and $$g_2 =  \prod_{P \in {{\mathscr{P}}}\setminus E} \delta_P, ~~ g = g_1g_2.$$ Then $f_2$ and $g_2$ are units at all $P \in C\cap E$. We claim that $f$ and $g$ have the required properties. Let $P \in {{\mathscr{P}}}$. Suppose $P \in C \cap E$. Then by the choices $f_2$ and $g_2$, they are units at $P$ and $m_P = (f_1, g_1)$. In particular $m_P  = (f, g)$ and $f, g$ define $C$ and $E$ respectively at $P$. Suppose that $P \not\in C$. Then $f_1$ and $g_2$ are units at $P$ and $f_2 = \pi_P u_P$ for some unit $u_P$ at $P$. Since $m_P = (\pi_P, g_1)$, we have $m_P = (f, g)$. Since $g_1$ defines $E$ at $P$ and $g_2$ is a unit at $P$, $g$ defines $E$ at $P$. Similarly if $P \not\in E$, then $m_P = (f, g)$ and $f$ defines $C$ at $P$.

\[theorem:bad\_char\] Let $K$ be a complete discretely valued field with residue field $\kappa$. Suppose that char$(K) = 0$ and char$(\kappa) = p > 0$ and $[\kappa : \kappa^p] = p^d$. Assume that $K$ contains a primitive $p^{th}$ root of unity. Then $p^{4d+4}$ is a $(2, p)$-uniform bound for $K(t)$.

Let $F$ be a finite 