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Quadratic forms with applications to algebraic geometry and topology. (English) Zbl 0847.11014
London Mathematical Society Lecture Note Series. 217. Cambridge: Cambridge Univ. Press. viii, 179 p. (1995).
The theory of quadratic forms over arbitrary fields was initiated by E. Witt in a paper of 1936 [J. Reine Angew. Math. 176, 31-44 (1936; Zbl 0015.05701)]. Since about 1965 there has been a dramatic development with the author as one of the main actors. Among the topics that have been studied consistently over the last 30 years are some of the principal numerical invariants of fields. The level of a field is the infimum of the number of summands needed to represent $$-1$$ as a sum of squares. The Pythagoras number is the supremum of the numbers $$p$$ such that there is a sum of squares which cannot be represented by fewer than $$p$$ squares. Both these invariants are concerned with the set of elements of the field representable by certain quadratic forms. A particularly important question about a quadratic form is whether $$0$$ can be represented or not. Accordingly, quadratic forms are called isotropic or anisotropic. For non-formally real fields the $$u$$-invariant is the largest number $$n$$ such that there exists an anisotropic quadratic form in $$n$$ variables. For formally real fields there is a slightly more elaborate definition of $$u$$-invariants. The book is built around these three invariants; the level, the Pythagoras number and the $$u$$-invariant. The level and the Pythagoras number have obvious generalizations for rings. For the level there is even a related notion for topological spaces.
All of these invariants have been computed in some instances, but are very hard to deal with in general. The book contains many examples along with the main results, old and new, about the invariants. In particular, relationships between the different invariants as well as with other mathematical topics are explored. An obvious connection exists with algebraic geometry. A quadratic form is isotropic over a field $$K$$ if and only if the corresponding quadratic has a $$K$$-rational point. This question can be generalized vastly by asking for the existence of $$K$$-rational points for systems of forms. The problem depends on several parameters: the field, the number of variables, the degrees of the forms, the number of forms. Answers can only be expected when there are restrictions on the parameters. For example, a homogeneous Nullstellensatz for $$p$$-fields is presented. The Nullstellensatz is used to prove the Borsuk-Ulam Theorem and the Brouwer Fixed Point Theorem. The Borsuk-Ulam Theorem yields the (topological) levels of the spheres. These, in turn, are related to the levels of certain rings of functions. Another type of restriction on the parameters leads into the Tsen-Lang theory of $$C_i$$-fields. Or, if one deals with only one quadratic form then there is a close connection with the $$u$$-invariant. For systems of quadratic forms there are variants of the $$u$$-invariant which can be bounded in terms of the number of quadratic forms and the $$u$$-invariant for a single form. The investigation of systems of quadratic forms is an important ingredient in the computation of the levels of projective spaces.
The author makes the book accessible to non-specialists by including some introductory material on quadratic forms, Witt rings and formally real fields. This does not mean that the book is self-contained. Numerous results are stated or used without a proof. In these cases precise references are always included. Altogether, the book gives a very readable account of a difficult and active part of the algebraic theory of quadratic forms and its relations to other mathematical theories. For anybody getting hooked on the subject there are enough references and open problems to keep reading or even to go into active research.

##### MSC:
 11E04 Quadratic forms over general fields 11-02 Research exposition (monographs, survey articles) pertaining to number theory 11E81 Algebraic theory of quadratic forms; Witt groups and rings 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) 12E15 Skew fields, division rings 55M99 Classical topics in algebraic topology