Analogs of Essential Singularities for Sequences of Polynomial Mappings

For a sequence of polynomial self-mappings of $\mathbb{C}^n$ and a given ball in $\mathbb{C}^n$, we state conditions guaranteeing that the union of images of any larger concentric ball is everywhere dense. Under slightly more severe conditions, one can use a sequence of concentric balls (one for each mapping) with radii tending to zero. The common center of these balls is, in a sense, an essential singularity of the sequence of mappings.