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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2017
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1703.01995 |
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| _version_ | 1866917572331438080 |
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| author | Berarducci, Alessandro Mantova, Vincenzo |
| author_facet | Berarducci, Alessandro Mantova, Vincenzo |
| contents | We show that Écalle's transseries and their variants (LE and EL-series) can be interpreted as functions from positive infinite surreal numbers to surreal numbers. The same holds for a much larger class of formal series, here called omega-series. Omega-series are the smallest subfield of the surreal numbers containing the reals, the ordinal omega, and closed under the exp and log functions and all possible infinite sums. They form a proper class, can be composed and differentiated, and are surreal analytic. The surreal numbers themselves can be interpreted as a large field of transseries containing the omega-series, but, unlike omega-series, they lack a composition operator compatible with the derivation introduced by the authors in an earlier paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1703_01995 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Transseries as germs of surreal functions Berarducci, Alessandro Mantova, Vincenzo Logic 03C64, 16W60, 04A10, 26A12, 13N15 We show that Écalle's transseries and their variants (LE and EL-series) can be interpreted as functions from positive infinite surreal numbers to surreal numbers. The same holds for a much larger class of formal series, here called omega-series. Omega-series are the smallest subfield of the surreal numbers containing the reals, the ordinal omega, and closed under the exp and log functions and all possible infinite sums. They form a proper class, can be composed and differentiated, and are surreal analytic. The surreal numbers themselves can be interpreted as a large field of transseries containing the omega-series, but, unlike omega-series, they lack a composition operator compatible with the derivation introduced by the authors in an earlier paper. |
| title | Transseries as germs of surreal functions |
| topic | Logic 03C64, 16W60, 04A10, 26A12, 13N15 |
| url | https://arxiv.org/abs/1703.01995 |