Composition laws of binary quadratic forms and isolations of quadratic forms

초록

A positive definite and integral quadratic form f is called irrecoverable if there is a quadratic form F such that it represents all proper subforms of f, whereas it does not represent f itself. In this case, F is called an isolation of f. In this article, we prove that there does not exist a binary isolation of any unary quadratic form. We also prove that there does not exist a ternary isolation of any binary quadratic form. Furthermore, if the form class group of a primitive binary quadratic form has no element of order 4, then the discriminant of any quaternary isolation of it, if exists, is a square of an integer. The composition laws of primitive binary quadratic forms play an essential role in the proofs of the results. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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