Draft:Kolchin Catenary Conjecture

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Instructions · What links here · Kolchin Catenary Conjecture (talk: + · bio) · (log) · Copyvios report · reFill · Citation Bot · (Search: Google, Bing, Wikipedia) · Submitted 28 days ago by DupuyTaylor (talk: D · +) · Last edited 25 days ago by Citation bot

Comment: Fails WP:GNG, requires significant coverage in multiple independent secondary sources. Dan arndt (talk) 01:23, 24 May 2024 (UTC)

The Kolchin Catenary Conjecture is one of Kolchin's Problems. It is a fundamental problem in open differential algebra related to dimension theory.

Statement[edit]

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

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]

References[edit]

^ 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.

[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.

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