The u-invariant of function fields in one variable
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916900718510080 |
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| author | Becher, Karim Johannes Daans, Nicolas Mehmeti, Vlerë |
| author_facet | Becher, Karim Johannes Daans, Nicolas Mehmeti, Vlerë |
| contents | The u-invariant of a field is the largest dimension of an anisotropic quadratic torsion form over the field. In this article we obtain a bound on the u-invariant of function fields in one variable over a henselian valued field with arbitrary value group and with residue field of characteristic different from 2. This generalises a theorem due to Harbater, Hartmann and Krashen and its extension due to Scheiderer. Their result covers the special case where the valuation is discrete.
We further give a new proof of a theorem due to Parimala and Suresh bounding by 8 the u-invariant of a function field in one variable over any henselian discretely valued field of characteristic 0 with perfect residue field of characteristic 2. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_13086 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The u-invariant of function fields in one variable Becher, Karim Johannes Daans, Nicolas Mehmeti, Vlerë Number Theory 11E04, 11E81, 12D15, 12J10, 14H05 The u-invariant of a field is the largest dimension of an anisotropic quadratic torsion form over the field. In this article we obtain a bound on the u-invariant of function fields in one variable over a henselian valued field with arbitrary value group and with residue field of characteristic different from 2. This generalises a theorem due to Harbater, Hartmann and Krashen and its extension due to Scheiderer. Their result covers the special case where the valuation is discrete. We further give a new proof of a theorem due to Parimala and Suresh bounding by 8 the u-invariant of a function field in one variable over any henselian discretely valued field of characteristic 0 with perfect residue field of characteristic 2. |
| title | The u-invariant of function fields in one variable |
| topic | Number Theory 11E04, 11E81, 12D15, 12J10, 14H05 |
| url | https://arxiv.org/abs/2502.13086 |