Local uniformization

**Local uniformization in positive characteristic.** Let $k$ be a field of characteristic $p > 0$, let $F$ be a finitely generated field extension of $k$, and let $\mathcal{O}$ be a valuation ring of $F$ containing $k$. Then $\mathcal{O}$ admits local uniformization over $k$.

This is open from dimension four on: it is known when the transcendence degree of $F \mid k$ is at most three, see local_uniformization_of_trdeg_le_three.