Waring’s theorem and the super Fermat problem for numbers and functions

Assume that , are positive intergers. Waring’s problem, first proved by Hilbert, asserts that every sufficiently large can be expressed as the sum of at most nth powers. The Super-Fermat problem asks for the smallest for which a solution of ( ) exists with of the form . Wiles has proved that We survey some known bounds for and and also for the corresponding quantities when the and are complex polynominals or entire or meromorphic functions in the plane.