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

TL;DR

This paper extends Hermite reduction theory to totally real number fields to analyze the growth of g-invariants of their rings of integers, providing the first sub-exponential bounds for these invariants beyond the integers.

Contribution

It generalizes HKZ-reduction to number fields and establishes sub-exponential bounds for g-invariants of rings of integers with class number one.

Findings

Growth of g-invariants is at most exponential of √n for class number 1 fields.

First sub-exponential upper bounds for g-invariants over rings of integers other than Z.

Extension of Hermite reduction theory to totally real number fields.

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 {\em $g$-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 $\mathbb Z$.