Does every power series have an inverse?

Question

I'm not even sure if every polynomial has an inverse, and what restrictions are required on them to have inverses. For example, the people in this question seem to suggest that the polynomial needs to be increasing to have an inverse.

My main question is, does every power series have an inverse?

Clearly not if it isn't bijective. And see Lagrange Inversion Theorem. — Simply Beautiful Art, Nov 20 '16 at 16:51
So as long as my function is analytic at some point, and also non-zero at that point, then I can find an inverse in a neighbourhood of the function at that point? — gypsydealer, Nov 20 '16 at 16:54
Yup. Also, under the examples of the Wikipedia you may find the expansion to the inverse of a polynomial degree 5. — Simply Beautiful Art, Nov 20 '16 at 17:03

score 1 · Answer 1 · answered Nov 20 '16 at 16:52

1

This link on the inversion of power series may help you: http://www.ams.org/mcom/1947-02-020/S0025-5718-1947-0022717-X/S0025-5718-1947-0022717-X.pdf

answered Nov 20 '16 at 16:52

score 1 · Answer 2 · answered Nov 20 '16 at 16:55

1

Every polynomial is a power series, just with infinitely many zero coefficients.

If you believe that not every polynomial has an inverse, then you already know that not every power series has an inverse, because the polynomials which lack inverses are already counterexamples.

answered Nov 20 '16 at 16:55

Chill2Macht

20,920

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What you write is false: the polynomial $1-x \in \mathbb Q[x]$ has no inverse in $\mathbb Q[x]$, but has an inverse seen as a power series, namely $1+x+x^2+...+x^n+...\in \mathbb Q[[x]]$. – Georges Elencwajg May 16 '20 at 11:07

score 1 · Accepted Answer · answered Nov 20 '16 at 17:34

According to Lagrange Inversion Theorem, the inverse of an analytic function near $x=f(a)$ is given by

$$f^{-1}(x)=a+\sum_{n=1}^\infty\left\lbrace\lim_{w\to a}\left[{(x-f(a))^n\over n!}{d^{n-1}\over dw^{n-1}}\left({w-a\over f(w)-f(a)}\right)^n\right]\right\rbrace$$

Does every power series have an inverse?

3 Answers3