Rosenlicht fields

Shackell, John

doi:10.1090/S0002-9947-1993-1085945-5

Rosenlicht fields
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by John Shackell

Trans. Amer. Math. Soc. 335 (1993), 579-595

DOI: https://doi.org/10.1090/S0002-9947-1993-1085945-5

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Abstract:

Let $\phi$ satisfy an algebraic differential equation over ${\mathbf {R}}$. We show that if $\phi$ also belongs to a Hardy field, it possesses an asymptotic form which must be one of a restricted number of types. The types depend only on the order of the differential equation. For a particular equation the types are still more restricted. In some cases one can conclude that no solution of the given equation lies in a Hardy field, and in others that a particular asymptotic form is the only possibility for such solutions. This therefore gives a new method for obtaining asymptotic solutions of nonlinear differential equations. The techniques used are in part derived from the work of Rosenlicht in Hardy fields.

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Rosenlicht fields and asymptotic forms