Creates a commutative algebra background for explicit and canonical resolution of singularities of algebraic varieties. The author's construction provides a coordinate-free definition of some Newton polyhedra related to germ of functions or an ideal.
This book creates a commutative algebra background for explicit and canonical resolution of singularities of algebraic varieties. The author's construction provides a coordinate-free definition of some Newton polyhedra related to a germ of functions or an ideal. In addition, the construction is significant as a step toward an explicit and canonical resolution of singularities in characteristic zero. The book is intended for researchers in algebraic geometry and commutative algebra.

(Contents List)
Part 1 Newton polyhedra without coordinatesfiltrations; contact and stably contact filtrations; the first derived filtration and its structure; change of the subring; references. Part 2 Newton polyhedra of ideals: standard bases and the main result; differential operators and principal parts; generalized fitting ideals; heuristics; generic position; fitting ideals and filtrations generated by standard bases; normalized standard bases; proof of the main theorem 2.7.