Polynomial‐exponential equations — Some new cases of solvability

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 with .