In the last decade much research has been done on questions related
to circle
packings, which are collections of circles in the plane with prescribed
patterns of
tangencies encoded by simplicial 2-complexes. The interest in circle packings
and their
connections with analytic functions was initiated by W. Thurston in 1985
when he
suggested a method for approximation of conformal mappings using circle
packings
[20]. This method was shown to work by B. Rodin &
D. Sullivan [15] later that year
(see also [17]). Since then several results
have been published (for example, [3, 4, 5, 10,
11, 14]) which seem to indicate the possibility of creating
a discrete analog of the
theory of analytic functions via circle packings. However, most of these
results deal
with univalent circle packings, that is, genuine
packings of circles. Influenced by the
idea of [3] that circle packings might be used to
construct discrete parallels of analytic
functions, we started in [7] to investigate
non-univalent circle packings, particularly
branched packings. We introduced there the notion of discrete Blaschke
products and
proved a branched version of the finite Riemann mapping theorem
(see [20, 15]). We
feel though that to have a solid foundation for a circle packing analog
of the theory
of analytic functions one needs something more. Since complex polynomials
form a
fundamental and very important class of analytic functions, it is natural
to try to
establish circle packing counterparts of these mappings. This is exactly
the main
theme of our paper.