A generalization of a lemma of Abhyankar and Moh (Q1073847)

The following result is proved: let k be a commutative ring with identity and \(\alpha\) be a surjection from the polynomial ring \(k[X_ 1,X_ 2,...,X_ n]\) to the polynomial ring \(k[Y_ 1,Y_ 2,...,Y_ m]\) with \(n\geq m\). If the kernel of \(\alpha\) is generated by n-m elements \((f_{m+1},f_{m+2},...,f_ n)\) and \(\alpha (X_ i)\) for each i is \(h_ i(Y_ 1,Y_ 2,...,Y_ m)\), then the ideal generated by the image of \(\partial (f_{m+1},f_{m+2},...,f_ n)/\partial (X_ 1,X_ 2,...,X_ n;\mu_ 1,...,\mu_ m)\) under \(\alpha\) where \(1\leq \mu_ 1<\mu_ 2<\mu_ 3...<\mu_ m\leq n\) is the same as the ideal generated by \(\partial (h_{\mu_ 1},h_{\mu_ 2},...,h_{\mu_ m})/\partial (Y_ 1,Y_ 2,...,Y_ m)\). This generalizes a result of Abhyankar and Moh who proved the above when \(m=1\). Here \(\partial (f_{m+1},f_{m+2},...,f_ n)/\partial (X_ 1,X_ 2,...,X_ n;\mu_ 1,...,\mu_ m)\) denotes the determinant of the matrix obtained from \([\partial f_ i/\partial x_ j]_{j=1,...,p}^{i=m+1,...,n}\) by deleting the columns \(\mu_ 1,...,\mu_ m\). The proof is quite elementary.