Isolations of the sum of two squares from its proper subforms

TL;DR

This paper investigates the concept of isolations of the sum of two squares quadratic form, establishing upper bounds on the number of such isolations in five variables and identifying specific forms that qualify.

Contribution

It proves that there are at most 231 quinary isolations of the sum of two squares and identifies 14 forms that are indeed isolations, advancing understanding of quadratic form representations.

Findings

Maximum of 231 quinary isolations of I_2

14 forms confirmed as isolations

No quaternary isolations of I_2 exist

Abstract

For a (positive definite and integral) quadratic form $f$, a quadratic form is said to be {\it an isolation of $f$ from its proper subforms} if it represents all proper subforms of $f$, but not $f$ itself. It was proved that the minimal rank of isolations of the square quadratic form $x^2$ is three, and there are exactly $15$ ternary diagonal isolations of $x^2$. Recently, it was proved that any quaternary quadratic form cannot be an isolation of the sum of two squares $I_2=x^2+y^2$, and there are quinary isolations of $I_2$. In this article, we prove that there are at most $231$ quinary isolations of $I_2$, which are listed in Table $1$. Moreover, we prove that $14$ quinary quadratic forms with dagger mark in Table $1$ are isolations of $I_2$.