Intersection of zero sets of real analytic functions of two variables

Question

The zero set of a real analytic function cannot contain an open set.

If we have two distinct real analytic functions of two variables, can they intersect in more than at isolated points?

Since the zero set of each is 'one-dimensional' at most, I was hoping that intersection of two such sets would be 'zero-dimensional' due to real analyticity. But, I have been unable to prove this.

Any counterexamples?

Answer 1

More than what?

But two distinct analytic functions can have the same zero set. Try $f(x,y)$ and $e^x f(x,y)$ for any $f$.