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arXiv · 1802.07811

Universal Quadratic Forms and Indecomposables over Biquadratic Fields

Martin Čech·Dominik Lachman·Josef Svoboda·Magdaléna Tinková·Kristýna Zemková
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Abstract

The aim of this article is to study (additively) indecomposable algebraic integers $\mathcal O_K$ of biquadratic number fields $K$ and universal totally positive quadratic forms with coefficients in $\mathcal O_K$. There are given sufficient conditions for an indecomposable element of a quadratic subfield to remain indecomposable in the biquadratic number field $K$. Furthermore, estimates are proven which enable algorithmization of the method of escalation over $K$. These are used to prove, over two particular biquadratic number fields $\mathbb{Q}(\sqrt{2}, \sqrt{3})$ and $\mathbb{Q}(\sqrt{6}, \sqrt{19})$, a lower bound on the number of variables of a universal quadratic forms, verifying Kitaoka's conjecture.

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Martin Čech, Dominik Lachman, Josef Svoboda, Magdaléna Tinková, Kristýna Zemková. 2018-02-21. Universal Quadratic Forms and Indecomposables over Biquadratic Fields. https://arxiv.org/abs/1802.07811

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