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arXiv · 2311.07486

Combing a hedgehog over a field

Alexey Ananyevskiy·Marc Levine
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Abstract

We investigate the question of the existence of a non-vanishing section of the tangent bundle on a smooth affine quadric hypersurface $Q^o$ over a given perfect field $k$. In case $Q^o$ admits a $k$-rational point, we give necessary and sufficient conditions for such existence. We apply these conditions in a number of examples, including the case of the algebraic $n$-sphere over $k$, $S^n_k\subset \mathbb{A}^{n+1}_k$, defined by the equation $\sum_{i=1}^{n+1}x_i^2=1$.

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Alexey Ananyevskiy, Marc Levine. 2023-11-13. Combing a hedgehog over a field. https://doi.org/10.2140/ant.2025.19.2409

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