ABHYANKAR 82nd BIRTHDAY CONFERENCE
SPEAKER: Shreeram Abhyankar (Purdue)
TITLE: Antiquadratic Transformations
ABSTRACT: This talk exemplifies my fondness for polynomial manipulations
which may be regarded as my primary field of interest. So we may think
of this as high-school algebra. But my speciality is that I see great
geometry in the interplay of terms caused by a change of variables.
A QDT is a Quadratic Transformation of the local plane which enables us
to resolve the singularities of a curve in it. It amounts to making the
simplest type of substitution in a bivariate polynomial. This was
initiated by Cremona in Italy, furthered by Max Noether in Germany, and
localized by Zariski in America. Thus, ring theoretically, we are passing
from a two dimentional regular local ring to a bigger one birationally
dominating it.
Seeing that Kuhlmann has written a long abstract, I decided to out-do him
by following H. F. Baker and completely avoid mathematical symbols to see
the geometry inside algebra.
But now, following Jacobi, I want to invert by walking backwards, i.e.,
from the dominating regular local ring to the original one. This is an
Antiqdt = Antiquadratic Transformation. It leads to the existence of
plane curves qith preassigned singularities. It also helps with the
Jacobian Problem discussed by Luengo, Sathaye, and Zhao. Nedless to say
that the Polygon of Newton gets the center stage.