Polynomial-exponential equations -- some new cases of solvability

TL;DR

This paper introduces new solvability results for polynomial-exponential equations, especially in two complex variables, using novel methods, and demonstrates the existence of infinitely many solutions for specific equations.

Contribution

It provides the first complete solutions for certain polynomial-exponential equations in two complex variables and introduces new proof techniques.

Findings

Complete solution to polynomial-exponential equations in two complex variables

Existence of infinitely many complex solutions for specific equations

New proof methods for solving polynomial-exponential equations

Abstract

Recently Brownawell and the second author proved a "non-degenerate" case of the (unproved) "Zilber Nullstellensatz" in connexion with "Strong Exponential Closure". Here we treat some significant new cases. In particular these settle completely the problem of solving polynomial-exponential equations in two complex variables. The methods of proof are also new, as is the consequence, for example, that there are infinitely many complex $z$ with $e^z+e^{1/z}=1$.