Some remarks on the Jacobian question. (Notes by Marius van der Put and William Heinzer, updated by Avinash Sathaye) (Q1338150)

The paper contains an updated version of Abhyankar's lectures of 1970- 1971 on the Jacobian problem. A version of them is contained in the author's book: ``Lectures on expansion techniques in algebraic geometry'', Tata Institute of Fundamental Research (Bombay 1977). The present article contains a lot of consistent additional material and it is an excellent invitation to a main open algebraic problem. It emphasizes the use of the Newton-Puiseux expansions and the study of corresponding value-semigroups. We notice the Newton polygon approach of the Jacobian problem for two variables, the Taylor resultant theorem, the Newton-Puiseux expansions for different weights, and an interesting formula for the conductor of a semigroup defined by nonpositive integers. The pseudoapproximate roots corresponding to a given weight are used for discussing the Jacobian theorem. Several solutions are described for special cases of the Jacobian problem. In particular a positive answer is given for the case of ``two characteristic pairs''.