I have calculated several homogeneous polynomials in 4,5 or 6 variables $t_1,\dots,t_6$.
I would like to rewrite them as a sum of products of specific lower degree polynomials, which have a meaning in the problem I'm solving.
For example, the simplest polynomial I have is $$ P_4(t_1,t_2,t_3,t_4) = t_1t_2+t_1(t_3+t_4)+t_2(t_3+t_4) $$ which I would like to express in terms of $t_1+t_2, t_2+t_3+t_4, t_1+t_3+t_4$ (only them, although not necessarily all of them). For example, it's easy to see that $$P_4(t_1,t_2,t_3,t_4) = (t_1+t_2)(t_2+t_3+t_4) - t_2^2$$ but I'd prefer something like $$P_4(t_1,t_2,t_3,t_4) = a(t_1+t_2)(t_2+t_3+t_4) + b(t_1+t_2)(t_1+t_3+t_4) + c(t_1+t_3+t_4)(t_2+t_3+t_4)$$
Question: Is there a way to ask Mathematica this sort of partial factorization?
Thank you in advance!
Note: I don't know if a solution exists for my exact polynomials (maybe I have to search for a different interpretation), but I think the general problem is interesting in its own right.
SolveAlways[ t1 t2 + t1 (t3 + t4) + t2 (t3 + t4) == a (t1 + t2) (t2 + t3 + t4) + b (t1 + t2) (t1 + t3 + t4) + c (t1 + t3 + t4) (t2 + t3 + t4), {t1, t2, t3, t4}]
. $\endgroup$p1 /. Solve[{p1==x y+x (w + z)+y (w + z), q==x+y, p==y+w+z, r==x+w+z}, {p1}, {x, y, w, z}] // Simplify
or e.g.GroebnerBasis[{x y + x (w + z) + y (w + z), x + y - q, y + w + z - p, x + w + z - r}, {p, q, r}, {x, y, w, z}] // Simplify
. Take a look at these answers Am I missing anything? and How to replace every possible A+B and AB in ... $\endgroup$