Global injectivity of planar non-singular maps polynomial in one variable

TL;DR

This paper investigates the injectivity of planar polynomial maps in one variable, establishing conditions under which such maps are composed of simpler maps and are injective, with implications for understanding their global behavior.

Contribution

It proves that polynomial maps with y-degree one are compositions of triangular and quasi-triangular maps, and establishes injectivity conditions for y-quadratic maps.

Findings

Y-degree one maps are compositions of triangular and quasi-triangular maps.

Non-singular y-quadratic maps are injective if a leading coefficient is non-zero.

Y-quadratic Jacobian maps decompose into a quasi-triangular map and three triangular maps.

Abstract

We consider non-singular and Jacobian maps whose components are polynomial in the variable y. We prove that if a map has y-degree one, then it is the composition of a triangular map and a quasi-triangular map. We also prove that non-singular y-quadratic maps are injective if one of the leading functional coefficients does not vanish. Moreover, y-quadratic Jacobian maps are the composition of a quasi-triangular map and 3 triangular maps. Other results are given for wider classes of non-singular maps, considering also injectivity on vertical strips I x R.