Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Differential dimension polynomials of finitely generated extensions


Author: William Sit
Journal: Proc. Amer. Math. Soc. 68 (1978), 251-257
MSC: Primary 12H05
MathSciNet review: 480353
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathcal{G} = \mathcal{F}\langle {\eta _1}, \ldots ,{\eta _n}\rangle $ be a finitely generated extension of a differential field $ \mathcal{F}$ with m derivative operators. Let d be the differential dimension of $ \mathcal{G}$ over $ \mathcal{F}$. We show that the numerical polynomial

$\displaystyle {\omega _{\eta /\mathcal{F}}}(X) - d\left( {\begin{array}{*{20}{c}} {X + m} \\ m \\ \end{array} } \right)$

can be viewed as the differential dimension polynomial of certain extensions. We then give necessary and sufficient conditions for this numerical polynomial to be zero. An invariant (minimal) differential dimension polynomial for the extension $ \mathcal{G}$ over $ \mathcal{F}$ is defined and extensions for which this invariant polynomial is $ d\left( {\begin{array}{*{20}{c}} {X + M} \\ m \\ \end{array} } \right)$ are characterised.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 12H05

Retrieve articles in all journals with MSC: 12H05


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1978-0480353-7
PII: S 0002-9939(1978)0480353-7
Keywords: Differential dimension polynomials, characteristic sets, differential prime ideals, differential polynomials, ranking, initial subsets
Article copyright: © Copyright 1978 American Mathematical Society