Is a function which is finitely multiple-valued in each variable separately, also finitely multiple-valued in all its variables jointly? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T00:34:47Z http://mathoverflow.net/feeds/question/14998 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14998/is-a-function-which-is-finitely-multiple-valued-in-each-variable-separately-also Is a function which is finitely multiple-valued in each variable separately, also finitely multiple-valued in all its variables jointly? Mark B Villarino 2010-02-11T15:36:04Z 2010-04-17T05:22:17Z <p>It is well known that under suitable conditions, a function which is:</p> <ol> <li><p>a polynomial in each variable separately is a polynomial in all its variables jointly.</p></li> <li><p>a rational function in each variable separately is a rational function.</p></li> <li><p>a holomorphic function in each variable separately is holomorphic in all its variables.</p></li> </ol> <p>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.</p> <p>Here is my question: <strong><em>is a complete analytic function, which is finitely multiple-valued in each variable separately, also finitely multiple-valued jointly?</em></strong></p> http://mathoverflow.net/questions/14998/is-a-function-which-is-finitely-multiple-valued-in-each-variable-separately-also/15002#15002 Answer by justlooking for Is a function which is finitely multiple-valued in each variable separately, also finitely multiple-valued in all its variables jointly? justlooking 2010-02-11T16:30:44Z 2010-02-11T16:30:44Z <p>Isn't the function x+y single-valued in each variable separately, but takes any value infinitely often? Or am I misunderstanding your definition?</p> http://mathoverflow.net/questions/14998/is-a-function-which-is-finitely-multiple-valued-in-each-variable-separately-also/15011#15011 Answer by Yaakov Baruch for Is a function which is finitely multiple-valued in each variable separately, also finitely multiple-valued in all its variables jointly? Yaakov Baruch 2010-02-11T17:07:56Z 2010-02-11T17:07:56Z <p>Engineer a function that has n>=2 values exactly in the set $S_n \backslash S_{n+1}$ where $S_n$={(x,y) | x>n, y>n, (x-n)*(y-n)>1}, and 1 value elsewehere. That should work as a counterexample, as far as $C^{\infty}$ functions go - analytic, not sure</p>