Isolations of cubic lattices from their proper sublattices

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 $\epsilon>0$.