Bounding the Pythagoras number of a field by $2^n+1$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866909100886982656 |
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| author | Becher, Karim Johannes Zaninelli, Marco |
| author_facet | Becher, Karim Johannes Zaninelli, Marco |
| contents | Given a positive integer $n$, a sufficient condition on a field is given for bounding its Pythagoras number by $2^n+1$. The condition is satisfied for $n=1$ by function fields of curves over iterated formal power series fields over $\mathbb{R}$, as well as by finite field extensions of $\mathbb{R}(\!(t_0,t_1)\!)$. In both cases, one retrieves the upper bound $3$ on the Pythagoras number. The new method presented here might help to establish more generally $2^n+1$ as an upper bound for the Pythagoras number of function fields of curves over $\mathbb{R}(\!(t_1,\dots,t_n)\!)$ and for finite field extensions of $\mathbb{R}(\!(t_0,\dots,t_n)\!)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_02384 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Bounding the Pythagoras number of a field by $2^n+1$ Becher, Karim Johannes Zaninelli, Marco Number Theory Commutative Algebra Algebraic Geometry 11E25 Given a positive integer $n$, a sufficient condition on a field is given for bounding its Pythagoras number by $2^n+1$. The condition is satisfied for $n=1$ by function fields of curves over iterated formal power series fields over $\mathbb{R}$, as well as by finite field extensions of $\mathbb{R}(\!(t_0,t_1)\!)$. In both cases, one retrieves the upper bound $3$ on the Pythagoras number. The new method presented here might help to establish more generally $2^n+1$ as an upper bound for the Pythagoras number of function fields of curves over $\mathbb{R}(\!(t_1,\dots,t_n)\!)$ and for finite field extensions of $\mathbb{R}(\!(t_0,\dots,t_n)\!)$. |
| title | Bounding the Pythagoras number of a field by $2^n+1$ |
| topic | Number Theory Commutative Algebra Algebraic Geometry 11E25 |
| url | https://arxiv.org/abs/2210.02384 |