Product of two simply connected compact sets in $\mathbb{C}$ is polynomially convex in $\mathbb{C}^2$?

Question

Is it true that the product of two simply connected compact sets in $\mathbb{C}$ is polynomially convex in $\mathbb{C}^2$ ?

Thanks !

Answer 1

Sure, assuming the two compact sets are nice enough (not just simply connected). Any points that is not in the product has either the first or the second coordinate not in one of these simple connected sets. So get a polynomial in the first or the second variable that is bigger at that point then on the compact set, and voila.

Of course, you're in trouble if the compact set in $\mathbb{C}$ is simply connected but its complement is not connected (that could happen if you for example loop one of the sides of the graph of $\sin(1/x)$ back on itself, it's simply connected because there is no continuous loop that goes around).

But if you instead do it with compact sets with connected complements, then you're fine.