Field discriminants of cyclotomic period equations

TL;DR

This paper reveals connections between cyclotomic period equations and quadratic map orbits, providing an explicit formula for their field discriminants that enhances understanding of number field structures and orbit organization.

Contribution

It introduces an analytical expression for the discriminant of cyclotomic period equations, linking number theory with dynamical systems and improving database construction methods.

Findings

Orbital equations coincide with cyclotomic period equations.

Explicit discriminant formula aids in number field database analysis.

Insights into periodic orbit organization of quadratic maps.

Abstract

We show that several orbital equations and orbital clusters of the quadratic (logistic) map coincide surprisingly with cyclotomic {\it period equations}, polynomials whose roots are Gaussian periods. An analytical expression for the field discriminant of period equations is obtained and applied to discover and to fill gaps in number field databases constructed by numerical search processes. Such expression allows easy assess to inessential divisors of conventional discriminants and sheds light into why numerical construction of databases is a hard problem. It also provides significant information about the organization of periodic orbits of the quadratic map.