# Perfect unary forms over real quadratic fields

Dan Yasaki[1]

• [1] Department of Mathematics and Statistics The University of North Carolina at Greensboro Greensboro, NC 27412, USA
• Volume: 25, Issue: 3, page 759-775
• ISSN: 1246-7405

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## Abstract

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Let $F=ℚ\left(\sqrt{d}\right)$ be a real quadratic field with ring of integers $𝒪$. In this paper we analyze the number ${h}_{d}$ of ${GL}_{1}\left(𝒪\right)$-orbits of homothety classes of perfect unary forms over $F$ as a function of $d$. We compute ${h}_{d}$ exactly for square-free $d\le 200000$. By relating perfect forms to continued fractions, we give bounds on ${h}_{d}$ and address some questions raised by Watanabe, Yano, and Hayashi.

## How to cite

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Yasaki, Dan. "Perfect unary forms over real quadratic fields." Journal de Théorie des Nombres de Bordeaux 25.3 (2013): 759-775. <http://eudml.org/doc/275681>.

@article{Yasaki2013,
abstract = {Let $F = \mathbb\{Q\}(\sqrt\{d\})$ be a real quadratic field with ring of integers $\mathcal\{O\}$. In this paper we analyze the number $h_d$ of $\operatorname\{GL\}_1(\mathcal\{O\})$-orbits of homothety classes of perfect unary forms over $F$ as a function of $d$. We compute $h_d$ exactly for square-free $d \le 200000$. By relating perfect forms to continued fractions, we give bounds on $h_d$ and address some questions raised by Watanabe, Yano, and Hayashi.},
affiliation = {Department of Mathematics and Statistics The University of North Carolina at Greensboro Greensboro, NC 27412, USA},
author = {Yasaki, Dan},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {quadratic forms; perfect forms; continued fractions; real quadratic fields; continued fraction},
language = {eng},
month = {11},
number = {3},
pages = {759-775},
publisher = {Société Arithmétique de Bordeaux},
title = {Perfect unary forms over real quadratic fields},
url = {http://eudml.org/doc/275681},
volume = {25},
year = {2013},
}

TY - JOUR
AU - Yasaki, Dan
TI - Perfect unary forms over real quadratic fields
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2013/11//
PB - Société Arithmétique de Bordeaux
VL - 25
IS - 3
SP - 759
EP - 775
AB - Let $F = \mathbb{Q}(\sqrt{d})$ be a real quadratic field with ring of integers $\mathcal{O}$. In this paper we analyze the number $h_d$ of $\operatorname{GL}_1(\mathcal{O})$-orbits of homothety classes of perfect unary forms over $F$ as a function of $d$. We compute $h_d$ exactly for square-free $d \le 200000$. By relating perfect forms to continued fractions, we give bounds on $h_d$ and address some questions raised by Watanabe, Yano, and Hayashi.
LA - eng
KW - quadratic forms; perfect forms; continued fractions; real quadratic fields; continued fraction
UR - http://eudml.org/doc/275681
ER -

## References

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11. François Sigrist, Cyclotomic quadratic forms. J. Théor. Nombres Bordeaux, 12(2) (2000), 519–530. Colloque International de Théorie des Nombres (Talence, 1999). Zbl0977.11029MR1823201
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