On the joint spectral radius

For a bounded subset $S$ of $d\times d$ complex matrices, the Berger-Wang theorem and Bochi's inequality allow to approximate the joint spectral radius of $S$ from below by the spectral radius of a short product of elements from $S$. Our goal is two-fold: we review these results, providing self-contained proofs, and we derive an improved version with explicit bounds that are polynomial in $d$. We also discuss other complete valued fields.