Let $f(z,w)$ be a holomorphic function on the polydisc $D\times D\subset \mathbb{C}^2$. Suppose that the zeroes of $f$ do not accumulate on $D\times\partial D$. Is it true that $$ \text{max}_{z\in D}\# \{w\in D\mid f(z,w)=0\}<\infty $$
By the identity principle, the quantity $ \#\{w\in D\mid f(z,w)=0\}$ is finite for every $z\in D$. Moreover, by Rouché's theorem it is locally constant in $z$. However, because $D$ is not compact, I do not see how to conclude.