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 PDF

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

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.

References

E. R. Kolchin, Differential algebra and algebraic groups, Pure and Applied Mathematics, Vol. 54, Academic Press, New York-London, 1973. MR 0568864
Joseph Johnson, Differential dimension polynomials and a fundamental theorem on differential modules, Amer. J. Math. 91 (1969), 239–248. MR 238822, DOI 10.2307/2373279

Systems of n partial differential equations in n unknown functions

Barbara A. Lando, Jacobi’s bound for the order of systems of first order differential equations, Trans. Amer. Math. Soc. 152 (1970), 119–135. MR 279079, DOI 10.1090/S0002-9947-1970-0279079-1

A generalised Jacobi conjecture for arbitrary systems of algebraic differential equations

William Yu Sit, Typical differential dimension of the intersection of linear differential algebraic groups, J. Algebra 32 (1974), no. 3, 476–487. MR 379455, DOI 10.1016/0021-8693(74)90153-7
William Yu Sit, Well-ordering of certain numerical polynomials, Trans. Amer. Math. Soc. 212 (1975), 37–45. MR 406992, DOI 10.1090/S0002-9947-1975-0406992-9