This picture is a view of the polynomial
The picture shows the curves Re(f(z)) sin(t) - IM(f(z)) cos(t) = 0
for various values of t . In his first proof of the
Fundamental Theorem of Algebra Gauss considered these curves for
t = 0, Pi/2 . He noted that these curves crossed only at
the roots of the polynomial, and thus to complete his proof it was
enough to show that these curves crossed. In these notes we consider
these curves for almost all (i.e. all but finitely many) values of
t and show that each (connected) componenent of these curves
passes through a root. On this picture the roots are clearly visible
as the intersection points of these curves. The points where different
(algebraic) components of the same curve cross are not roots but
critical points. One ironic point is that Gauss' proof has often been
cited as one of the first non-constructive existence proof in mathematics,
this picture clearly shows that in fact Gauss' proof is now quite constructive,
albeit not an efficient solution technique.
More interesting is the fact that these curves are precisely the curves
that Newton's method tends to follow on the way to the roots. Thus this
picture can be interpreted as a picture of the approximate basins of
attraction of Newton's method. However, we also know that this is not
precisely true since the boundaries of the Newton Basins are not
algebraic curves but fractals. The following picture shows the
actual basin
of attraction of the root 0 for this polynomial. The
different colors show how many iterations are necessary to hit the
black target area.
Finally the two pictures (the curves and the basin) are combined
to emphasize the relation between the geometry and the analysis.
