Reciprocal polynomial
In mathematics , a reciprocal polynomial is a polynomial whose coefficients are symmetric in a suitable sense:
A polynomial
of degree is called reciprocal if for k = 0, ..., n holds (the sequence of coefficients is thus mirror-symmetric).
This is the case if and only if
- .
Sometimes the polynomial is called the reciprocal polynomial of .
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
- The circle division polynomials are reciprocal.
- 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 .
properties
For example, reciprocal polynomials have the following properties:
- If there is a zero of a reciprocal polynomial, then there is also a zero.
- 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.
- If the degree of a reciprocal polynomial is even, it can be written as
- with a uniquely determined polynomial of degree . So the zeros of are exactly the solutions of
for the zeros of .
variants
version 1
The symmetry condition can be modified as follows: Polynomials
of the degree for which
- for k = 0, ..., n
holds, have similar properties as reciprocal polynomials:
- They are exactly the polynomials of degree that satisfy.
- If there is a zero, so too . Every such polynomial has the root 1.
Variant 2
We assume that the basic body used does not have characteristic 2.
One can use polynomials
from the degree consider their coefficient
- for k = 0, ..., n
fulfill. Non-trivial polynomials of this kind are only possible for even .
They have the following properties:
- They are characterized by
- If there is a zero, so too
- 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.
- 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 .
literature
- Meyer's great arithmetic, Bibliographisches Institut, Mannheim, 1961.
- Helmut Meyn, Werner Götz: Self-reciprocal Polynomials Over Finite Fields , [1]
Individual evidence
- ↑ For complex polynomials,, one usually uses a similar symmetry condition, namely for k = 0, ..., n (the coefficients are conjugated ).
- ↑ See for example the article by Meyn / Götz.
- ↑ One could sketch the proof here, or give a reference.