Ragsdale conjecture
The Ragsdale conjecture is a
Formulation of the conjecture
Ragsdale's dissertation, "On the Arrangement of the Real Branches of Plane Algebraic Curves," was published by the American Journal of Mathematics in 1906. The dissertation was a treatment of Hilbert's sixteenth problem, which had been proposed by Hilbert in 1900, along with 22 other unsolved problems of the 19th century; it is one of the handful of Hilbert's problems that remains wholly unresolved. Ragsdale formulated a conjecture that provided an upper bound on the number of topological circles of a certain type,[2] along with the basis of evidence.
Conjecture
Ragsdale's main conjecture is as follows.
Assume that an algebraic curve of degree 2k contains p even and n odd ovals. Ragsdale conjectured that
She also posed the inequality
and showed that the inequality could not be further improved. This inequality was later proved by Petrovsky.
Disproving the conjecture
The conjecture was held of very high importance in the field of real algebraic geometry for most of the twentieth century. Later, in 1980, Oleg Viro[3] introduced a technique known as "patchworking algebraic curves"[1] and used to generate a counterexample to the conjecture.
In 1993, Ilia Itenberg
The problem of finding a sharp upper bound remains unsolved.
References
- ^ .
- ^ De Loera, Jesús; Wicklin, Frederick J. "Biographies of Women in Mathematics: Virginia Ragsdale". Anges Scott College. Retrieved 22 March 2019.
- Zbl 0422.14032.
- Zbl 1162.14300.