# Sets of uniqueness in $\mathbb{C}^2$

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!

• Isn't sufficient to check that these sets are non-discrete? Holomorphic non-constant maps should have discrete sets of zeroes (if I am not wrong this is true in several complex variables as in one variable). – Crostul Mar 20 '16 at 11:16
• Not in SCV. The zero set of a holomorphic function on a domain $D \subset \mathbb{C}^n$, $n \geq 2$ contains no isolated points. – K.Reeves Mar 20 '16 at 11:31
• @Crostul Note $f(z,w) = z$ is zero on $\{0\}\times \mathbb C.$ – zhw. Mar 20 '16 at 23:08

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?
• Not really...I don't really see how $g = 0$ in this hyperplane helps. – K.Reeves Mar 20 '16 at 23:53
• For each $c$, $g(z) = 0$ for all $z$. But this implies that $f=0$ everywhere. – mrf Mar 21 '16 at 13:43
• @mrf How do you characterise this hyperplane. Shebat's text defines a real hyperplane as the set of all vectors in $\mathbb{C}^n$ that are real orthogonal to some vector $z^0$. How have you used that here? – user412674 Mar 10 '17 at 1:07
• @mrf Also, Shabat states that if a function $f \in \mathcal{O}(D)$ vanishes in a real neighbourhood of a point $a \in D$, then $f \equiv 0$ in $D$. Would we not then simply assert that for every point on a real hyperplane, there is a real neighbourhood about that point? Similarly for (b)? – user412674 Mar 10 '17 at 1:15