Kolchin's Problems
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Kolchin's problems are a set of
unsolved problems List of unsolved problems may refer to several notable conjectures or open problems in various academic fields: Natural sciences, engineering and medicine * Unsolved problems in astronomy * Unsolved problems in biology * Unsolved problems in ch ...
in
differential algebra In mathematics, differential algebra is, broadly speaking, the area of mathematics consisting in the study of differential equations and differential operators as algebraic objects in view of deriving properties of differential equations and op ...
, outlined by
Ellis Kolchin Ellis Robert Kolchin (April 18, 1916 – October 30, 1991) was an American mathematician at Columbia University. He earned a doctorate in mathematics from Columbia University in 1941 under supervision of Joseph Ritt. Shortly after he served in t ...
at the
International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...
in 1966 (Moscow)


Kolchin Catenary Conjecture

The Kolchin Catenary Conjecture is a fundamental open problem in
differential algebra In mathematics, differential algebra is, broadly speaking, the area of mathematics consisting in the study of differential equations and differential operators as algebraic objects in view of deriving properties of differential equations and op ...
related to
dimension theory In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
.


Statement

"Let \Sigma be a differential
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...
of
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
d. By a ''long gap chain'' we mean a chain of irreducible differential subvarieties \Sigma_0 \subset \Sigma_1 \subset \Sigma_2 \subset \cdots of
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
length \omega^m \cdot d." Given an irreducible differential variety \Sigma of dimension d > 0 and an arbitrary point p \in \Sigma , does there exist a long gap chain beginning at p and ending at \Sigma? The positive answer to this question is called the Kolchin catenary conjecture.


References

{{reflist Differential algebra Dimension theory