  We study systems of parameters over finite fields from a probabilistic
perspective, and use this to give the first effective Noether normalization
result over a finite field. Our central technique is an adaptation of Poonen's
closed point sieve, where we sieve over higher dimensional subvarieties, and we
express the desired probabilities via a zeta function-like power series that
enumerates higher dimensional varieties instead of closed points. This also
yields a new proof of a recent result of Gabber-Liu-Lorenzini and
Chinburg-Moret-Bailly-Pappas-Taylor on Noether normalizations of projective
families over the integers.
