A set $M$ is called a set of uniqueness for functions of a class $\mathcal{F}$ if any function $f \in \mathcal{F}$, equal to $0$ on $M$, is identically equal to $0$. Prove that the following sets are sets of uniqueness for functions holomorphic on $\mathbb{C}^2$:
(a) a real hyperplane in $\mathbb{C}^2$;
(b) the real two-dimensional plane $\{z_{1} = \bar{z_2} \}$;
(c) the arc $\{z_2 = \bar{z_1}, y_1 = x_1 \text{ sin }(1/x_1)\}$
Any help with these would be great. I was wondering specifically if there is a systematic way to address each of these proofs. If so, some help with part (a) would be great, and I can attempt (b) and (c). Thanks!
Sets of uniqueness for (subclasses of) holomorphic functions in several variables is fairly tricky, and not completely understood. For example there is no known simple, general characterisation of sets of uniqueness for bounded holomorphic functions in several variables.
For your specific question, for a) you can assume that the hyperplane is $\operatorname{Im} z_2 = 0$. Assume that $f = 0$ on the hyperplane and look at functions $$ g(z) = f(c,z) $$ for a fixed $c$. Then $g$ is entire and vanishes for $\operatorname{Im} z = 0$, hence everywhere. Can you take it from there?