Computational Aspects of the Resolution of Singularities

The task of resolution of singularities has been one of the central topics in Algebraic Geometry for many decades. After results in low dimension in the first half of the 20th century, it was Hironaka's monumental article in 1964 which solved the porblem in the general case in chareacteristic zero. The case of characteristic $p > 0$ is still unsolved except in partial results in low dimension. But Hironaka's proof did not put an end to the interest in characteric zero, instead it shifted the focus toward the task of finding a more constructive approach. Such algorithmic approaches appeared at the end of the 1980's independently by Villamayor and by Bierstone and Milman. In this talk we consider the computational tasks arising from Villamayor's algorithm and present an implementation.