Certain linear combinations of two Pfister forms and the isotropy problem (Q5949811)

Let \(F\) be a field of characteristic \(\neq 2\). The author studies the problem of when an anisotropic quadratic form \(\varphi\) over \(F\) becomes isotropic over the function field \(F(\psi)\) of another form \(\psi\) over \(F\). This problem has been studied in numerous papers by various authors in the past (including the author himself, the reviewer, Izhboldin, Karpenko and others), most notably in cases where \(\varphi\) is of small dimension or of a special type. In this article, the author considers the case where \(\varphi\) can be written as the orthogonal sum of two forms that are similar to an \(n\)-fold and an \(m\)-fold Pfister form, respectively, where \(n\geq m\). The idea is to define an auxiliary form \(\eta\) related to \(\varphi\) and to show that \(\varphi\) becomes an anisotropic Pfister neighbor of a Pfister form \(\pi_0\) defined over \(F\) after passing to a certain field \(F_{\epsilon}\) in the generic splitting tower of \(\eta\). The isotropy of \(\varphi\) over \(F(\psi)\) implies that \(\psi\) becomes similar to a subform of \(\pi_0\) over \(F_{\epsilon}\). A descent argument then yields the desired information on the form \(\psi\). One of the main results characterizes those forms \(\psi\) of dimension \(\geq 2^{n+1}\) such that \(\varphi\) becomes isotropic over \(F(\psi)\). More precise results are obtained in the cases where \(\varphi\) is divisible by an \((m-1)\)-fold Pfister form and for \(n\leq 3\). The author also provides partial results for orthogonal sums of two forms similar to the pure parts of Pfister forms by reducing this case to the previous one. NEWLINENEWLINENEWLINESome of the techniques used in the proofs depend on Galois cohomological results related to the Milnor conjecture, which are well-established in small degrees and which, for higher degrees, appear in preprints by \textit{V. Voevodsky} [On \(2\)-torsion in motivic cohomology, \(K\)-theory Preprint Archives 502 (2001)] and \textit{D. Orlov}, \textit{A. Vishik}, and \textit{V. Voevodsky} [An exact sequence for Milnor's \(K\)-theory with applications to quadratic forms, \(K\)-theory Preprint Archives 454 (2000)].