Some reductions on Jacobian problem in two variables. (Q1428111)

The first main result of the paper is an equivalent formulation of the Jacobian conjecture in the two-dimensional case. It follows from the following general result on arbitrary polynomial mappings. Let \((f_{1},f_{2} ):\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}\) be a polynomial mapping with a finite set of zeros. By simple modifications (i.e. by using compositions with polynomial automorphisms) the author can assume that \(f_{1}(x,y)\) and \(f_{2}(x,y)\) have the form \(x^{n}+(\)terms of lower degree) and that the intersection points of the curves \(f_{1}=0\) and \(f_{2}=0\) lie on the \(x\)-axis. Then (1) there exists a unique polynomial solution \(g=(g_{1},g_{2})\) of the equation \(y\text{Jac}(f)=f_{1}g_{1}+f_{2}g_{2}\), \(\deg g_{i}=n-1\), and (2) the total intersection number (counting multiplicity) of the affine curves \(f_{1}=0\) and \(f_{2}=0\) equals to the coefficient of \(x^{n-1}\) of \(g_{2}.\) In particular if \(\text{Jac}(f)\equiv1\) then to prove the Jacobian conjecture it suffices to check that the coefficient in \(2.\) is equal to \(1.\) The second main result \ gives an explicite form for \((f_{1},f_{2})\) under the assumption that all intersection points of the curves \(f_{1}=0\) and \(f_{2}=0\) are normal crossings. In the proofs of the results the author uses the multidimensional residue theory.