A probabilistic approach to systems of parameters and Noether normalization

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

Vol. 13, No. 9, 2019

Juliette Bruce and Daniel Erman

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors