Compositions of quadratic forms
type: Item
instance of: scholarly article
title: Compositions of quadratic forms
zbMATH Open document ID: 0954.11011
author: Daniel B. Shapiro
published in: De Gruyter Expositions in Mathematics
publication date: 1992-09-18T00:00:00Z
review text: The main theme of this book concerns the composition of sums of squares over \(\mathbb{R}\), it goes back to A. Hurwitz. The author considers all facets, variations and generalisations of this theme in great detail. Many different ideas and methods from algebra, geometry, combinatorics and topology have been used and are presented here. The choice of material shows good taste and reveals the author's love for this subject. A lot of exercises and 25 pages of references to the original literature complete the book. Hurwitz's original question was the following: \[ \begin{cases} \text{For which triples }(r,s,n) \text{ is there a normed bilinear pairing}\\ F:\mathbb{R}^r \times\mathbb{R}^s \to\mathbb{R}^n \\ \text{with }|f(x,y)|= |x|\cdot|y|,\text{ i.e. } \sum^r_1 x^2_i\cdot \sum^s_1 y^2_j= \sum^n_1 f^2_k(x,y)? \end{cases} \tag{*} \] He solved this question for the case \(s=n\). Part I (Chapters 1-11) of the present book is devoted to this ``classical'' case and its natural generalisations replacing \(\mathbb{R}\) by another field \(F\) (always \(\text{char} F\neq 2)\) or sums of squares by other quadratic forms. The main results are: (1) Theorem of Radon-Hurwitz: Such an \([r,n,n]\)-formula exists (over \(\mathbb{R}\) or any other field \(F)\) iff \(r\leq\rho(n)\). For \(n=2^{4a+b}\cdot n_0\) (with \(0\leq b\leq 3\), \(n_0\) odd) the number \(\rho(n)\) is given by the famous equation \(\rho (n)=8a+ 2^b\leq n\). In particular: \(\rho(n)=n \Leftrightarrow n=1,2,4,8\). (2) Let \(\sigma,q\) be regular quadratic forms over \(F\), \(\dim\sigma=s\), \(\dim q=n\). We say that \(\sigma\) allows composition with \(q\) if \(\sigma(X) q(Y)=q(Z)\) where \(Z\) is a \(F\)-bilinear expression in \(X,Y\). In other words: \(\sigma(X)\) is a ``similarity'' of \(q\), in short terms: \(\sigma<\text{Sim}(q)\). Cor. 2.12 states: If \(\sigma< \text{Sim} (q)\) then \(s\leq\rho(n)\). (3) The author considers in particular the case \(n=2^m\), \(s=\rho(n)\). He puts up a conjecture PC\((m)\), whi…
Mathematics Subject Classification ID: 11Exx
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 12D15
zbMATH DE Number: 53030
zbMATH Keywords: intercalate matrices
zbMATH Keywords: Hopf construction
zbMATH Keywords: hidden maps
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: theorem of Radon-Hurwitz
zbMATH Keywords: similarity
zbMATH Keywords: Pfister factor conjecture
zbMATH Keywords: \(m\)-fold multiplicative quadratic form
zbMATH Keywords: composition of sums of squares
zbMATH Keywords: Clifford algebras
zbMATH Keywords: involutions
zbMATH Keywords: topology
zbMATH Keywords: differential geometry
reviewed by: Albrecht Pfister
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 53030