Dimensions of anisotropic indefinite quadratic forms. II. (Q612981)

Let \(F\) be a field of characteristic not 2. Among the classical invariants of \(F\) related to quadratic forms are: \[ \begin{aligned} u(F) &=\sup\{ \dim\varphi\mid \text{\(\varphi\) is an anisotropic torsion form over \(F\)}\}\\ \tilde{u}(F) &=\sup\{ \dim \varphi\mid \text{\(\varphi\) is an anisotropic totally indefinite form over \(F\)}\}\\ p(F) &=\inf\{ n\mid \text{every sum of squares in \(F\) is a sum of \(n\) squares}\}. \end{aligned} \] The author gives several new bounds on \(\tilde{u}(F)\), for example: \[ \tilde{u}(F)\leq\frac{p(F)}{2}(u(F)+2). \] This is a substantial improvement of the known bounds of \textit{R. Elman} and \textit{A. Prestel} [Am. J. Math. 106, 1237--1260 (1984; Zbl 0573.10015)] and \textit{E. A. M. Hornix} [Indag. Math. 47, 305--312 (1985; Zbl 0595.10016)], although it is not known just how good the new bounds are since the only known examples of fields with \(u(F)<\tilde{u}(F)\) have \(u(F)=2n,\,\tilde{u}(F)=2n+2\) or \(u(F)=8,\,\tilde{u}(F)=12\). The author says \(F\) has the Pfister neighbor property \(\text{PN}(n),\, n\geq 0\) if every form of dimension \(2^n+1\) is a Pfister neighbor. He shows that if \(F\) has \(\text{PN}(n)\) for \(n\geq 2\) then either \(\tilde{u}(F)\leq 2^n\) or \(2^{n+1}\leq \tilde{u}(F)\leq 2^{n+1}+2^n-2\). He conjectures that, in fact, either \(\tilde{u}(F)\leq 2^n\) or \(u(F)=\tilde{u}(F)=2^{n+1}\). A consequence of his result is that \(\tilde{u}(F)<\infty\) iff \(F\) has \(\text{PN}(n)\) for some \(n\geq 2\). It is remarkable that these improvements are proven using only results known for 30 years. He gives an interpretation of \(\text{PN}(n)\) in terms of Rost projectors on Chow groups, hinting that further improvements can be made using more recent work. For Part I, see Doc. Math., J. DMV Extra Vol., 183--200 (2001; Zbl 1025.11007).