  Suppose R is a Noetherian local ring with prime characteristic p>0. In this
article, we show the existence of a local numerical invariant, called the
F-signature, which roughly characterizes the asymptotic growth of the number of
splittings of the iterates of the Frobenius endomorphism of R. This invariant
was first formally defined by C. Huneke and G. Leuschke and has previously been
shown to exist only in special cases. The proof of our main result is based on
the development of certain uniform Hilbert-Kunz estimates of independent
interest. Additionally, we analyze the behavior of the F-signature under finite
ring extensions and recover explicit formulae for the F-signatures of finite
quotient singularities.
