Nash resolution for binomial varieties as Euclidean division. A priori termination bound, polynomial complexity in essential dimension 2

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).