Akinari Hoshi

Rationality problem of three-dimensional purely monomial group actions: the last case

A k-automorphism σ of the rational function field k (x 1, ..... x n ) is called purely monomial it a sends every variable x i to a monic Laurent monomial in the variables x 1, ...., x n . Let G be a finite subgroup of purely monomial k-automorphisms of k(x 1 ,...., x n ). The rationality problem of the G-action is the problem of whether the G-fixed field k(x 1 ,....,x n ) G is k-rational, i.e., purely transcendental over k, or not. In 1994, M. Hajja and M. Kang gave a positive answer for the rationality problem of the three-dimensional purely monomial group actions except one case. We show that the remaining case is also affirmative.

A Geometric Framework for the Subfield Problem of Generic Polynomials Via Tschirnhausen Transformation

Let k be an arbitrary field. We study a general method to solve the subfield problem of generic polynomials for the symmetric groups over k via Tschirnhausen transformation. Based on the general result in the former part, we give an explicit solution to the field isomorphism problem and the subfield problem of cubic generic polynomials for \( \mathfrak{S} \)3 and C3 over k. As an application of the cubic case, we also give several sextic generic polynomials over k.

Families of cyclic polynomials obtained from geometric generalization of Gaussian period relations

A general method of constructing families of cyclic polynomials over Q with more than one parameter will be discussed, which may be called a geometric generalization of the Gaussian period relations. Using this, we obtain explicit multi-parametric families of cyclic polynomials over Q of degree 3 < e ≤ 7. We also give a simple family of cyclic polynomials with one parameter in each case, by specializing our parameters.

Rationality Problem for Algebraic Tori

We give the complete stably rational classification of algebraic tori of dimensions

Noether’s problem and ℚ-generic polynomials for the normalizer of the 8-cycle in ₈ and its subgroups

4

ON THE FIELD INTERSECTION PROBLEM OF SOLVABLE QUINTIC GENERIC POLYNOMIALS

and

On the Simplest Quartic Fields and Related Thue Equations

5

SOME DIOPHANTINE PROBLEMS ARISING FROM THE ISOMORPHISM PROBLEM OF GENERIC POLYNOMIALS

over a field

Noether’s problem and unramified Brauer groups

k

Quasi-monomial actions and some 4-dimensional rationality problems ☆

. In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank