Constructiveness of Hironaka's resolution

Reduction of the singularities of codimension one singular foliations in dimension three

“…In particular, simple singularities are a generalization of functions locally given by a monomial. Note that getting locally a monomial is the main objective in the problem of reduction of singularities of varieties, both in zero and positive characteristic [15], [1], [28], [29], [4]. However for the saddle-nodes, that correspond to λ = 0, the leaves are far from being comparable to the level sets of a function.…”

Toward Resolution of Singularities over a Field of Positive Characteristic—Part I. Foundation; the language of the idealistic filtration

“…We can repeat the process only finitely many times, since after each blowup the value of the strand of invariants strictly drops and since the set of its values satisfies the descending chain condition, leading to the termination of the algorithm. (See, e.g., [Vil89] Mk07] for details of the construction of the strand of invariants and the corresponding modifications in the classical setting. )…”

A new Proof of Desingularization over fields of characteristic zero

“…Different constructive proofs, following Hironaka's approach, appear in [7], [16], and in [27] (see also [28]). Each one of these proofs provides an algorithm of desingularization which indicates where to blow-up in order to eliminate the singularities in a step by step procedure.…”