A geometric approach to some systems of exponential equations
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866916102115688448 |
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| author | Aslanyan, Vahagn Kirby, Jonathan Mantova, Vincenzo |
| author_facet | Aslanyan, Vahagn Kirby, Jonathan Mantova, Vincenzo |
| contents | Zilber's Exponential Algebraic Closedness conjecture (also known as Zilber's Nullstellensatz) gives conditions under which a complex algebraic variety should intersect the graph of the exponential map of a semiabelian variety.
We prove the special case of the conjecture where the variety has dominant projection to the domain of the exponential map, for abelian varieties and for algebraic tori. Furthermore, in the situation where the intersection is 0-dimensional, we exhibit structure in the intersection by parametrizing the sufficiently large points as the images of the period lattice under a (multivalued) analytic map. Our approach is complex geometric, in contrast to a real analytic proof given by Brownawell and Masser just for the case of algebraic tori. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2105_12679 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A geometric approach to some systems of exponential equations Aslanyan, Vahagn Kirby, Jonathan Mantova, Vincenzo Complex Variables Logic Number Theory Primary: 11G10, 14K20, Secondary: 03C60, 12L12 Zilber's Exponential Algebraic Closedness conjecture (also known as Zilber's Nullstellensatz) gives conditions under which a complex algebraic variety should intersect the graph of the exponential map of a semiabelian variety. We prove the special case of the conjecture where the variety has dominant projection to the domain of the exponential map, for abelian varieties and for algebraic tori. Furthermore, in the situation where the intersection is 0-dimensional, we exhibit structure in the intersection by parametrizing the sufficiently large points as the images of the period lattice under a (multivalued) analytic map. Our approach is complex geometric, in contrast to a real analytic proof given by Brownawell and Masser just for the case of algebraic tori. |
| title | A geometric approach to some systems of exponential equations |
| topic | Complex Variables Logic Number Theory Primary: 11G10, 14K20, Secondary: 03C60, 12L12 |
| url | https://arxiv.org/abs/2105.12679 |