A geometric reduction theory for indefinite binary quadratic forms over $\mathbb{Z}[\lambda]$ Article Swipe
Gauss' classical reduction theory for indefinite binary quadratic forms over $\mathbb{Z}$ has originally been proven by means of purely algebraic and arithmetic considerations. It was later discovered that this reduction theory is closely related to a certain symbolic dynamics for the geodesic flow on the modular surface, and hence can also be deduced geometrically. In this article, we use certain symbolic dynamics for the geodesic flow on Hecke triangle surfaces (also the non-arithmetic ones) to develop reduction theories for the indefinite binary quadratic forms associated to Hecke triangle groups. Moreover, we propose an algorithm to decide for any $g\in {\rm PSL}_2(\mathbb{R})$ whether or not $g$ is contained in the Hecke triangle group under consideration, and provide an upper estimate for its run time.
Related Topics
Concepts
Mathematics
Binary quadratic form
Reduction (mathematics)
Algebraic number
Binary number
Quadratic equation
Pure mathematics
Quadratic field
Geodesic
Gauss
Quadratic form (statistics)
Modular form
Combinatorics
Discrete mathematics
Arithmetic
Quadratic function
Mathematical analysis
Geometry
Physics
Quantum mechanics
Metadata
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/1512.08090
- https://arxiv.org/pdf/1512.08090
- OA Status
- green
- Cited By
- 1
- References
- 6
- Related Works
- 20
- OpenAlex ID
- https://openalex.org/W2261496677
All OpenAlex metadata
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- OpenAlex ID
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https://openalex.org/W2261496677Canonical identifier for this work in OpenAlex
- Title
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A geometric reduction theory for indefinite binary quadratic forms over $\mathbb{Z}[\lambda]$Work title
- Type
-
preprintOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2015Year of publication
- Publication date
-
2015-12-26Full publication date if available
- Authors
-
Anke Pohl, Verena SpratteList of authors in order
- Landing page
-
https://arxiv.org/abs/1512.08090Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/1512.08090Direct link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
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greenOpen access status per OpenAlex
- OA URL
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https://arxiv.org/pdf/1512.08090Direct OA link when available
- Concepts
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Mathematics, Binary quadratic form, Reduction (mathematics), Algebraic number, Binary number, Quadratic equation, Pure mathematics, Quadratic field, Geodesic, Gauss, Quadratic form (statistics), Modular form, Combinatorics, Discrete mathematics, Arithmetic, Quadratic function, Mathematical analysis, Geometry, Physics, Quantum mechanicsTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
1Total citation count in OpenAlex
- Citations by year (recent)
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2018: 1Per-year citation counts (last 5 years)
- References (count)
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6Number of works referenced by this work
- Related works (count)
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20Other works algorithmically related by OpenAlex
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| abstract_inverted_index.algorithm | 93 |
| abstract_inverted_index.classical | 1 |
| abstract_inverted_index.contained | 106 |
| abstract_inverted_index.quadratic | 7, 82 |
| abstract_inverted_index.reduction | 2, 29, 76 |
| abstract_inverted_index.arithmetic | 21 |
| abstract_inverted_index.associated | 84 |
| abstract_inverted_index.discovered | 26 |
| abstract_inverted_index.indefinite | 5, 80 |
| abstract_inverted_index.originally | 12 |
| abstract_inverted_index.$\mathbb{Z}$ | 10 |
| abstract_inverted_index.consideration, | 113 |
| abstract_inverted_index.geometrically. | 53 |
| abstract_inverted_index.non-arithmetic | 72 |
| abstract_inverted_index.considerations. | 22 |
| abstract_inverted_index.PSL}_2(\mathbb{R})$ | 100 |
| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 2 |
| citation_normalized_percentile |