A Springer theorem for higher degree forms
type: Item
instance of: scholarly article
title: A Springer theorem for higher degree forms
zbMATH Open document ID: 1162.11023
author: Patrick Morandi
published in: Mathematische Zeitschrift
publication date: 2007-05-31T00:00:00Z
review text: \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.
reviewed by: Osman Bizim
Mathematics Subject Classification ID: 11E76
Mathematics Subject Classification ID: 12J10
zbMATH DE Number: 5159764
zbMATH Keywords: Springer theorem
zbMATH Keywords: cubic forms
MaRDI profile type: Publication
full work available at URL: s00209-006-0070-7
OpenAlex ID: W2073751436
cites work: Q5577214
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cites work: Q3708857
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DOI: 10.1007/S00209-006-0070-7
description: scientific article; zbMATH DE number 5159764