From list to multivariate polynomial

Question

From a list of integers of length $2^d$ where $d\in \mathbb{N}$ I would like to generate a homogeneous bivariate polynomial of degree $d$. To fix ideas, let $$ l=\{a,b,c,d,e,f,g,h\}$$ ( length $2^3$) where $a,...,h \in \mathbb{R}$. I would like to define a function MultiPolGenerator such that $$\text{MultiPolGenerator[l,d]}= ax^3 + \frac{b+c+d}{3} x^2y + \frac{e+f+g}{3} xy^2 + h y^3 $$

Equivalently if it suits you best, let $$ l=\{a,b,b,b,c,c,c,d\},$$ I would like to define a function MultiPolGenerator such that $$\text{MultiPolGenerator[l,d]}= ax^3 + b x^2y + c xy^2 + d y^3 $$

An answer to either the first or the second question will be accepted. I have tried to use Internal`FromCoefficientList but the output isn't what I would expect.

Side note : it would be great if, instead of a definite number of variables (2 in this case), the function would work for an indefinite number of variables $N$

Answer 1

We may generate names for the different constants using "Array" The number of constants belonging to a monomial can be generated by "Binomial". The constants can then be partitioned using "TakeList". Then we need to divide the constants by the corresponding divisor. The different monomials can be generated using "Table". Then having the constants and the monomials, we only need to assemble the homogeneous function using "MapThread":

d = 3;
bin = Table[Binomial[d, i], {i, 0, d}];
const = TakeList[Array[Subscript[a, #] &, 2^d], bin];
const = #/Length[#] & /@ const;
monomials = Table[x^i y^(d - i), {i, 0, d}];
Plus @@ MapThread[Total[Times[#1, #2]] &, {monomials, const}]