Customer Review

July 29, 2006
As in his book "On Riemann's Theory of Algebraic Functions and Their Integrals", Klein takes Riemann surfaces as the starting point of complex function theory. Complex functions are then introduced in terms of potential flows on these surfaces. Approaching algebraic functions from this foundation "has the great advantage that the existence and number of the algebraic functions on the Riemann surfaces is known in advance and we need only seek their algebraic presentations. The above mentioned authors [Weierstrass et al.] on the other hand face the double task of finding the algebraic presentations and from there to arrive at the theorems on the existence and the number of the functions. Because of this their approach is ... very laborious and abstract." (p. 72). This development of the theory of algebraic functions, abelian integrals, etc. is the first part of the book. The second part treats the deep harmony with the theory of algebraic curves. So for instance the Riemann-Roch theorem counting the number of linearly independent rational functions with prescribed poles (pp. 63-65) provides an approach to the "intersection point theorems" describing the interrelations of the intersection points of two algebraic curves (pp. 141-147), which "cuts through the entire subject of algebraic curves" (p. 147) and "from which the most remarkable geometric theorems can be derived" (p. 119). The a priori Riemann surface point of view is the way to go also in algebraic curve theory: "It is essential that we characterise the algebraic functions independently of their analytic presentations, and in particular that we understand the importance of the number p [the genus] as a number that measures the 'connectivity' of the algebraic object. When Brill and Noether ... introduce p otherwise, the theorem on the invariance of p under bijective transformations appears as a curious side effect, and not, as in Riemann, as something thoroughly obvious." (p. 192).
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