On near-neighbor quadratic forms (Q1770287)

A quadratic form \(\varphi\) over \(F\) is called a quasi-neighbor if \((\varphi_{F(\varphi)})_{an}\) is \(F(\varphi)\)-similar to an \(F\)-form \(\psi\). In this case the possible forms \(\psi\) (up to \(F\)-similarity) make up a set \(S(\psi)\). An easy determinant argument shows that any quasi-neighbor \(\psi\) of odd dimension is a neighbor with \(S(\varphi)= \{\varphi'\}\) where \(\varphi'\) is the complementary form of \(\varphi\). The main result of the paper is a full (resp. partial) description of all quasi-neighbors \(\varphi\) with \(\dim\varphi\leq 8\) (resp. \(\leq 12\)). In particular, \(\varphi\) is not always a neighbor. Furthermore, the problem \(| S(\varphi)|= 1?\) is studied and solved for certain cases of small height \(h(\varphi)\leq 3\). The paper is well written. The proofs use earlier results of the author and of Hoffmann as well as methods and results of Izhboldin (on motivic equivalence), Izhboldin-Kersten (on excellent forms), Kahn (on descent of quadratic forms) and others.