Vol. 13, No. 9, 2019

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A probabilistic approach to systems of parameters and Noether normalization

Juliette Bruce and Daniel Erman

Vol. 13 (2019), No. 9, 2081–2102
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 and Lorenzini (2015) and Chinburg, Moret-Bailly, Pappas and Taylor (2017) on Noether normalizations of projective families over the integers.

Keywords
Noether normalization, system of parameters, closed point sieve
Mathematical Subject Classification 2010
Primary: 13B02
Secondary: 11G25, 14D10, 14G10, 14G15
Milestones
Received: 23 May 2018
Revised: 18 December 2018
Accepted: 27 June 2019
Published: 7 December 2019
Authors
Juliette Bruce
Department of Mathematics
University of Wisconsin
Madison, WI
United States
Daniel Erman
Department of Mathematics
University of Wisconsin
Madison, WI
United States