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 be a differential algebraic variety of dimension By a long gap chain we mean a chain of irreducible differential subvarieties of ordinal number length ."
Given an irreducible differential variety of dimension and an arbitrary point , does there exist a long gap chain beginning at and ending at ?
The positive answer to this question is called the Kolchin catenary conjecture.[1][2][3][4]
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.
- ^ 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.
- ^ 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.