Polynomial on each variable

Problem: Suppose {f\in \mathcal{O}(\mathbb{C}^2)}. If {f(z_1,z_2)} is a polynomial of {z_1} for every fixed {z_2} and a polynomial of {z_2} for every fixed {z_1}. Then {f} must be a polynomial.

Proof: Since {f} is holomorphic on the {\mathbb{C}^2}, we have the power series expansion

\displaystyle f(z_1,z_2)=\sum_{i,j\geq 0}a_{ij}z_1^iz_2^j=\sum_{i\geq 0}a_i(z_1)z_2^i

where {a_i(z_1)} is a holomorphic function on {\mathbb{C}} for any {i\geq 0}. Suppose there exists a sequence {i_1,i_2,\cdots} with {\lim_{k\rightarrow \infty}i_k=\infty} satisfying {a_{i_k}(z_1)\neq 0}. Since the set of roots of {a_{i_k}} are countable in {\mathbb{C}}, then all the union of these sets are also countable. Therefore, {\exists\, w\in\mathbb{C}} such that {a_{i_k}(w)\neq 0} for all {k\geq 0}. Then {f(w,z_2)} can never be a polynomial. So

\displaystyle f(z_1,z_2)=\sum_{i=0}^Ma_i(z_1)z_2^i=\sum_{i\geq 0}b_i(z_2)z_1^i

where every {b_i} is polynomial of {z_2}. Repeating the above proof we get this is also a finite sum, which means {f} is a polynomial. \Box

Remark: Exercise from Shabat, 1.10. Idea comes from Hanlong Fang.

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