Kolchin's problems explained

Kolchin's problems are a set of unsolved problems in differential algebra, outlined by Ellis Kolchin at the International Congress of Mathematicians in 1966 (Moscow)

Kolchin Catenary Conjecture

The Kolchin Catenary Conjecture is a fundamental open problem in differential algebra related to dimension theory.

Statement

"Let

\Sigma be a differential algebraic variety of dimension d By a long gap chain we mean a chain of irreducible differential subvarieties \Sigma _{0} \subset \Sigma _{1} \subset \Sigma _{2} \subset … of ordinal number length \omega ^{m} ⋅ 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.^{[1] [2] [3] [4]}

Notes and References

1. Kolchin, Ellis Robert, Alexandru Buium, and Phyllis Joan Cassidy. Selected works of Ellis Kolchin with commentary. Vol. 12. American Mathematical Soc., 1999. (pg 607)
2. On Linear Dependence Over Complete Differential Algebraic Varieties. James. Freitag. Omar León. Sánchez. William. Simmons. June 2, 2016. Communications in Algebra. 44. 6. 2645–2669. CrossRef. 10.1080/00927872.2015.1057828. 1401.6211 .
3. A notion of krull dimension for differential rings. Joseph. Johnson. December 1, 1969. Commentarii Mathematici Helvetici. 44. 1. 207–216. Springer Link. 10.1007/BF02564523.
4. Specializations in differential algebra. Azriel. Rosenfeld. May 26, 1959. Transactions of the American Mathematical Society. 90. 3. 394–407. www.ams.org. 10.1090/S0002-9947-1959-0107642-2.