The Pythagoras number of fields of transcendence degree $1$ over $\mathbb{Q}$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913925865406464 |
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| author | Benoist, Olivier |
| author_facet | Benoist, Olivier |
| contents | We show that any sum of squares in a field of transcendence degree $1$ over $\mathbb{Q}$ is a sum of $5$ squares, answering a question of Pop and Pfister. We deduce this result from a representation theorem, in $k(C)$, for quadratic forms of rank $\geq 5$ with coefficients in $k$, where $C$ is a curve over a number field $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21380 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Pythagoras number of fields of transcendence degree $1$ over $\mathbb{Q}$ Benoist, Olivier Algebraic Geometry Number Theory 11E25, 11E12, 14G25, 14G12 We show that any sum of squares in a field of transcendence degree $1$ over $\mathbb{Q}$ is a sum of $5$ squares, answering a question of Pop and Pfister. We deduce this result from a representation theorem, in $k(C)$, for quadratic forms of rank $\geq 5$ with coefficients in $k$, where $C$ is a curve over a number field $k$. |
| title | The Pythagoras number of fields of transcendence degree $1$ over $\mathbb{Q}$ |
| topic | Algebraic Geometry Number Theory 11E25, 11E12, 14G25, 14G12 |
| url | https://arxiv.org/abs/2506.21380 |