Isolations of cubic lattices from their proper sublattices

TL;DR

This paper investigates the minimal rank of quadratic forms called isolations that represent all subforms of a sum of squares except the form itself, providing exact values for small cases and bounds for general n.

Contribution

It establishes the exact minimal ranks for isolations of sums of squares and derives bounds, including asymptotic growth rates, for all positive integers n.

Findings

Iso(I_2)=5

Iso(I_3)=6

Iso(I_n) grows at least as fast as n^{3/2 - epsilon}

Abstract

A (positive definite and integral) quadratic form is called {\it an isolation} of a quadratic form $f$ if it represents all subforms of $f$ except for $f$ itself. The minimum rank of isolations of a quadratic form $f$ is denoted, if it exists, by $\text{Iso}(f)$. In this article, we show that $\text{Iso}(I_2)=5$ and $\text{Iso}(I_3)=6$, where $I_n=x_1^2+\dots+x_n^2$ is the sum of $n$ squares for any positive integer $n$. After proving that there always exists an isolation of $I_n$ for any positive integer $n$, we provide some explicit lower and upper bounds for $\text{Iso}(I_n)$. In particular, we show that $\text{Iso}(I_n) \in \Omega(n^{\frac32-\epsilon})$ for any $\epsilon>0$.