Vol. 86, No. 1, 1980

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Differential valuations

Maxwell Alexander Rosenlicht

Vol. 86 (1980), No. 1, 301–319

A notion of “differential valuation” is defined for ordinary differential fields of characteristic zero by postulating for a given valuation of the field a natural analogue of the elementary L’Hospital’s rule. Such valuations occur implicitly in classical analysis, for example in Hardy’s orders of infinity and in the study of singular points of systems of ordinary differential equations. The fundamental properties of differential valuations are worked out in this paper, numerous examples are discussed, and it is shown that a differential valuation can always be extended to an algebraic extension field. Applications are anticipated to the study of singularities of algebraic differential equations.

Mathematical Subject Classification 2000
Primary: 12H05
Received: 21 September 1978
Published: 1 January 1980
Maxwell Alexander Rosenlicht