On the dth Roots of Exponential Polynomials and Related Problems Arising from the Green–Griffiths–Lang Conjecture

Abstract

We show that if an exponential polynomial $$\sum _{i=1}^m P_i(z)e^{Q_i(z)}$$ , where $$P_i$$ , $$Q_i\in \mathbb C[z]$$ , is a dth power, $$d\ge 2$$ , of an entire function g, then g itself is also an exponential polynomial. We also study when a multivariable polynomial with moving targets of slow growth evaluated at unit arguments can be a dth power of an entire function. Finally, we formulate a boundary case of the Green–Griffiths–Lang conjecture for projective spaces with moving targets.