Details:
Title  Nash resolution for binomial varieties as Euclidean division. A priori termination bound, polynomial complexity in essential dimension 2  Author(s)  Dima Grigoriev, Pierre D. Milman  Type  Article in Journal  Abstract  We establish a (novel for desingularization algorithms) a priori bound on the length of resolution of singularities by means of the compositions of the normalizations with Nash blowings up, albeit that only for affine binomial varieties of (essential) dimension 2. Contrary to a common belief the latter algorithm turns out to be of a very small complexity (in fact polynomial). To that end we prove a structure theorem for binomial varieties and, consequently, the equivalence of the Nash algorithm to a combinatorial algorithm that resembles Euclidean division in dimension ≥2 and, perhaps, makes the Nash termination conjecture of the Nash algorithm particularly interesting. A bound on the length of the normalized Nash resolution of a minimal surface singularity via the size of the dual graph of its minimal desingularization is in the Appendix (by M. Spivakovsky).  Keywords  Nash blowings up, Binomial varieties, Desingularization, A priori bound, Euclidean multidimensional algorithm, Polynomial complexity  ISSN  00018708 
URL 
http://www.sciencedirect.com/science/article/pii/S0001870812003179 
Language  English  Journal  Advances in Mathematics  Volume  231  Number  6  Pages  3389  3428  Year  2012  Edition  0  Translation 
No  Refereed 
No 
