# On a Waring's problem for integral quadratic and hermitian forms

## TL;DR

This paper establishes that the minimal number of squares needed to represent integral quadratic and hermitian forms grows at most exponentially with the square root of the number of variables, improving previous bounds.

## Contribution

It provides new upper bounds on the growth of the Waring's problem functions for quadratic and hermitian forms, reducing from exponential in n to exponential in √n.

## Key findings

- Growth of g_Z(n) is at most exponential of √n
- Growth of g_O*(n) is at most exponential of √n when class number is 1
- Improves bounds on s-integral lattices by Conway-Sloane and Kim-Oh

## 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 as $n$ goes to infinity, 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 in 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 results of Conway-Sloane and Kim-Oh on $s$-integral lattices.