On linear chains of blow-ups related to the Jacobian conjecture (Q1889662)

The following theorem is proved. If the Jacobian of a pair of polynomials \(P(x,y)\) an \(Q(x,y)\) is equal to 1, \(n\) is an arbitrary positive integer, and the parameters \(p_1,p_2,\dots,p_n\) take any values, with the restriction \(p_2\neq 0\), then, after the substitution \(x\mapsto xy^n+p_n y^{n-1}+\dots p_2y+p_1\), \(y\mapsto 1/y\), at least one of the functions \(P(x,y)\) or \(Q(x,y)\) is no longer polynomial (in the new variables). This is a generalization of the similar result obtained by A. G. Vitushkin for \(n\leq 3\) without the restriction \(p_2\neq 0\). In the paper there is also described a geometric construction having a direct application to the Jacobian conjecture.