Julia Robinson numbers and arithmetical dynamic of quadratic polynomials

TL;DR

This paper investigates the structure of Robinson numbers for rings of totally real algebraic integers, constructing infinitely many fields where the set of Robinson numbers forms an interval different from the previously known cases.

Contribution

It introduces new examples of fields with Robinson number sets that are intervals not equal to [4,+∞), expanding understanding of their arithmetical properties.

Findings

Constructed infinitely many fields with Robinson number sets as intervals.

Demonstrated the Robinson number set can differ from the known forms.

Extended the classification of Robinson numbers for totally real algebraic integer rings.

Abstract

For rings $\mathcal{O}_K$ of totally real algebraic integers, J. Robinson defined a set which is always $\{+\infty\}$ or of the form $[\lambda,+\infty)$ or $(\lambda,+\infty)$ for some real number $\lambda\ge4$. All known examples give either $\{+\infty\}$ or $[4,+\infty)$. In this paper, we construct infinitely many fields such that the set is an interval, but not equal to $[4,+\infty)$.