Ragsdale 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

${\displaystyle p\leq {\tfrac {3}{2}}k(k-1)+1\quad {\text{and}}\quad n\leq {\tfrac {3}{2}}k(k-1).}$

She also posed the inequality

${\displaystyle |2(p-n)-1|\leq 3k^{2}-3k+1,}$

and showed that the inequality could not be further improved. This inequality was later proved by Petrovsky.

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^[4] produced additional counterexamples to the Ragsdale conjecture, so Viro and Itenberg wrote a paper in 1996 discussing their work on disproving the conjecture using the "patchworking" technique.^[1]

The problem of finding a sharp upper bound remains unsolved.