Necessary and sufficient conditions for a function to be separable

摘要

The necessary and sufficient conditions for a function F(x1, … , xn) 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 Ln∣F∣ depends only on the corresponding variable. The conditions of separability are expressed neatly in terms of the matrix FF,ij − F,iF,j 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 in two variables, 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. The latter results are extended to polynomials in n variables.