Toric decomposition in algebraic groups

Over an arbitrary field $\mathbb{F}$, we construct $n+1$ maximal tori $T_1,\dots,T_{n+1}$ in $\operatorname{PGL}_n(\mathbb{F})$ so that the product $T_1\dots T_{n+1}$ is almost the whole $\operatorname{PGL}_n(\mathbb{F})$ and every $g\in T_1\dots T_{n+1}$ can be expressed uniquely as $g=t_1\dots t_{n+1}$ where $t_i\in T_i$.

The construction is optimal, as the number of tori with this property attains a general upper bound for connected reductive groups over an algebraically closed field, as well as over finite fields.

We also show that $n+2$ suitably chosen maximal tori $T_1,\dots,T_{n+2}$ are enough to cover the whole group, i.e. $\operatorname{PGL}_n(\mathbb{F})=T_1\dots T_{n+2}$, provided $|\mathbb{F}|>n^2$.

This is optimal over a finite field and is conjecturally optimal over algebraically closed fields, i.e. the number of such tori is as small as possible.