Reciprocal polynomial

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

A polynomial

Is the reciprocal of the degree if k = 0, ..., N is considered ( the sequence of the coefficients is so mirror-symmetrical ).

This is exactly the case when

Sometimes the polynomial is called the reciprocal polynomial of.

In this case we call polynomials which satisfy the symmetry condition, self - reciprocal - this is the usual way of speaking in the English literature.

Reciprocal polynomials are often used over finite fields.

Examples

Properties

Reciprocal polynomials for example, have the following properties:

• If a zero of a reciprocal polynomial, so is a zero.
• It follows that the degree of a reciprocal polynomial is odd, then is a root. Then is divisible by. The quotient (see polynomial ) is a reciprocal polynomial again.
• Is the degree of polynomial reciprocal straight, it can be written as

For the zeros of.

Variants

Variant 1

You can modify the symmetry condition as follows: Polynomials

The degree to which

Applies, have properties similar reciprocal polynomials:

• You are exactly the polynomials of degree that meet.
• Is a zero, so also. Each such polynomial has a zero of 1

Variant 2

We assume that the base used does not have characteristic 2.

You can polynomials

Consider the degree whose coefficients

. meet Non-trivial polynomials of this kind are possible only for straight.

You have the following properties:

• They are characterized by

• Is a zero, so also
• Is not divisible by 4, then and zeros. Such a polynomial is therefore divisible by; The quotient polynomial is again of the same type whose degree is divisible by 4.
• Is divisible by 4, so can be such a polynomial as writing with a uniquely determined polynomial of degree. The zeros of are thus the solutions of the equations for the zeros of.