Is the degree function well behaved over power series? For non-zero formal polynomials $x$ and $y$ it holds that $\deg(xy)=\deg x + \deg y$. Allowing for infinite degrees, does this formula hold for arbitrary non-zero power series? And is there a definition that extends $\deg$ such that the formula holds for arbitrary non-zero Laurent series?

A not so satisfying answer: $-\deg$ is a valuation on the field $K(t)$ of rational functions in the variable $t$. $K(t)$ is a subfield of the field $K((t))$ of formal Laurent series. Hence the valuation $-\deg$ can be extended to a valuation of $K((t))$ (in fact there are infinitely many such extensions) - in particular the formula under discussion holds for that extension. However, as far as I know these extensions do not have an obvious interpretation.
