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.
