I was trying to convert a rational function RR(X) to a polynomial of type RR|X,X^{-1}| b

Jupellodseple804

Jupellodseple804

Answered question

2022-02-18

I was trying to convert a rational function R(X) to a polynomial of type R|X,X1| but I failed. I searched internet and R|X,X1| has its own name!: "Laurent polynomials".
a. Every Laurent polynomial is a combination of rational functions. Is every Laurent polynomial possible to be equal to a single rational function?
b. Are there methods to convert rational functions to Laurent polynomials? How are they?

Answer & Explanation

jorgegalar0xk

jorgegalar0xk

Beginner2022-02-19Added 5 answers

a. Yes. A Laurent polynomial is an expression of the type
anXn+an+1Xn+1++an+kXn+k,   (1)
with nZ and kN. This is a polynomial (and therefore a rational function) is n0. Otherwise
(1)=an+an+1Xn++an+kXkXn
which is a rational function.
2. No. For instance, 11+X is a rational function, but you can't express it as an element of R[X,X1].
Vikki Chapman

Vikki Chapman

Beginner2022-02-20Added 8 answers

A rational function f(X) can be a Laurent polynomial if and only if Xmf(x) is an ordinary polynomial for some positive integer m.
Clearly one can find rational functions failing this condition.

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