Kolchin's problems

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 ${\displaystyle \Sigma }$ be a differential algebraic variety of dimension ${\displaystyle d}$. By a long gap chain we mean a chain of irreducible differential subvarieties ${\displaystyle \Sigma _{0}\subset \Sigma _{1}\subset \Sigma _{2}\subset \cdots }$ of ordinal number length ${\displaystyle \omega ^{m}\cdot d}$."

Given an irreducible differential variety ${\displaystyle \Sigma }$ of dimension ${\displaystyle d>0}$ and an arbitrary point ${\displaystyle p\in \Sigma }$ , does there exist a long gap chain beginning at ${\displaystyle p}$ and ending at ${\displaystyle \Sigma }$?

The positive answer to this question is called the Kolchin catenary conjecture. ^{[1] [2] [3] [4]}

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. Freitag, James; Sánchez, Omar León; Simmons, William (June 2, 2016). "On Linear Dependence Over Complete Differential Algebraic Varieties". Communications in Algebra. 44 (6): 2645–2669. arXiv: 1401.6211 . doi:10.1080/00927872.2015.1057828 – via CrossRef.
3. Johnson, Joseph (December 1, 1969). "A notion of krull dimension for differential rings". Commentarii Mathematici Helvetici . 44 (1): 207–216. doi:10.1007/BF02564523 – via Springer Link.
4. Rosenfeld, Azriel (May 26, 1959). "Specializations in differential algebra". Transactions of the American Mathematical Society. 90 (3): 394–407. doi:10.1090/S0002-9947-1959-0107642-2 – via www.ams.org.