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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 coincide with its test ideals, but Frobenius powers appear to be a more refined measure of singularities than test ideals 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.
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