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arXiv · 2105.08860

Pythagoras numbers of orders in biquadratic fields

Jakub Krásenský·Martin Raška·Ester Sgallová
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Abstract

We examine the Pythagoras number $\mathcal{P}(\mathcal{O}_K)$ of the ring of integers $\mathcal{O}_K$ in a totally real biquadratic number field $K$. We show that the known upper bound $7$ is attained in a large and natural infinite family of such fields. In contrast, for almost all fields $\mathbb{Q}(\sqrt5, \sqrt{s})$ we prove $\mathcal{P}(\mathcal{O}_K)=5$. Further we show that $5$ is a lower bound for all but seven fields $K$ and $6$ is a lower bound in an asymptotic sense.

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Jakub Krásenský, Martin Raška, Ester Sgallová. 2021-05-19. Pythagoras numbers of orders in biquadratic fields. https://doi.org/10.1016/j.exmath.2022.06.002

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