Some reductions on Jacobian problem in two variables

Let f = (f(1), f(2)) be a regular sequence of affine curves in C-2. Under some reduction conditions achieved by composing with some polynomial automorphisms of C-2, we show that the intersection number of curves (f(i)) in C-2 equals to the coefficient of the leading term x(n-1) in g(2), where n = deg f(i) (i = 1,2) and (g(1), g(2)) is the unique solution of the equation yg(f) = g(1)f(1) + g(2)f(2) with deg gi less than or equal to 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. (C) 2003 Elsevier B.V. All rights reserved.