Integer valued polynomials on lower triangular integer matrices
Let \(D\) be an integral domain with quotient field \(K\), and denote by \(T_n(D)\) the set of \(n\)-dimensional lower triangular matrices with entries from \(D\). The authors study the set \(\mathrm{Int}(T_n(D))\) of integer valued polynomials, i.e. polynomials \(f\in K(x)\) satisfying \(f(T_n(D))\subset T_n(D)\). First they consider the case \(D=Z\), \(n=2\) and show (Theorem 3) that \(\mathrm{Int}(T_2(Z))\) equals the set of polynomials \(f\in Q[x]\) for which \(f(Z)\subset Z\) and \((f(x)-f(y))/(x-y)\) maps \(Z\times Z\) into \(Z\). In the general case the authors prove (Theorem 16) that \[ \mathrm{Int}(T_n(D))=\{f\in K[x]:\;\Phi^k(f)\text{ maps } D^{k+1} \text{ into } D\text{ for } k=1,2,\dots,n-1\}, \] where the \(k\)-th divided difference \(\Phi^k(f)\) of \(f\) is defined recursively by \[ \Phi^0(f)(x_0)=f,\;\Phi^1(f)(x_0,x_1)={f(x_0)-f(x_1)\over x_1-x_0}, \] \[ \Phi^{k+1}(f)(x_0,\dots,x_{k+1})={\Phi^k(f)(x_1,\dots,x_k)-\Phi^k(f)(x_1,\dots,x_{k+1})\over x_k-x_{k+1}}. \] In the last section the authors show (Theorem 27) that if \(D\) is the ring of \(S\)-integers of a global field, \(S\) being a finite set of primes, then the largest set containing \(T_n(D)\) and having the same set of integer valued polynomials equals the set of \(n\)-dimensional matrices over \(D\) whose characteristic polynomials are products of linear polynomials over \(D\).
scientific article; zbMATH DE number 6162231