The effect of rational maps on polynomial maps (Q2724116)

Let \(\phi: \mathbb C^2_{(v,w)}\to\mathbb C^2_{(x,y)}\) be the rational map defined by NEWLINE\[NEWLINE x=\frac{1}{v^n},\quad y=A_1v^n+\dots+A_{m-1}v^{n(m-1)}+v^{nm-k}w,\;n,m,k\in\mathbb N,\;k<n. NEWLINE\]NEWLINE In the paper the authorNEWLINENEWLINENEWLINE1. describes the polynomials \(P\in\mathbb C[x,y]\) such that \(P\circ \phi(v,w)\in\mathbb C[v,w]\),NEWLINENEWLINENEWLINE2. characterizes polynomials obtained in the above way (i.e. those \(Q\in\mathbb C[x,y]\) which have the form \(P\circ \phi(v,w)\in\mathbb C[v,w]\)).NEWLINENEWLINENEWLINEThese polynomials \(P,Q\) are closely related to many interesting examples of polynomials, known in algebraic geometry and theory of singularities.NEWLINENEWLINENEWLINEThe author illustrates her results by considering:NEWLINENEWLINENEWLINE1. Briançon's example of a polynomial with smooth and irreducible fibers -- the author gives a new family of such polynomials.NEWLINENEWLINENEWLINE2. The Pinchuk's example -- a counterexample to the real jacobian conjecture.NEWLINENEWLINENEWLINE3. The bad field generators (they are defined as polynomials \(f\in\mathbb C[x,y]\) such that there exists \(g\in\mathbb C(x,y)\) for which \(\mathbb C(x,y)= \mathbb C(f,g)\) and there is no \(g\in\mathbb C[x,y]\) with the same property \(\mathbb C(x,y)= \mathbb C(f,g))\) -- the author gives new examples of such polynomials.