Differential dimension polynomials of finitely generated extensions

Sit, William

doi:10.1090/S0002-9939-1978-0480353-7

Differential dimension polynomials of finitely generated extensions
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by William Sit

Proc. Amer. Math. Soc. 68 (1978), 251-257

DOI: https://doi.org/10.1090/S0002-9939-1978-0480353-7

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