I'm trying to figure out if there's an obvious reason why the following set of $\mathbb{C}$-subspaces of $\mathbb{C}(t)$ does not form a filtration. For a rational function $h(t) = \frac{f(t)}{g(t)}\in \mathbb{C}(t)$ where $f$ and $g$ are coprime, define the degree as $$\deg(h) = \deg_t(f)-\deg_t(g).$$ Then define $F_i = \{h\in \mathbb{C}(t)\mid \deg(h)\leq i\}$.
Obviously this is not an $\mathbb{N}$-filtration as by my definition functions can have negative degree. But I cannot see why this isn't a filtration - maybe the notion of degree is poorly defined in some sense?