Universal Quadratic Forms and Indecomposables over Biquadratic Fields

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 quadra...

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Published in:arXiv.org 2018-02
Main Authors: Čech, Martin, Lachman, Dominik, Svoboda, Josef, Tinková, Magdaléna, Zemková, Kristýna
Format: Article
Language:English
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Integers
Lower bounds
Mathematical analysis
Number theory
Quadratic forms
Queuing theory
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Lachman, Dominik
Svoboda, Josef
Tinková, Magdaléna
Zemková, Kristýna
description 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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subjects Integers
Lower bounds
Mathematical analysis
Number theory
Quadratic forms
Queuing theory
title Universal Quadratic Forms and Indecomposables over Biquadratic Fields
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