The Ragsdale conjecture is a

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...

that concerns the possible arrangements of real

algebraic curves In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...

embedded in the

projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that ...

. It was proposed by Virginia Ragsdale in her dissertation in 1906 and was disproved in 1979. It has been called "the oldest and most famous conjecture on the topology of real algebraic curves".

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 The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United ...

in 1906. The dissertation was a treatment of

Hilbert's sixteenth problem Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, as part of his list of 23 problems in mathematics. The original problem was posed as the ''Problem of the topolog ...

, which had been proposed by

Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...

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, along with the basis of evidence.

Conjecture

Ragsdale's main conjecture is as follows. Assume that an

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...

of degree 2''k'' contains ''p'' even and ''n'' odd ovals. Ragsdale conjectured that :

p \le \tfrac32 k(k-1) + 1 \quad\text\quad n \le \tfrac32 k(k-1).

She also posed the inequality :

,  2(p-n)-1 ,  \le 3k^2 - 3k + 1,

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

Petrovsky Petrovsky (masculine) and its feminine form Petrovskaya are Russian-language surnames. People with the surname include: People * Adolf Petrovsky (1887–1937), Soviet diplomat *Boris Petrovsky (1908–2004), Soviet surgeon and politician * Grigo ...

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 Oleg Yanovich Viro (russian: Олег Янович Виро) (b. 13 May 1948, Leningrad, USSR) is a Russian mathematician in the fields of topology and algebraic geometry, most notably real algebraic geometry, tropical geometry and knot theory. C ...

introduced a technique known as "patchworking algebraic curves" and used to generate a

counterexample A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is ...

to the conjecture. In 1993, Ilia Itenberg produced additional

counterexamples A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is ...

to the Ragsdale conjecture, so Viro and Itenberg wrote a paper in 1996 discussing their work on disproving the conjecture using the "patchworking" technique. The problem of finding a sharp upper bound remains unsolved.

References

{{Disproved conjectures Disproved conjectures Real algebraic geometry