Isotropy of quadratic forms over function fields in characteristic 2 (Q6600719)

An important aspect of the algebraic theory of quadratic forms over fields is the study of the isotropy behaviour of a quadratic form defined over a base field \(F\) when passing to a field extension \(K\). Of particular interest is the case where \(K\) is the function field of another quadratic form, or, more generally, of an irreducible polynomial (a quadratic form may be considered as a homogeneous polynomial of degree \(2\)). There is an extensive body of literature on that problem, mainly in the case where the characteristic of \(F\) is not \(2\). More recently, this question has also been studied in characteristic \(2\) and the present paper is concerned with extending known important results from characteristic not \(2\) to characteristic \(2\). One then would like to systematically include the case of singular quadratic forms which causes additional complications and necessitates more refined arguments.\N\NThe main result of the present paper reads as follows. Let \(F\) be a field of characteristic \(2\), \(\varphi\) be a nondefective quadratic form over \(F\) representing \(1\) (where nondefective means having an anisotropic radical). For a field extension \(E/F\), let \(D_E(\varphi)\subset E^*\) denote the set of nonzero elements represented by \(\varphi\) over \(E\), \(D_E(\varphi)^m\) the set of \(m\)-fold products of such elements and \(\langle D_E(\varphi)\rangle\) the subgroup of \(E^*\) generated by such elements. Let \(f\) be a nonzero polynomial in \(F[X]\) (where \(X=(X_1,\ldots,X_n)\)) with leading coefficient \(a\in F^*\) (with respect to the lexicographical ordering of monomials). Then \(f\in \langle D_{F(X)}(\varphi)\rangle\) if and only if \(a\in \langle D_F(\varphi)\rangle\) and \(\varphi\) is isotropic over the function field \(F(g)\) for every monic irreducible \(g\in F[X]\) that appears in a factorization of \(f\) to an odd power (Theorem 3.7). In characteristic not \(2\), this result is originally due to \textit{S. Roussey} [J. Algebra 616, 49--67 (2023; Zbl 1506.11047)].\N\NApplied to the case where \(f=\psi(X)\) is the homogeneous polynomial of a nondefective quadratic form \(\psi\) of dimension \(n\), one deduces that \(\varphi\) becomes isotropic over \(F(\psi)\) if and only if \(a\psi(X)\in D_{F(X)}(\varphi)^2\) for every \(a\in D_F(\psi)\). This yields a nice criterion for stable birational equaivalence of these quadratic forms \(\varphi\) and \(\psi\), where stable birational equivalence can be characterized by the fact that \(\varphi\) is isotropic over \(F(\psi)\) and \(\psi\) is isotropic over \(F(\varphi)\). This now holds if and only if \(D_{F(X)}(\varphi)^2=D_{F(X)}(\psi)^2\) (where \(X=(X_1,\ldots,X_n)\) with \(n\geq\max\{\dim\varphi,\dim\psi\}\)), or equivalently, that \(D_{E}(\varphi)^2=D_{E}(\psi)^2\) for every field extension \(E/F\) over which \(\varphi\) and \(\psi\) stay nondefective.\N\NA further application is the following nice result. Let \(\varphi\) and \(\psi\) be quadratic forms and \(\pi\) be a bilinear Pfister form over \(F\). If \(\varphi\) is isotropic over \(F(\psi)\), then \(\pi\otimes\varphi\) is isotropic over \(F(\pi\otimes\psi)\).