A probabilistic approach to systems of parameters and Noether normalization

TL;DR

This paper introduces a probabilistic method for systems of parameters over finite fields, leading to the first effective Noether normalization over finite fields and providing new proofs for related algebraic geometry results.

Contribution

It develops a novel probabilistic approach using a zeta function-like series to achieve effective Noether normalization over finite fields, extending previous work.

Findings

First effective Noether normalization over finite fields.

Probabilistic framework using higher-dimensional sieve and zeta functions.

New proof of Noether normalization results for projective families over integers.

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.