The Pythagoras number of a rational function field in two variables
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | |
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| _version_ | 1866912027317895168 |
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| author | Becher, Karim Johannes Daans, Nicolas Grimm, David Manzano-Flores, Gonzalo Zaninelli, Marco |
| author_facet | Becher, Karim Johannes Daans, Nicolas Grimm, David Manzano-Flores, Gonzalo Zaninelli, Marco |
| contents | We prove that every sum of squares in the rational function field in two variables $K(X,Y)$ over a hereditarily pythagorean field $K$ is a sum of $8$ squares. More precisely, we show that the Pythagoras number of every finite extension of $K(X)$ is at most $5$. The main ingredients of the proof are a local-global principle for quadratic forms over function fields in one variable over a complete rank-$1$ valued field due to V. Mehmeti and a valuation theoretic characterization of hereditarily pythagorean fields due to L. Bröcker. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_11425 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Pythagoras number of a rational function field in two variables Becher, Karim Johannes Daans, Nicolas Grimm, David Manzano-Flores, Gonzalo Zaninelli, Marco Number Theory 11E81, 12D15, 12J10 We prove that every sum of squares in the rational function field in two variables $K(X,Y)$ over a hereditarily pythagorean field $K$ is a sum of $8$ squares. More precisely, we show that the Pythagoras number of every finite extension of $K(X)$ is at most $5$. The main ingredients of the proof are a local-global principle for quadratic forms over function fields in one variable over a complete rank-$1$ valued field due to V. Mehmeti and a valuation theoretic characterization of hereditarily pythagorean fields due to L. Bröcker. |
| title | The Pythagoras number of a rational function field in two variables |
| topic | Number Theory 11E81, 12D15, 12J10 |
| url | https://arxiv.org/abs/2302.11425 |