Transcendence degrees of fields generated by exponentials of products

TL;DR

This paper investigates the transcendence degree of fields generated by exponentials of products of real numbers, providing a weaker proven estimate and linking the problem to a major conjecture in algebraic geometry.

Contribution

It proves a weaker lower bound for the transcendence degree and reduces the main conjecture to a well-known problem on subvariety intersections in split tori.

Findings

Established a lower bound for the transcendence degree T.

Reduced the main conjecture to a problem on intersections of subvarieties with subgroups.

Connected transcendence degree estimates to a prominent conjecture in algebraic geometry.

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.