This book contains a series of lectures given by the author at the Tata Institute during Fall 1975 and Spring 1976. These lectures are divided into two parts. The first one contains the fundamental theorem on meromorphic curves and applications, and it is, essentially, a simplification of the original proof of the theorem, given by the author and T. T. Moh J. Reine Angew. Math. 260 (1973), 47--83; ibid. 261 (1973), 29--54; MR0337955 (49 #2724). The second part contains the statement of the so-called Jacobian problem, and a partial solution to it.
For the content of the first part, the reader is referred to the above-mentioned review op. cit.. Some applications of the fundamental theorem are given in Chapter IV. We quote the following: (1) Epimorphism theorem, first formulation: If ; is a field, ; lphacolon kX,Yrightarrow k ; is an epimorphism and ; extchar, ; does not divide ; extg.c.d.(degalpha(X),degalpha(Y) ; then ; egalpha(X ; divides ; egalpha(Y ; or ; egalpha(Y ; divides ; egalpha(X ; (2) Epimorphism theorem, second formulation: Let ; lpha,betacolon kX,Yrightarrow k ; be epimorphisms such that neither ; lph ; nor ; et ; is wild ; lph ; is wild if and only if ; lpha(X)
eq ; lpha(Y)
eq ; and ; extchar, ; divides both ; egalpha(X ; and ; egalpha(Y ; ; then there exists an automorphism ; igm ; of ; X, ; such that (i) ; etasigma=alph ; (ii) ; igm ; is a finite product of automorphisms ; arph ; of the forms ; arphi(X)= ; arphi(Y)=Y+P(X ; arphi(X)=X+P(Y ; arphi(Y)= ; arphi(X)=a_1X+b_1Y+c_ ; arphi(Y)=a_2X+b_2Y+c_ ; (in this case, we say that ; igm ; is a tame automorphism). (3) Automorphism theorem: Every automorphism of ; X, ; is tame. (4) Some applications to affine curves are also given. In particular, it is proved that, if ; is the coordinate ring of an affine curve over a ground field of characteristic zero, with only one place at ; nft ; then the number of equivalence classes of embeddings of the curve in the affine plane is finite. The end of the first part, that is, Chapter V of the book, contains some irreducibility criteria for polynomials in ; ((X)) ; and approximate roots, plus a note on Newton's polygons.
The Jacobian problem is stated as follows: Let ; =kx_1,x_ ; be a polynomial ring in two indeterminates over a field ; of characteristic zero, and let ; ,gin ; be such that the Jacobian of ; f,g ; with respect to ; _1,x_ ; is a nonzero constant. Does it follow that ; =kf, ; In Chapter VI, equivalent formulations of the Jacobian problem are given, some of them using Newton-Puiseux expan- sions. At the end of the book, a partial solution is given (Theorem 21.11), by valuation-theoretic methods. This result amounts to the following: If ; f,g ; is such that the Jacobian is a nonzero constant, and if ; (x_1,x_2 ; is a Galois extension of ; (f,g ; then ; f,g=kx_1,x_ ;
The reviewer has the feeling that further simplifications of these developments can be achieved, and thinks, sharing the opinion of M. Fried in his review MR0337955 (49 #2724), that it is worthwhile to try it.
Reviewed by Josộ L. Vicente