Representation of polynomials as products of two values of a quadratic form
Let \(k\) be a field of characteristic different from \(2\) and \(p\) a monic irreducible polynomial over \(k\). For a quadratic form \(\varphi\) over \(k\), the author considers the following statement (A): if \(\varphi\) is isotropic over the residue field \(k_p = k[t]/p(t)\), then the form \(\varphi \perp -p(t) \varphi\) is isotropic over the rational function field \(k(t)\). While (A) holds true for quadratic polynomials and for odd degree polynomials, the author proves that for a suitably chosen base field \(k\) and any \(n \geq 2\) there exists an irreducible monic polynomial \(p\) of degree \(2n\) for which (A) is false. In addition, the anisotropic part of the form \(\varphi_{k_p}\) is not defined over \(k\), so that \(k_p/k\) is a nonexcellent field extension of degree \(2n\). A weaker variant of (A), to which the author does not know any counterexamples, is also considered. This amounts to requiring that, if \(\varphi\) becomes isotropic over a finite field extension \(\ell\) of \(k\), then there exists a monic irreducible polynomial \(p\) over \(k\) such that \(\ell = k_p\) and the form \(\varphi \perp -p(t) \varphi\) is isotropic over \(k(t)\). It is proved that this holds when \(\ell\) is a biquadratic extension of \(k\) and \(\varphi\) is a form of dimension \(4\).
scientific article; zbMATH DE number 6108400