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Let § = SXij],..., n"> be a finitely generated extension of a differential field 9 with m derivative operators. Let d be the differential dimension of â over W. We show that the numerical polynomial 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 S over 3F is defined and extensions for which thisdoi:10.1090/s0002-9939-1978-0480353-7 fatcat:b3de6ifg3ffbbnabzktych63ua