Nash resolution for binomial varieties as Euclidean division. A priori termination bound, polynomial complexity in essential dimension 2
The authors establish for affine binomial varieties of dimension 2 an a priori bound on the lenght of resolution of singularities by means of the compositions of the normalizations with Nash blowings up. For this purpose they prove a structure theorem for binomial varieties, i.e. isomorphisms of the irreducible components of any binomial variety with its toric component. Then they establish the equivalence of Nash blowings up to an Euclidean \(m\)-dimensional division. Finally a combinatorial version of the composition of normalizations with Nash blowings up for \(m=2\) yields a sharp a priori bound and the algorithm is of a very small complexity. In an Appendix by M. Spivakovsky a bound on the length of the normalized Nash resolution of a minimal singularity via the size of the dual graph of its minimal desingularization is given. The reviewer means that it would be interesting to search for a connection with Hirzebruch-Jung continued fractions and also with Gröbner fans.
scientific article; zbMATH DE number 6108933
scientific article