  The necessary and sufficient conditions for a function to be totally or
partially separable are derived. It is shown that a function is totally
separable if and only if each component of the gradient vector of depends only
on the corresponding variable. The conditions of separability are expressed
neatly in terms of the matrix which has to be diagonal if the function is to be
totally separable, and has to assume a diagonal block form in order that the
function is partially separable. The conditions of separability are also given
without using derivatives. For polynomials, the conditions of separability are
shown to hold if and only if the product of the first column and the first row
of the coefficients matrix is equal to the matrix itself. This promotes an easy
computational scheme for checking, and actually carrying out, variable
separation.
