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Many valued “functions” play an important role in complex analysis. In many cases of interest, these may converted to honest functions through the application of a differential operator. For example, the complex logarithm is only well defined up to the addition of an integral multiple of twice pi times the square root of negative one, but the logarithmic derivative which takes a function f(z) to the derivative of log(f(z)) is a well-defined function from (nowhere zero, differentiable, complex valued) functions to functions. Curiously, the logarithmic derivative is actually a differential rational function. That is, it is given by an expression which is a rational function of its argument and some of the derivatives of the argument. Other functions obtained by applying differential operators to eliminate the ambiguity of multivalued functions share this feature. I will describe a general context in which multivalued functions arising as inverses to certain analytic covering maps may be converted to well-defined differential algebraic functions by composing them with a well chosen differential operator.

While the theorem itself has some nice consequences in the theories of functional transcendence and differential algebra, in this lecture I will focus on two ideas from logic underlying the proof. First, the notion of elimination of imaginaries in the theories of algebraically closed and differentially closed fields will supply the requisite differential operator. Secondly, a beautiful theorem of Peterzil and Starchenko on definable complex analytic sets will be used to show that the classical analytic construction of such functions is necessarily algebraic.