Polynomials defined within a simplex can be written in terms of barycentric coordinates. Such expressions are not unique because the coordinates add up to unity. I am deriving basis functions for high-order finite elements. Sometimes one form of the expression will jump out as superior in elegance (brevity or symmetry). Sometimes a bit of thought will reveal an elegant form, but I may be missing some. Is there any systematic way to go about this? Possibly some form of condition number might be involved but at the moment I am concerned only with aesthetics.
ATTEMPT AT CLARIFICATION I have a homogeneous polynomial in the variables ${x_1, x_2,\ldots x_n}$. It can be written in a variety of forms because there is a constraint that $\sum_ix_i=1$, Usually one of the $x_i$ is distinguished and the polynomial is symmetric in the others. Will there be, in any sense, a "most elegant" way to write the polynomial? and is there a systematic way to find it? I do not need to retain homogeneity.