The Newton map N for finding the roots of two polynomial equations P(x,y)=0, Q(x,y)=0 has incredibly complicated dynamics because the system simultaneously has points of indeterminacy and topological degree > 1. In this mini-course we will examine the simplest non-trivial case, the Newton map to find the roots of x(x-1)=0 and y^2+Bx-y=0.

Theorem: If |B-1|>1, then the basins of attraction for the roots (0,0) and (0,1) have infinitely generated first homology and the basins of attraction for the roots (1,0) and (1,1-B) have either trivial first homology, or infinitely generated first homology, depending on the parameter B.

The proof of this theorem uses many different methods/techniques including blow-ups, Morse theory, linking numbers, and invariant currents. We will introduce the necessary details about these techniques in the mini-course.