I have a large homogenous multivariate polynomial in, say, 5 variables $a,b,c,d,e$. As an example take the polynomial $$a^4+2abcd+a^2 b^2+e^4+cde^2.$$ Now I would like to replace $k$-th power of any variable by $c_k$, so the above example would become
$$c_4+2c_1^4+c_2^2+c_4+c_1^2c_2.$$ At the moment I am using the CoefficientRules
command which gives in the above example
coef={{4, 0, 0, 0, 0} -> 1, {2, 2, 0, 0, 0} -> 1, {1, 1, 1, 1, 0} -> 2,
{0, 0, 1, 1, 2} -> 1, {0, 0, 0, 0, 4} -> 1}.
Then I sum over the rules in this list by
Sum[coef[[All, 2]][[k]]*
Product[c[coef[[All, 1]][[k, j]]], {j, 1, Length[coef[[All, 1]][[k]]]}],
{k, 1, Length[coef]}]
which is very inefficient. Any suggestions?