Algebrability of the set of everywhere surjective functions on $\mathbb{C}$

Abstract

We show that the set ${\mathcal L}$ of complex-valued everywhere surjective functions on $\mathbb{C}$ is algebrable. Specifically, ${\mathcal L}$ contains an infinitely generated algebra every non-zero element of which is everywhere surjective. We also give a technique to construct, for every $n \in \mathbb N$, $n$ algebraically independent everywhere surjective functions, $f_1, f_2, \dots, f_n$, so that for every non-constant polynomial $P \in \mathbb{C}[z_1, z_2, \dots, z_n]$, $P(f_1, f_2, \dots, f_n)$ is also everywhere surjective.