## Number Theory Seminar: David Masser (Univ. of Basel)

**Solving polynomial-exponential equations.**

Inspired by Schanuel's Conjecture, Boris Zilber has proposed a ``Nullstellensatz'' (also conjectural) asserting which sorts of polynomial-exponential equations in several variables have a complex solution. Last year Dale Brownawell and I published a proof in the situation which can be regarded as ``typical''. But it does not cover all situations for two variables, some of which involve simply stated problems in one variable like finding complex $z \neq 0$ with $e^z+e^{1/z}=1$. Recently Vincenzo Mantova and I have settled the general case of two variables. We describe our methods -- for example, to solve

$$e^z+e^{\root 9 \of {1-z^9}}=1$$

one approach uses theta functions on ${\bf C}^{28}$.

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