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

Let RR be a ring. The Special Automorphism Group SAutRR[x1,…,xn] is the set of all automorphisms with determinant of the Jacobian equal to 1. It is shown that the canonical map of SAutR[t]R[t][x1,…,xn] to SAutRmRm[x1,…,xn] where Rm≔R[t]/(tm)Rm≔R[t]/(tm) and Q⊂RQ⊂R is surjective. This result is used to study a particular case of the following question: if AA is a subring of a ring BB and f∈A[n]f∈A[n] is a coordinate over BB does it imply that ff is a coordinate over AA? It is shown that if A=R[tm,tm+1,…]⊂R[t]=BA=R[tm,tm+1,…]⊂R[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.