Next: Combinatorial Look on Patchworking
Patchworking Algebraic Curves
Ilia Itenberg and Oleg Viro
Real algebraic curves seem to be quite distant from
combinatorial geometry. In this paper we intend to demonstrate how to build
algebraic curves in a combinatorial fashion: to
patchwork them from pieces which essentially are lines.
One can trace related constructions back
to Newton's consideration of branches at a singular
point of a curve. Nonetheless an explicit formulation does not look
familiar for mathematicians outside of a narrow community of
specialists in topology of real algebraic varieties.
This technique was
developed by the second author in the beginning of eighties. Using it,
the first author has recently found counter-examples to the oldest and
most famous conjecture on the topology of real algebraic curves. The
conjecture was formulated as early as 1906 by V. Ragsdale 
on the basis of experimental material provided by A. Harnack's and D. Hilbert's constructions , .