Vincenzo Mantova

The celebrated Schanuel conjecture predicts a lower bound for the transcendence degree of the values of the complex exponential function. A less celebrated conjecture by Schanuel is a sort of converse statement. Using deep model-theoretic tools, Zilber refined and reformulated the converse conjecture as the "strong exponential-algebraic closure conjecture": all systems of polynomial-exponential equations have solutions of large transcendence degree, as long as they do not contradict Schanuel conjecture.
    I will discuss what is known about the converse conjecture and focus on the one-variable case, for which we have an essentially positive answer. In this case, the statement can be reduced to the finiteness of the rational solutions of certain polynomial-exponential equations, which in turn can be proved using tools from diophantine geometry.