A Layman's Perspective of Abhyankar's Conjecture

by Ganapathy Sundaram

Consider a polynomial in two variables X and Y. This can be treated as a polynomial of degree n in Y, whose coefficients are polynomials in X. Now plug in values for X, and it can be seen that for most values of X, we get n-corresponding roots for Y. But there are some places where there are fewer than n-roots for Y. These special places are called discriminant points. If there are no discriminant points, then f factors completely. This was noticed by the famous mathematician Riemann in the mid-1800's by shrinking the plane to a point. Prior to Riemann, the famous French mathematician Galois made the same observation by looking at permutations of the roots of the equations. The set of all permutations of the roots forms a group and is now referred to as the Galois group of the polynomial. Historically speaking, Galois was interested in studying the group of permutations of the roots in order to prove that one "cannot" solve a polynomial of degree greater than or equal to five. (Formulas for solving quadratic equations were discovered by Indian mathematicians around the 4th century A.D. Formulas for solving degree four and five equations were discovered over a thousand years later by Italian mathematicians.) In essence, the Galois group of a polynomial tells us how far we can "solve" a polynomial.

Abhyankar's work in Galois theory started with his Harvard thesis in the early 1950's under the able guidance of the famous mathematician, Oscar Zariski. He was interested in polynomials in three variables (i.e., surfaces), and he was working on removing so-called singularities (beak-like points) of the surface over modular fields (i.e., fields obtained by fixing a prime number and considering remainders of every integer after dividing by that fixed prime). This problem also has its origins in Riemann's work, but the three variable case over "usual" fields (i.e., complex numbers) was solved by Jung in 1908. In his thesis, Abhyankar first showed that over modular ground fields, Jung's classical method of changing the surface to get rid of singularities does not work because of strange properties of Galois groups. In particular, he discovered that things were different over modular ground fields because a polynomial with coefficients in a modular field need not factor completely even if it has no discriminant points. As a result, in 1957 he launched a systematic attack on understanding Galois groups of polynomials (in two variables and later in more variables) and conjectured what Galois groups can be obtained over modular ground fields. Really what he wrote down was more than a conjecture-he laid out a philosophy. What Harbater and Raynaud have proved are special cases of this philosophy pertaining to polynomials in two variables, i.e., curves. This is only the tip of the iceberg, but the excitement is due to the fact that one now has a structured approach to understanding Galois groups of various equations.

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