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arXiv · 2506.21380

The Pythagoras number of fields of transcendence degree $1$ over $\mathbb{Q}$

Olivier Benoist
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Abstract

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 $k$.

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Olivier Benoist. 2025-06-26. The Pythagoras number of fields of transcendence degree $1$ over $\mathbb{Q}$. https://arxiv.org/abs/2506.21380

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