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Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations. (English) Zbl 0816.34040
Summary: 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 $${\mathbb{R}}(x)$$.

##### MSC:
 34E05 Asymptotic expansions of solutions to ordinary differential equations 26A12 Rate of growth of functions, orders of infinity, slowly varying functions 13N99 Differential algebra
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