# Constructing minimal periods of quadratic irrationalities in Zagier's reduction theory

## TL;DR

This paper introduces a new map for Zagier's reduction theory of quadratic irrationalities, creating a bridge with Dirichlet's Gauss reduction and revealing that Zagier's theory refines Gauss's approach.

## Contribution

It defines an analogue of the minimal period map for Zagier-reduced forms and establishes a near-embedding of Gauss's reduction structure into Zagier's, enhancing understanding of their relationship.

## Key findings

- A new map for Zagier-reduced forms is constructed.
- A near-embedding from Gauss to Zagier reduction structures is established.
- Zagier's reduction is shown to be a refinement of Gauss's reduction.

## Abstract

Dirichlet's version of Gauss's reduction theory for indefinite binary quadratic forms includes a map from Gauss-reduced forms to strings of natural numbers. It attaches to a form the minimal period of the continued fraction of a quadratic irrationality associated with the form. When Zagier developed his own reduction theory, parallel to Dirichlet's, he omitted an analogue of this map. We define a new map on Zagier-reduced forms that serves as this analogue. We also define a map from the set of Gauss-reduced forms into the set of Zagier-reduced forms that gives a near-embedding of the structure of Gauss's reduction theory into that of Zagier's. From this perspective, Zagier-reduction becomes a refinement of Gauss-reduction.