Isolations of the sum of two squares from its proper subforms

초록

For a (positive definite and integral) quadratic form f, a quadratic form is said to be 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 x2 is three, and there are exactly 15 ternary diagonal isolations of x2. Recently, it was proved that any quaternary quadratic form cannot be an isolation of the sum of two squares I2 = x2 + y2, and there are quinary isolations of I2. In this article, we prove that there are at most 231 quinary isolations of I2, which are listed in Table 1. Moreover, we prove that 14 quinary quadratic forms with dagger mark in Table 1 are isolations of I2. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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