A basis for formal laurent series

Question

I was looking for a countable $A$-shauder's basis for the Laurent formal series in two variables $ \mathbb{C}[[t,s]][(ts)^{-1}]$. $A=\mathbb{C}[[t-s]]$. For example $\{t^ns^n, t^{n+1}s^n\}_{n \in \mathbb{Z}} $ but if I would see the expansion of a generic element in this basis there is a typical way? I've just tried with a combinatorial method but it's not satisfactory but sufficent

Answer 1

The natural Schauder basis of $$\mathbb{C}[[t,s]][(ts)^{-1}]$$ is $$\{s^i t^j\}_{i,j\in \Bbb{Z} }$$ And $$ \sum_{i,j\in \Bbb{Z}} c_{i,j}s^i t^j \in \mathbb{C}[[t,s]][(ts)^{-1}]$$ iff there is $M$ such that $c_{i,j}=0$ for $\min(i,j)<M$.