$

literature

• Meyer's great arithmetic, Bibliographisches Institut, Mannheim, 1961.
• Helmut Meyn, Werner Götz: Self-reciprocal Polynomials Over Finite Fields , [1]

Individual evidence

1. ↑ For complex polynomials,, one usually uses a similar symmetry condition, namely for k = 0, ..., n (the coefficients are conjugated ). ${\displaystyle f\in \mathbb {C} [X]}$ ${\displaystyle c_{k}={\overline {c_{n-k}}}}$
2. ↑ See for example the article by Meyn / Götz.
3. ↑ One could sketch the proof here, or give a reference.