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Consider the power series ring $\mathbf{Z}_p[[T]]$, where $\mathbf{Z}_p$ denotes the $p$-adic integers. I'll call a function $f(T) \in \mathbf{Z}_p[[T]]$ a rational function if I can write it as:

$$f(T) = \dfrac{g(T)}{h(T)} $$

where $g, h \in \mathbf{Z}_p[[T]]$ are polynomials. (Note that $g(T)/h(T)$ is shorthand for $g(T)$ times the multiplicative inverse of $h(T)$.)

My question is: is it possible to characterize which elements of $\mathbf{Z}_p[[T]]$ are rational functions? If I am given $f(T) \in \mathbf{Z}_p[[T]]$, and I write it as $f(T) = \sum a_nT^n$, can I tell whether $f(T)$ is a rational function just by looking at the coefficients $a_n$?

Adithya Chakravarthy
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    See https://mathoverflow.net/questions/249541/formal-power-series-is-taylor-expansion-of-rational-function-iff-hankel-determin . (This post deals with coefficients in a field, but the criteria will be the same for an integral domain.) – Ira Gessel May 12 '22 at 02:08
  • yes, it does. thank you! – Adithya Chakravarthy May 12 '22 at 03:47

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