I have been looking at hyperelliptic curves over finite field, and there is something I don't figure out. In characteristic different from two, I think it's rather "ok". But in the lecture I am following, we consider some irreducible monic polynomial $\phi$ of odd degree, and the hyperelliptic curve given by : $$ Y^2+Y = \frac{1}{\phi(T)}$$ But, the definition I have for hyperelliptic curves in characteristic two is given by equations of the form : $$ y^2+H(x,z)y = F(x,z) $$ where $H, F$ are homogenous of degrees $\deg(H(x,z)) = g+1$ and $\deg(F(x,z)) = 2g+2$ (and then the genus of the curve is $g$), and $H$, $F$ squarefree. We can divide by $H(x,z)^2$ in order to obtain an equation of the form : $$y^2+y = \frac{F(x,z)}{H(x,z)} $$ So, if $z \neq 0$ and $u=\frac{y}{z^{g+1}}, v=\frac{x}{z}$, we have : $$ u^2 + u = \frac{F(v, 1)}{H(v,1)} = \frac{f(v)}{h(v)}$$
But It doesn't give me what I want. I mean, $\frac{f(v)}{h(v)}$ is not any rational functions, we have some hypothesis on the degrees, for example. How can I relate the equation of the hyperelliptic curves of the beginning with the definition I have of hyperelliptic curve ?
Thank you !