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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
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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.

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Keywords: Differential dimension polynomials, characteristic sets, differential prime ideals, differential polynomials, ranking, initial subsets
Article copyright: © Copyright 1978 American Mathematical Society