A probabilistic approach to systems of parameters and Noether normalization

Abstract

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.

20 pages. Minor revisions to exposition

References in corpus (6)

• Hypersurfaces in projective schemes and a moving lemma
• Bertini irreducibility theorems over finite fields
• $K(\pi ,1)$-neighborhoods and comparison theorems
• Furstenberg sets and Furstenberg schemes over finite fields
• Noether normalizations, reductions of ideals, and matroids
• Lines on cubic hypersurfaces over finite fields

Cited by in corpus (2)

• Interpolation over ZZ and torsion in class groups
• Effective Bounds on the Dimensions of Jacobians Covering Abelian Varieties