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Let $K$ be a field. For $m>1$, elements in the ring of power series in $m$ variables over $K$, i.e., $K[[X_1, \cdots, X_m]]$ look like - $$ \displaystyle\sum_{(i_1, \cdots, i_m)\in (\mathbb N_{0})^m} a_{i_1, \cdots, i_m} X_1^{i_1} \cdots X_m^{i_m}$$ where $\mathbb N_0$ is the set of non-negative integers.

Is there an explicit expression for elements in its quotient field, i.e., the field of meromorphic series in $m$ variables?

Using the geometric series, we know that for $m=1$, the elements look like $\displaystyle\sum_{i \geq r} a_i X^i$ for some integer $r$.

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I don't have a complete answer, but one typically does not have a unique series expression without imposing additional constraints. For example one may embed $K((X,Y))$ in either $K((X))((Y))$ or $K((Y))((X))$ by expanding quotients with one term order or another. This yields different series expressions of $1/(X-Y)$. – S. Carnahan Feb 28 '12 at 5:54
Yes, you can assume an embedding. For every permutation on $m$ symbols, you have an ordering of $\mathbb Z^m$ and you may assume that $K((X,Y))$ is contained in the field of functions from $\mathbb Z^m$ to $K$ with well-ordered support under this ordering. – Abhishek Parab Feb 28 '12 at 16:59

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