ABHYANKAR 82nd BIRTHDAY CONFERENCE

SPEAKER: Avinash Sathaye (Kentucky)

TITLE: Root approximators and the Jacobian problem

ABSTRACT: Given a plane curve $f(X,Y)=0$, we study its generic translate
$f(X,Y)+\lambda $ and especially its fundamental differential $\omega =
dx/f_Y = du/J(f,u) $, where $u$ is any polynomial in $X,Y$ without a
common component with $f(X,Y)+\lambda $.

We analyze the values of this differential at various branches of
$f(X,Y)+\lambda$ at infinity, using the root approximators to help us
deal with the indeterminate $\lambda$.

The Jacobian hypothesis implies that the maximum value of the
denominators $J(f,u)$ at any of these valuations at infinity is zero and
the Jacobian theorem would follow if the values of $\omega$ itself are
shown to be always negative.