Lengths of central linear generalized polynomials in matrix algebras

Let D be a division algebra finite-dimensional over its center C and let A=Mm(D), the m×m matrix ring over D. By the length of a linear generalized polynomial (GP) ϕ(X), we mean the least positive integer n such that ϕ(X) can be represented in the form for some . We denote by L(ϕ)=n the length of ϕ. By a central linear GP for A we mean a nonzero linear GP with central values on A. In this paper we characterize all central linear GPs for A and determine the lengths of all central linear GPs for A.