Negative solutions to three-dimensional monomial Noether problem

TL;DR

This paper investigates conditions under which the three-dimensional monomial Noether problem has negative solutions for certain groups, using Galois cohomology to establish necessary and sufficient criteria.

Contribution

It provides the first complete characterization of when negative solutions occur in the three-dimensional monomial Noether problem based on coefficient conditions.

Findings

Identifies necessary and sufficient conditions for negative solutions.

Uses Galois cohomology to derive irrationality criteria.

Classifies groups with negative solutions in three dimensions.

Abstract

Three-dimensional monomial Noether problem can have negative solutions for 8 groups by the suitable choice of the coefficients. We find the necessary and sufficient condition for the coefficients to have a negative solution. The results are obtained by two criteria of irrationality using Galois cohomology.