Galois theory on the line in nonzero characteristic

The author surveys Galois theory of function fields with non-zero caracteristic and its relation to the structure of finite permutation groups and matrix groups.Comment: 66 pages. Abstract added in migration

Similarity method for the Jacobian problem

This is an expository article giving a modified version of my talk at the October 2006 Conference in Hanoi

Some thoughts on the Jacobian Conjecture, Part II

AbstractIn Abhyankar's Purdue Lectures of 1971, the bivariate Jacobian Conjecture was settled for the case of two plus epsilon characteristic pairs. In the published version, the epsilon part got left out. Now we take care of the omission by preparing for a sharper result with full proof in Part III. The Jacobian Method is applied to giving a new simple proof of Jung's Automorphism Theorem. A detailed description of the Degreewise Newton Polygon is given. Some thoughts on the multivariate Jacobian Conjecture are included

Existence of dicritical divisors revisited

We characterize the dicriticals of special pencils. We also initiate higher dimensional dicritical theory

On the uniqueness of the coefficient ring in a polynomial ring

This article does not have an abstract

On the uniqueness of the coefficient ring in a polynomial ring

This article does not have an abstract

Construction techniques for Galois coverings of the affine line

For constructing un ramified coverings of the affine line in characteristicp, a general theorem about good reductions modulop of coverings of characteristic zero curves is proved. This is applied to modular curves to realize SL(2, Z/nZ)/{± 1}, with GCD(n, 6) = 1, as Galois groups of unramified coverings of the affine line in characteristicp, for p = 2 or 3. It is applied to the Klein curve to realize PSL(2, 7) for p = 2 or 3, and to the Macbeath curve to realize PSL(2, 8) for p = 3. By looking at curves with big automorphism groups, the projective special unitary groups PSU(3, pv) and the projective special linear groups PSL(2, pv) are realized for allp, and the Suzuki groups Sz(22v+1) are realized for p = 2. Jacobian varieties are used to realize certain extensions of realizable groups with abelian kernels