It is well known that under suitable conditions, a function which is:
a polynomial in each variable separately is a polynomial in all its variables jointly.
a rational function in each variable separately is a rational function.
a holomorphic function in each variable separately is holomorphic in all its variables.
A complete analytic function can be single-valued or multiple-valued according as it does not have, or does have, branch points. The algebraic functions are examples of the latter.
Here is my question: is a complete analytic function, which is finitely multiple-valued in each variable separately, also finitely multiple-valued jointly?
$(f(x_0,y_0))$
could take an infinite number of values, if you were to change only one variable in a continuous manner (as in moving around a branch point) and returning to$(x_0,y_0)$
, you can reach only a finite number of the values of$(f(x_0,y_0))$
, whereas you can reach an infinite number by moving both coordinates together? $\endgroup$