Some invariants of a field of characteristic 2 associated to the \(\hat{u}\)-invariant (Q888462)

The \(\hat{u}\)-invariant of a field \(F\) of characteristic 2 is the maximal dimension of an anisotropic \(F\)-quadratic form. The author introduces a new type of \(u\)-invariant, namely the \(\tilde{u}\)-invariant, defined as the maximal dimension of an anisotropic not totally singular \(F\)-quadratic form. The introduction of this new invariant is motivated by the fact that most of the known values of the \(\hat{u}\)-invariant are realized by totally singular quadratic forms. The author compares the \(\tilde{u}\)-invariant with other \(u\)-invariants in characteristic 2 as the ones studied in [\textit{R. Baeza}, Bol. Soc. Bras. Mat. 13, No. 1, 105--114 (1982; Zbl 0573.10014); \textit{P. Mammone} et al., Math. Z. 208, No. 3, 335--348 (1991; Zbl 0733.11011); Math. Ann. 290, No. 1, 109--128 (1991; Zbl 0713.12002)] and obtains results about their values. In particular he shows that the \(\tilde{u}\)-invariant cannot take the values 3, 5, 7.