Some results on the Jacobian conjecture in higher dimension (Q1331909)

Let \(k\) be a field of characteristic 0 and \(k[x]\) the polynomial ring in \(n\) variables \(x_ 1, \dots, x_ n\) over \(k\), \(F = (F_ 1,\dots, F_ n)\), \(F_ i \in k[x]\), \(i = 1, \dots,n\). The authors state that if \(F_ i = xg_ i\), \(g_ i \in k[x]\) and \(\text{det} (\partial F_ i/ \partial x_ j) \in k^*\), then \(F_ i = \alpha_ i x_ i\) with \(\alpha_ i \in k^*\). They use this result to prove Keller's Jacobian conjecture when each \(F_ i = x_ i + \alpha_ i M_ i\) where \(\alpha_ i \in k\) and \(M_ i\) are monomials. In the proofs of proposition 6 and lemma 7 there are some fragments not clear for the reviewer.