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

Barry R. Smith

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arXiv · 1703.01993

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

Barry R. Smith

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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.

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Barry R. Smith. 2017-03-06. Constructing minimal periods of quadratic irrationalities in Zagier's reduction theory. https://arxiv.org/abs/1703.01993

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