Revisiting second-order linear differential equations over Hardy fields
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866914363384791040 |
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| author | Aschenbrenner, Matthias Dries, Lou van den van der Hoeven, Joris |
| author_facet | Aschenbrenner, Matthias Dries, Lou van den van der Hoeven, Joris |
| contents | We review second-order homogeneous linear differential equations with coefficient functions whose germs lie in a Hardy field (and hence are strongly non-oscillating). We prove a conjecture of Boshernitzan (1982): the oscillating solutions to such an equation are given by amplitude and phase functions with germs in a bigger Hardy field, and hence oscillate in a very regular way. We give sharp conditions for the uniqueness of such germs, study their asymptotic behavior, and use this to obtain information about the zeros and critical points of oscillating solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_02013 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Revisiting second-order linear differential equations over Hardy fields Aschenbrenner, Matthias Dries, Lou van den van der Hoeven, Joris Classical Analysis and ODEs Logic We review second-order homogeneous linear differential equations with coefficient functions whose germs lie in a Hardy field (and hence are strongly non-oscillating). We prove a conjecture of Boshernitzan (1982): the oscillating solutions to such an equation are given by amplitude and phase functions with germs in a bigger Hardy field, and hence oscillate in a very regular way. We give sharp conditions for the uniqueness of such germs, study their asymptotic behavior, and use this to obtain information about the zeros and critical points of oscillating solutions. |
| title | Revisiting second-order linear differential equations over Hardy fields |
| topic | Classical Analysis and ODEs Logic |
| url | https://arxiv.org/abs/2603.02013 |