Lengths of central linear generalized polynomials in matrix algebras (Q716433)

Let \(D\) be a division algebra finite-dimensional over its center \(C\) and let \(A=M_{m}(D)\), the \(m\times m\) matrix ring over \(D\). By the length of a linear generalized polynomial (GP) \(\phi (X)\), the authors mean the least positive integer \(n\) such that \(\phi (X)\) can be represented in the form \(\sum_{i=1}^n a_iXb_i\) for some \(a_i,b_i \in A\). Denote by \(L(\phi )=n\) the length of \(\phi \). By a central linear GP for \(A\) the authors mean a nonzero linear GP with central values on \(A\). The authors also characterize all central linear GPs for \(A\) and determine the lengths of all central linear GPs for \(A\).