S. Encinas

Interests:

Resolution of Singularities and applications.

Resolution of singularities

Some easy examples:

Blowing up of the origin in the plane, here.
Three singular curves and its blowing up at the origin, here.
Curves: $x^3-y^2$ , $x^5-y^2$ , $x^2(x+1)-y^2$ .
Resolution of the cusp $x^3-y^2$ , here.

Some examples of resolution of singularities. Computed with Villamayor's algorithm of resolution of singularities and using the program implemented by G. Bodnár y J. Schicho from RISC Institut, Linz (Austria).

The cusp $x_1^2-x_2^3$ .
Whitney umbrella $x_{1}^2-x_{2}^2 x_{3}$ .
Basic object $(x_{1}^2-x_{2}^2 x_{3},2)$ .
Basic object $\left(\left(x_{3}^2-x_{1}^3\right)^2-2 x_{1}^4 x_{2}\left(x_{1}^3+x_{3}^2\right),4\right)$ .
Basic object $\left(\left(x_{3}^2-x_{1}\right)^2-2 x_{1}^2 x_{2}\left(x_{1}+x_{3}^2\right),2\right)$ , obtained from the blowing up of previuos equation with center $(x_{1},x_{3})$ and taking chart $x_{1}$ .

Łojasiewicz exponent:

The Łojasiewicz exponent can be computed using an algorithm of resolution of singularities.
Here you can download a Singular program for computing Lojasiewicz exponent. You need to install the package desing implemented by G. Bodnár y J. Schicho.