  This article extends the notion of a Frobenius power of an ideal in prime
characteristic to allow arbitrary nonnegative real exponents. These generalized
Frobenius powers are closely related to test ideals in prime characteristic,
and multiplier ideals over fields of characteristic zero. For instance, like
these well-known families of ideals, Frobenius powers also give rise to jumping
exponents that we call critical Frobenius exponents. In fact, the Frobenius
powers of a principal ideal coincides with its test ideals, but appear to be a
more refined measure of singularities in general. Herein, we develop the theory
of Frobenius powers in regular domains, and apply it to study singularities,
especially those of generic hypersurfaces. These applications illustrate one
way in which multiplier ideals behave more like Frobenius powers than like test
ideals.
