# Local uniformization

For a local ring, this is the determination of a regular local ring birationally equivalent to it. For an irreducible algebraic variety (cf. Irreducible variety) $V$ over a field $k$ a resolving system is a family of irreducible projective varieties $\{V_\alpha\}$ birationally equivalent to $V$ (that is, such that the rational function fields $k(V_\alpha)$ and $k(V)$ are isomorphic) and satisfying the following condition: For any valuation (place) $v$ of $k(V)$ there is a variety $V'\in\{V_\alpha\}$ such that the centre $P'$ of $v$ on $V'$ is a non-singular point. The existence of a resolving system (the local uniformization theorem) was proved for arbitrary varieties over a field of characteristic zero (see ), and also for two-dimensional varieties over any field and three-dimensional varieties over an algebraically closed field of characteristic other than 2, 3 or 5 (see ). The existence of a resolving system for $V$ consisting of a single variety implies resolution of the singularities of $V$ and can be obtained from the local uniformization theorem in dimension $\leq3$. In the general case the local uniformization theorem implies the existence of a finite resolving system (see ).