While the equations x(x-1)=0, y^2+Bxy-y=0 are easy to solve, the dynamics of the Newton map N(x,y) for finding the four roots is quite complicated. In particular, N is many-to-one and N has points of indeterminacy.

The two vertical lines x=0 and x=1 are invariant under N and super-attracting. Within these lines the ''circles'' Re(y) = 1/2 and Re(y) = (1-B)/2, respectively, are hyperbolically repelling with multiplier 2. In this talk we will prove that these circles have superstable manifolds of real dimension 3 using the technique of holomorphic motions. These manifolds extend to all points with Re(x) < 1/2 and Re(x) > 1/2 respectively and provide insight into the topology of the basins of attraction for the four roots.

This work follows the ideas of John H. Hubbard and Sebastien Krief.