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 Σ {\displaystyle \Sigma } be a differential algebraic variety of dimension d {\displaystyle d} . By a long gap chain we mean a chain of irreducible differential subvarieties Σ 0 ⊂ Σ 1 ⊂ Σ 2 ⊂ ⋯ {\displaystyle \Sigma _{0}\subset \Sigma _{1}\subset \Sigma _{2}\subset \cdots } of ordinal number length ω m ⋅ d {\displaystyle \omega ^{m}\cdot d} ."
Given an irreducible differential variety Σ {\displaystyle \Sigma } of dimension d > 0 {\displaystyle d>0} and an arbitrary point p ∈ Σ {\displaystyle p\in \Sigma } , does there exist a long gap chain beginning at p {\displaystyle p} and ending at Σ {\displaystyle \Sigma } ?
The positive answer to this question is called the Kolchin catenary conjecture.1234
References
Kolchin, Ellis Robert, Alexandru Buium, and Phyllis Joan Cassidy. Selected works of Ellis Kolchin with commentary. Vol. 12. American Mathematical Soc., 1999. (pg 607) ↩
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. http://www.tandfonline.com/doi/full/10.1080/00927872.2015.1057828 ↩
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. https://doi.org/10.1007/BF02564523 ↩
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. https://www.ams.org/tran/1959-090-03/S0002-9947-1959-0107642-2/ ↩