Togliatti systems and Galois coverings

TL;DR

This paper investigates cyclic group-invariant homogeneous ideals in three-variable polynomial rings, showing they are monomial Togliatti systems, with minimality conditions and connections to Ceva configurations, especially for prime or prime power degrees.

Contribution

It characterizes invariant ideals as monomial Togliatti systems, determines minimality criteria, and links these systems to geometric Ceva configurations, providing a complete description for prime or prime power degrees.

Findings

All such ideals are monomial Togliatti systems.

Minimality holds under specific diagonal actions.

Complete classification for prime or prime power degrees.

Abstract

We study the homogeneous artinian ideals of the polynomial ring $K[x,y,z]$, generated by the homogenous polynomials of degree $d$ which are invariant under an action of the cyclic group $\mathbb Z/d\mathbb Z$, for any $d\geq 3$. We prove that they are all monomial Togliatti systems, and that they are minimal if the action is defined by a diagonal matrix having on the diagonal $(1, e, e^a)$, where $e$ is a primitive $d$-th root of the unity. We get a complete description when $d$ is prime or a power of a prime. We also establish the relation of these systems with linear Ceva configurations.