Zariski density of the complex exponential graph

The graph $\{(x,e^x):x\in \mathbb{C}\}$ is Zariski dense in ${\mathbb{A}}_{\mathbb{C}}^2$. If ${\sum }_j{p}_j(x)e^{jx}=0$, repeated application of $D-m$ removes the largest exponential term while acting injectively on the others, proving inductively that every $p_j$ vanishes.