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 | References | Similar Articles | Additional Information
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