文件名称：Algebraic Functions
文件大小：2.61MB
文件格式：DJVU
更新时间：2012-10-06 08:47:30
math algebraic function rational functions
This book, immediately striking for its conciseness, is one of the most remarkable works ever produced on the subject of algebraic functions and their integrals. The distinguishing feature of the book is its third chapter, on rational functions, which gives an extremely brief and clear account of the theory of divisors. Here the integrands of the three elementary types of abelian integrals are set up by the arithmetic methods of Dedekind and Weber. The arithmetic treatment is definitely simpler and more elegant than the potential-theoretic method of Riemann, or the geometric method of Brill and Noether which is based on the reduction of the singularities of algebraic curves. The theory of divisors, hitherto available chiefly in the ponderous classic treatise of Hensel and Landsberg, is presented by Bliss in hardly more than thirty pages. A very readable account is given of the topology of Riemann surfaces and of the general properties of abelian integrals. Abel's theorem is presented, with some simple applications. The inversion problem is studied for the cases of genus zero and genus unity. The chapter on the reduction of singularities is very noteworthy. Here one finds a thorough treatment of the problem of reducing the singularities of an algebraic curve, by means of a birational transformation, to double points with distinct tangents. This question is examined for the projective plane, and also for the space of analysis. In this chapter, as in the chapter on divisors, the influence of Bliss's own researches is seen. A final chapter illustrates the general theory with some examples. In particular, constructive methods are given for treating algebraic relations which are of the third degree in one of the variables. There is a good bibliography. The arithmetic theory of algebraic functions is a good thing. In making its study easy, Bliss has performed a service which will win him the gratitude of an ever increasing number of readers.