Growth Orders Occurring In Expansions Of Hardy-Field Solutions Of Algebraic Differential-Equations

We consider the asymptotic growth of Hardy-field solutions of algebraic differential equations, e.g. solutions with no oscillatory component, and prove that no 'sub-term' occurring in a nested expansion of such can tend to zero more rapidly than a fixed rate depending on the order of the differential equation. We also consider series expansions. An example of the results obtained may be stated as follows. Let g be an element of a Hardy field which has an asymptotic series expansion in x, e(x) and lambda, where lambda tends to zero at least as rapidly as some negative power of exp(e(x)). If lambda actually occurs in the expansion, then g cannot satisfy a first-order algebraic differential equation over R(x).