next up previous
Next: Combinatorial Look on Patchworking

Patchworking Algebraic Curves
Disproves the Ragsdale Conjecture

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 [14] on the basis of experimental material provided by A. Harnack's and D. Hilbert's constructions [5], [6].