Lectures on forms in many variables. (English) Zbl 0185.08304

Greenberg, Marvin J.

Mathematics Lecture Note Series. New York-Amsterdam: W.A. Benjamin, Inc. ix, 167 p. (1969).

There are many contributions by many mathematicians on the existence of non-trivial solutions of a homogeneous equation when the number of variables is large enough compared to its degree, and one typical contribution is the theory of \(C_i\)-fields formulated by Lang. The article gives a good treatment of such existence problem.
The text begins with a historical survey (Chapter 1), then gives the Chevalley-Warning theorem which asserts that a finite field is \(C_1\) (Chapter 2). Then the cases of function fields and complete valuation rings are discussed and there are results heavily due to Lang (Chapter 3–6). Then the case of a \(p\)-adic number field is discussed in Chapter 7; there is explained a counter-example to a conjecture of Artin that a \(p\)-adic number field is \(C_2\) (by Terjanian and Schanuel). In Chapter 8, the author exposes a theorem of Brauer and Birch, which asserts a similar result as in the \(C_i\)-field case but without assuming \(C_i\)-property and good for forms of odd degrees. Finally in Chapter 9, recent works of Ax, Kochen and Paul Cohen are discussed.

Reviewer: Masayoshi Nagata (Kyoto)

MSC:

11-02	Research exposition (monographs, survey articles) pertaining to number theory
11D72	Diophantine equations in many variables
11D88	\(p\)-adic and power series fields
11E99	Forms and linear algebraic groups

Keywords:

forms in many variables; existence of non-trivial solutions of homogeneous equation; theory of \(C_i\)-fields; Chevalley-Warning theorem; finite field; function fields; complete valuation rings; \(p\)-adic number field; counter-example to a conjecture of Artin; theorem of Brauer and Birch; forms of odd degrees