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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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On a Waring’s problem for integral quadratic and Hermitian forms
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by Constantin N. Beli, Wai Kiu Chan, María Inés Icaza and Jingbo Liu PDF
Trans. Amer. Math. Soc. 371 (2019), 5505-5527 Request permission

Abstract:

For each positive integer $n$, let $g_\mathbb {Z}(n)$ be the smallest integer such that if an integral quadratic form in $n$ variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of $g_\mathbb {Z}(n)$ squares of integral linear forms. We show that every positive definite integral quadratic form is equivalent to what we call a balanced Hermite–Korkin–Zolotarev-reduced form and use it to show that the growth of $g_\mathbb {Z}(n)$ is at most an exponential of $\sqrt {n}$. Our result improves the best known upper bound on $g_\mathbb {Z}(n)$ which is on the order of an exponential of $n$. We also define an analogous number $g_{\mathcal O}^*(n)$ for writing Hermitian forms over the ring of integers $\mathcal O$ of an imaginary quadratic field as sums of norms of integral linear forms, and when the class number of the imaginary quadratic field is $1$, we show that the growth of $g_{\mathcal O}^*(n)$ is at most an exponential of $\sqrt {n}$. We also improve on results of both Conway and Sloane and Kim and Oh on $s$-integrable lattices.
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Additional Information
  • Constantin N. Beli
  • Affiliation: Institute of Mathematics Simion Stoilow of the Romanian Academy, Calea Grivitei 21, RO-010702 Bucharest, Romania
  • MR Author ID: 718695
  • Email: Constantin.Beli@imar.ro
  • 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
  • Jingbo Liu
  • Affiliation: Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong
  • MR Author ID: 1166897
  • Email: jliu02@hku.hk
  • Received by editor(s): April 3, 2017
  • Received by editor(s) in revised form: October 4, 2017
  • Published electronically: September 28, 2018
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 371 (2019), 5505-5527
  • MSC (2010): Primary 11E12, 11E25, 11E39
  • DOI: https://doi.org/10.1090/tran/7571
  • MathSciNet review: 3937301