Abstract
We determine the indexes of all orthogonal Grassmannians of a generic 16-dimensional quadratic form in \(I^3_q\). This is applied to show that the 3-Pfister number of the form is \(\ge \)4. Other consequences are: a new and characteristic-free proof of a recent result by Chernousov–Merkurjev on proper subforms in \(I^2_q\) (originally available in characteristic 0) as well as a new and characteristic-free proof of an old result by Hoffmann–Tignol and Izhboldin–Karpenko on 14-dimensional quadratic forms in \(I^3_q\) (originally available in characteristic \(\ne \)2). We also suggest an extension of the method, based on investigation of the topological filtration on the Grothendieck ring of a maximal orthogonal Grassmanian, which applies to quadratic forms of dimension higher than 16.
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This work has been accomplished during author’s stay at the Universität Duisburg-Essen; it has been supported by a Discovery Grant from the National Science and Engineering Board of Canada.
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Karpenko, N.A. Around 16-dimensional quadratic forms in \(I^3_q\) . Math. Z. 285, 433–444 (2017). https://doi.org/10.1007/s00209-016-1714-x
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DOI: https://doi.org/10.1007/s00209-016-1714-x