A Springer theorem for higher degree forms
\textit{T. A. Springer} [Nederl. Akad. Wet., Proc. Ser. A 58 ( Indag. Math. 17), 512--516 (1955; Zbl 0067.27701)] considered anisotropic cubic forms over a complete discretely valued field, showing that associated to such a form are three anisotropic cubic forms over the residue field, which allowed him to prove that the \(u\)-invariant of cubic forms satisfies the inequality \(u(3,F)\leq 3u(3,\overline{F})\). But his methods do not generalize to degree greater than 3. In this paper the author considered an analogue of Springer's theorem in the case of degree \(d\)-forms. The theorem holds for certain classes of forms, including diagonal forms, but does not hold in general. The difficulty is that, since higher degree forms need not decompose into one-dimensional forms, it is not clear how to associate a series of residue forms to a given form.
scientific article; zbMATH DE number 5159764
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