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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Hermite reduction and a Waring’s problem for integral quadratic forms over number fields


Authors: Wai Kiu Chan and María Inés Icaza
Journal: Trans. Amer. Math. Soc. 374 (2021), 2967-2985
MSC (2020): Primary 11E12, 11E25, 11E39
DOI: https://doi.org/10.1090/tran/8298
Published electronically: February 2, 2021
MathSciNet review: 4223039
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Abstract: We generalize the Hermite-Korkin-Zolotarev (HKZ) reduction theory of positive definite quadratic forms over $\mathbb Q$ and its balanced version introduced recently by Beli-Chan-Icaza-Liu to positive definite quadratic forms over a totally real number field $K$. We apply the balanced HKZ-reduction theory to study the growth of the $g$-invariants of the ring of integers of $K$. More precisely, for each positive integer $n$, let $\mathcal {O}$ be the ring of integers of $K$ and $g_{\mathcal {O}}(n)$ be the smallest integer such that every sum of squares of $n$-ary $\mathcal {O}$-linear forms must be a sum of $g_{\mathcal {O}}(n)$ squares of $n$-ary $\mathcal {O}$-linear forms. We show that when $K$ has class number 1, the growth of $g_{\mathcal {O}}(n)$ is at most an exponential of $\sqrt {n}$. This extends the recent result obtained by Beli-Chan-Icaza-Liu on the growth of $g_{\mathbb Z}(n)$ and gives the first sub-exponential upper bound for $g_{\mathcal {O}}(n)$ for rings of integers $\mathcal {O}$ other than $\mathbb Z$.


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Additional Information

Wai Kiu Chan
Affiliation: Department of Mathematics and Computer Science, Wesleyan University, Middletown, Connecticut 06459
MR Author ID: 336822
Email: wkchan@wesleyan.edu

María Inés Icaza
Affiliation: Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile
Email: icazap@inst-mat.utalca.cl

Keywords: Waring’s problem, integral quadratic forms, sums of squares, reduction theory
Received by editor(s): January 23, 2020
Received by editor(s) in revised form: September 4, 2020
Published electronically: February 2, 2021
Dedicated: In memory of John Hsia, a mentor and a friend, who taught us everything we know about quadratic forms.
Article copyright: © Copyright 2021 American Mathematical Society