Some Reductions on Jacobian Problem in Two Variables

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