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.
References
