Some reductions on Jacobian problem in two variables

Let f=(f1,f2) be a regular sequence of affine curves in C2. Under some reduction conditions achieved by composing with some polynomial automorphisms of C2, we show that the intersection number of curves (fi) in C2 equals to the coefficient of the leading term xn−1 in g2, where n=degfi(i=1,2) and (g1,g2) is the unique solution of the equation yJ(f)=g1f1+g2f2 with deggi⩽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.