In this talk we will see the main ingredients to construct an algorithm of resolution of singularities. In particular, we will construct a specific algorithm of resolution of singularities for binomial ideals in arbitrary characteristic.

To resolve binomial ideals we define a modified order function, $E$-order, as the order along a normal crossing divisor $E$. With this $E$-order function we construct a resolution function that drops after blowing up and which provides only combinatorial centers. This kind of centers preserve the binomial structure of the ideal after blowing up. The output of this procedure is a locally monomial ideal that can be easily resolved.

References:

E. Bierstone and P. Milman, Desingularization of toric and binomial varieties.
Journal of Algebraic Geometry 15, 443--486, 2006.

S. Encinas and H. Hauser. Strong resolution of singularities in characteristic zero.
Commentarii Mathematici Helvetici 77 (4), 821--845, 2002.

S. Encinas and O. Villamayor. A course on Constructive Desingularization and Equivariance.
In Resolution of Singularities, A research textbook in tribute to Oscar Zariski, H. Hauser, J. Lipman, F. Oort, and A. Quir\'os, Eds. 147--227. Progress in Math. 181, Birkhäuser, Basel, 2000.

J. Wodarczyk. Simple Hironaka resolution in characteristic zero. Preprint, 2004, arXiv:math/0401401v1.