The pole assignability property in polynomial rings over GCD-domains
Let R be a commutative ring with F and G matrices over R of sizes \(n\times n\) and \(n\times m\), respectively. The pair (F,G) is said to be reachable iff the R-module generated by the columns of the matrix \([G,FG,...,F^{n-1}G]\) is \(R^ n\). The ring R is said to have the PAF- property iff each reachable pair (F,G) is ''pole-assignable'' - i.e., given arbitrary elements \(r_ 1,...,r_ n\) in R, there exists an \(m\times n\) matrix K such that the characteristic polynomial of the matrix \(F+GK\) is \((x-r_ 1)...(x-r_ n)\). The main result of this paper is proposition 3: Let D be a GCD domain and X a set of indeterminates. Suppose there is a finitely generated prime ideal Q of D[X] such that D[X]/Q has an ideal I satisfying: (i) I is generated by two elements, (ii) I is not principal, and (iii) the square of I (as an ideal) is principal. Then D[X] does not have the PAF-property. As a consequence, one obtains the well-known facts that k[x,y] and \({\mathbb{Z}}[x]\) are not PAF-rings for k a field and \({\mathbb{Z}}\) the ring of integers. One question left open by the authors is the following: If V is a Noetherian valuation domain, does V[x] have the PAF-property?
scientific article; zbMATH DE number 3916430