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arXiv · 2506.01123

Transcendence degrees of fields generated by exponentials of products

Heinrich Massold
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Abstract

Let $\theta=(\theta_1,\ldots,\theta_m) \in \R^m, \kappa=(\kappa_1,\ldots,\kappa_n) \in \R^n$ be two tuples of real numbers each linearly independent over $\Q$, and $T$ the transcendence degree of the field generated by $\{\exp(\theta_i \kappa_j) | i=1,\ldots,m, \; j=1,\ldots,n \}$ 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 $\theta$ and $\kappa$. 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.

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Heinrich Massold. 2025-06-01. Transcendence degrees of fields generated by exponentials of products. https://arxiv.org/abs/2506.01123

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