Algorithmic equiresolution of possibly non-reduced families of singularities

This paper studies the concept of algorithmic equiresolution of a family of embedded varieties or ideals, which means a simultaneous resolution of such a family compatible with a given (suitable) algorithm of resolution in characteristic zero. The paper's approach is more indirect: it primarily considers the more general case of families of basic objects (or marked ideals). A definition of algorithmic equiresolution is proposed, which applies to families whose parameter space T may be non-reduced, e.g., the spectrum of a suitable artinian ring. Other definitions of algorithmic equiresolution are also discussed. These are geometrically very natural, but the parameter space T of the family must be assumed regular. It is proven that when T is regular all the proposed definitions are equivalent.