Transcendence degrees of fields generated by exponentials of products
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915317806006272 |
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| author | Massold, Heinrich |
| author_facet | Massold, Heinrich |
| contents | Let $θ=(θ_1,\ldots,θ_m) \in \R^m, κ=(κ_1,\ldots,κ_n) \in \R^n$ be two tuples of real numbers each linearly independent over $\Q$, and $T$ the transcendence degree of the field generated by $\{\exp(θ_i κ_j) | i=1,\ldots,m, \; j=1,\ldots,n \}$ over $\Q$. The estimate $T \geq \frac{mn}{m+n} -1$ has been conjectured for some time but could only be proved under additional hypotheses for $θ$ and $κ$. This paper proves a weaker estimate for $T$ while also reducing the strong estimate to a prominent conjecture on intersections of subvarieties of split tori with subgroups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01123 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Transcendence degrees of fields generated by exponentials of products Massold, Heinrich Number Theory 11J85, 11J81, 14C17 Let $θ=(θ_1,\ldots,θ_m) \in \R^m, κ=(κ_1,\ldots,κ_n) \in \R^n$ be two tuples of real numbers each linearly independent over $\Q$, and $T$ the transcendence degree of the field generated by $\{\exp(θ_i κ_j) | i=1,\ldots,m, \; j=1,\ldots,n \}$ over $\Q$. The estimate $T \geq \frac{mn}{m+n} -1$ has been conjectured for some time but could only be proved under additional hypotheses for $θ$ and $κ$. This paper proves a weaker estimate for $T$ while also reducing the strong estimate to a prominent conjecture on intersections of subvarieties of split tori with subgroups. |
| title | Transcendence degrees of fields generated by exponentials of products |
| topic | Number Theory 11J85, 11J81, 14C17 |
| url | https://arxiv.org/abs/2506.01123 |