The special automorphism group of R[t]/(tm)[x1,…,xn] and coordinates of a subring of R[t][x1,…,xn]

Let R be a ring. The Special Automorphism Group image is the set of all automorphisms with determinant of the Jacobian equal to 1. It is shown that the canonical map of image to image where Rmcolon, equalsR[t]/(tm) and image is surjective. This result is used to study a particular case of the following question: if A is a subring of a ring B and fset membership, variantA[n] is a coordinate over B does it imply that f is a coordinate over A? It is shown that if A=R[tm,tm+1,…]subset ofR[t]=B then the answer to this question is “yes”. Also, a question on the Vénéreau polynomial is settled, which indicates another “coordinate-like property” of this polynomial.