On Appelgate-Onishi's lemmas (Q1112880)

The paper contains straightforward proofs of two lemmas stated by \textit{H. Appelgate} and \textit{H. Onishi} [J. Pure Appl. Algebra 37, 215--227 (1985; Zbl 0571.13005)] in their proof of a particular case of the Jacobian conjecture. The lemmas concern properties of two polynomials \(f,g\in K[x,y]\) (\(K\) is a field of characteristic zero) satisfying the Jacobian conjecture assumption, i.e. the Jacobian of \((f,g)\) is a non-zero constant.