Similarity of quadratic forms and half-neighbors
The quadratic forms \(\phi\) and \(\psi\) of dimension \(2^n\) are called half-neighbors iff \(\phi \perp -a \psi\) is similar to an \((n + 1)\)-fold Pfister form for some scalar \(a\). If \(n \leq 2\) then every half-neighbors are similar but for \(n \geq 3\) counterexamples have been given by Izhboldin and by the author. So the aim of the paper under review is to explain under which circumstances half-neighbors of dimension \(2^n\) are similar for \(n \geq 3\). The results are: If \(\widetilde u(F) < 2^n\), \(n-2 \geq m \geq 1\) and \(I_t^{m + 1} F = 0\) then if one of the half-neighbors is congruent to an \(m\)-fold Pfister form modulo \(I^{m + 1}F\) then they are similar. If \(n \geq m \geq 2\) and \(F\) is \(m\)-linked, then every half-neighbors in \(I^mF\) of dimension \(2^n\) are similar. Moreover, the author discusses dependencies between various equivalence relations for quadratic forms: similarity, birational equivalence, stable birational equivalence and half-neighbor equivalence. Some partial results are here established.
scientific article; zbMATH DE number 1218696
scientific article