Partial properness and real planar maps

The Asymptotic Variety of a Pinchuk Map as a Polynomial Curve

“…The polynomial Q varies for different Pinchuk maps, but always has the form Q = −t 2 − 6th(h + 1) − u(f, h), where u is an auxiliary polynomial in f and h, chosen so that j(P, Q) = t 2 + (t + f (13 + 15h)) 2 + f 2 . As in [Cam96,Ess00], choose specifically…”

“…The points (−1, −163/4) and (0, 0) of A(F ) have no inverse image under F , all other points of A(F ) have one inverse image, and all points of the image plane not on A(F ) have two. See [Cam96,Cam08].…”

“…Eqn (4) was first circulated by Arno van den Essen in an email message to a number of colleagues in June 1994. I made two blunders and a significant typographical error in describing the associated Pinchuk map F in [Cam96]. The errors were corrected in [Cam98] and belatedly for the official record in [Cam08].…”

“…Let the polynomial map F = (P, Q) : R 2 → R 2 be the Pinchuk map of total degree 25 considered in [Cam96,Ess00,Cam98,Cam08]. Its asymptotic variety, A(F ), a closed curve in the image (P, Q)-plane, was computed in [Cam96]. It is depicted below using differently scaled axes.…”

Erratum to the Pinchuk map description in “Partial properness and real planar maps” [Appl. Math. Lett. 9 (1996) 99–105]

“…The subject work [1] uses the standard Pinchuk map as an example. The map is the simplest in a family of counterexamples discovered by Pinchuk [6].…”