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

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- by Wai Kiu Chan and María Inés Icaza PDF
- Trans. Amer. Math. Soc.
**374**(2021), 2967-2985 Request permission

## 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
- Received by editor(s): January 23, 2020
- Received by editor(s) in revised form: September 4, 2020
- Published electronically: February 2, 2021
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 2967-2985 - MSC (2020): Primary 11E12, 11E25, 11E39
- DOI: https://doi.org/10.1090/tran/8298
- MathSciNet review: 4223039

Dedicated: In memory of John Hsia, a mentor and a friend, who taught us everything we know about quadratic forms.