Entire functions with infinitely many zeros

Question

If a non zero entire function has infinitely many zeros, does that always mean that it is transcendental? Converse is not true and we can see that $e^z$ is the counter example.

Answer 1

Assume $P(f(z),z)\equiv 0$ for some non-zero polynomial $P(x,y)=\sum_{i,j}a_{i,j}x^iy^j$. Pick such $P$ of minimal degree. There are infinitely many $z_k$ where $f(z_k)=0$ and hence $P(0,z_k)=0$. It follows that $P(0,z)$ viewed as a polynomial of $z$ alone is the zero polynomial, i.e., $a_{0,j}=0$ for all $j$. Then $P(x,y)=xQ(x,y)$ with $Q$ of lesser degree. For all $z$ with $f(z)\ne 0$, we have $Q(f(z),z)=\frac1{f(z)}P(f(z),z)=0$, and by continuity (the roots of $f$ are isolated!) $Q(f(z),z)\equiv 0$, contradicting minimality of $\deg P$.