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Non-oscillating solutions of a differential equation and Hardy fields. (Solutions non 0scillantes d’une èquation diffèrentielle et corps de Hardy.) (French. English summary) Zbl 1133.34007

Summary: Let \(\varphi :x\mapsto\varphi (x),x\gg 0\) be a solution of an algebraic differential equation of order \(n, P(x,y,y',\dots,y^{ (n) })=0\). We establish a geometric criterion so that the germs at infinity of \(\varphi \) and the identity function on \(\mathbb R\) belong to a common Hardy field. This criterion is based on the concept of nonoscillation.

MSC:

34A26 Geometric methods in ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
37C10 Dynamics induced by flows and semiflows
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