Some Reductions on Jacobian Problem in Two Variables

Abstract

Let $f=(f_1,f_2)$ be a regular sequence of affine curves in $\bC^2$. Under some reduction conditions achieved by composing with some polynomial automorphisms of $\bC^2$, we show that the intersection number of curves $(f_i)$ in $\bC^2$ equals to the coefficient of the leading term $x^{n-1}$ in $g_2$, where $n=°f_i$ $(i=1,2)$ and $(g_1,g_2)$ is the unique solution of the equation $y\mathcal{J}(f)=g_1{f}_1+g_2{f}_2$ with $°g_i\le n-1$. So the well-known Jacobian problem is reduced to solving the equation above. Furthermore, by using the result above, we show that the Jacobian problem can also be reduced to a special family of polynomial maps.