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Compositions of quadratic forms. (English) Zbl 0954.11011
de Gruyter Expositions in Mathematics. 33. Berlin: Walter de Gruyter. xiii, 417 p. DM 248.00 (2000).

The main theme of this book concerns the composition of sums of squares over $ℝ$, 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:

$\left\{\begin{array}{c}\text{For}\phantom{\rule{4.pt}{0ex}}\text{which}\phantom{\rule{4.pt}{0ex}}\text{triples}\phantom{\rule{4.pt}{0ex}}\left(r,s,n\right)\phantom{\rule{4.pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{there}\phantom{\rule{4.pt}{0ex}}\text{a}\phantom{\rule{4.pt}{0ex}}\text{normed}\phantom{\rule{4.pt}{0ex}}\text{bilinear}\phantom{\rule{4.pt}{0ex}}\text{pairing}\hfill \\ F:{ℝ}^{r}×{ℝ}^{s}\to {ℝ}^{n}\hfill \\ \text{with}\phantom{\rule{4.pt}{0ex}}|f\left(x,y\right)|=|x|·|y|,\phantom{\rule{4.pt}{0ex}}\text{i.e.}\phantom{\rule{4.pt}{0ex}}\sum _{1}^{r}{x}_{i}^{2}·\sum _{1}^{s}{y}_{j}^{2}=\sum _{1}^{n}{f}_{k}^{2}\left(x,y\right)?\hfill \end{array}\right\\phantom{\rule{2.em}{0ex}}\left(*\right)$

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 $ℝ$ by another field $F$ (always $\text{char}F\ne 2\right)$ or sums of squares by other quadratic forms. The main results are:

(1) Theorem of Radon-Hurwitz: Such an $\left[r,n,n\right]$-formula exists (over $ℝ$ or any other field $F\right)$ iff $r\le \rho \left(n\right)$. For $n={2}^{4a+b}·{n}_{0}$ (with $0\le b\le 3$, ${n}_{0}$ odd) the number $\rho \left(n\right)$ is given by the famous equation $\rho \left(n\right)=8a+{2}^{b}\le n$. In particular: $\rho \left(n\right)=n⇔n=1,2,4,8$.

(2) Let $\sigma ,q$ be regular quadratic forms over $F$, $dim\sigma =s$, $dimq=n$. We say that $\sigma$ allows composition with $q$ if $\sigma \left(X\right)q\left(Y\right)=q\left(Z\right)$ where $Z$ is a $F$-bilinear expression in $X,Y$. In other words: $\sigma \left(X\right)$ is a “similarity” of $q$, in short terms: $\sigma <\text{Sim}\left(q\right)$. Cor. 2.12 states: If $\sigma <\text{Sim}\left(q\right)$ then $s\le \rho \left(n\right)$.

(3) The author considers in particular the case $n={2}^{m}$, $s=\rho \left(n\right)$. He puts up a conjecture PC$\left(m\right)$, which he calls “Pfister Factor Conjecture”: This is possible iff $q$ is a scalar multiple of an $m$-fold multiplicative quadratic form in the sense of the reviewer. He proves PC$\left(m\right)$ for $m\le 5$ in Chapter 9 using results on Clifford algebras and involutions. – [Remark: Recent results of Izhboldin-Karpenko and Hoffmann-Tignol (see the Notes in Ch. 9, p. 175) show that $m$-fold forms can have rather bad properties for $m\ge 4$. Therefore it is not obvious to me that PC$\left(m\right)$ should hold for all $m$ and all fields $F$.]

Part II (Chapters 12-16) is more difficult, the results are less complete, some proofs are only outlined. The methods used range from algebra to combinatorics (“intercalate matrices”), topology (“Hopf construction”, “hidden maps”), $K$-theory and differential geometry. The main topic concerns computations or estimates for the following three numbers:

$\begin{array}{cc}\hfill r*s& =min\left\{n:\exists \phantom{\rule{4.pt}{0ex}}\text{normed}\phantom{\rule{4.pt}{0ex}}\text{bilinear}\phantom{\rule{4.pt}{0ex}}\text{map}\phantom{\rule{4.pt}{0ex}}f:{ℝ}^{r}×{ℝ}^{s}\to {ℝ}^{n}\phantom{\rule{4.pt}{0ex}}\text{as}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}\left(*\right)\right\}\hfill \\ \hfill r#s& =min\left\{n:\exists \phantom{\rule{4.pt}{0ex}}\text{nonsingular}\phantom{\rule{4.pt}{0ex}}\text{bilinear}\phantom{\rule{4.pt}{0ex}}\text{map}\phantom{\rule{4.pt}{0ex}}f:{ℝ}^{r}×{ℝ}^{s}\to {ℝ}^{n}\right\}\hfill \\ \hfill r\circ s& =min\left\{n:\left(\genfrac{}{}{0pt}{}{n}{k}\right)\phantom{\rule{4.pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{even}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}k\phantom{\rule{4.pt}{0ex}}\text{with}\phantom{\rule{4.pt}{0ex}}n-r

By a famous theorem of Stiefel and Hopf we have $r\circ s\le r#s$, and $r#s\le r*x\le r·s$ is trivial. Similar invariants can be defined over other fields $F$ instead of $ℝ$, and even over $ℤ$. The best original results are due to K. Y. Lam, S. Yuzvinsky and P. Yiu, e.g. $16{*}_{ℤ}16=32$.

##### MSC:
 11Exx Forms and linear algebraic groups 11-02 Research monographs (number theory) 12D15 Formally real fields