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.