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].

- Combinatorial Look on Patchworking
- Digression on Real Plane Algebraic Curves
- T-curves
- Ragsdale Conjecture
- Patchworking Harnack Curves
- Counter-Examples to Ragsdale Conjecture
- Bibliography