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

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.

### 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