## "Nash desingularization for binomial varieties as Euclidean multidimensional division

Pierre Milman

Toronto University, Canada.

Abstract

We establish a (novel for desingularization algorithms) a priori bound on the length of resolution of singularities by means of the composites of 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 (including in dimension > 1 ) and, perhaps, makes Nash termination conjecture of the Nash algorithm particularly interesting. (joint work with Dima Grigoriev) The second talk will elaborate on the geometric aspects and on the structure of toric varieties.