Quantifying singularities with differential operators.

The F -signature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties. For example, this invariant detects (non)singularity and strong F -regularity. However, it is very difficult to compute. Motivated by different aspects of the F -signature, we define a numerical invariant for rings of characteristic zero or p > 0 that exhibits many of the useful properties of the F -signature. We also compute many examples of this invariant, including cases where the F -signature is not known. We also obtain a number of results on symbolic powers and Bernstein-Sato polynomials. [ABSTRACT FROM AUTHOR]