The special automorphism group of R[t]/(t(m))[x₁,…,x(n)] and coordinates of a subring of R[t][x₁,…,x(n)]

Let R be a ring. The Special Automorphism Group SAut(R)R[x(1),..., x(n)] is the set of all automorphisms with determinant of the Jacobian equal to 1. It is shown that the canonical map of SAut(R[t]) R[t][x(1),...,x(n)] to SAutR(m) R-m[x(1),...,x(n)] where R-m := R[t]/(t(m)) and Q subset of R 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 f is an element of A vertical bar n vertical bar is a coordinate over B does it imply that f is a coordinate over A? It is shown that if A = R[t(m), t(m+1),...] subset of R[t] = B then the answer to this question is "yes". Also, a question on the Venereau polynomial is settled, which indicates another "coordinate-like property" of this polynomial.