Assumptions on symbolic expressions

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Thanks, makes sense. What if the base field is more complicated, though? E.g. E.<w> = CyclotomicField(3) and I would like to assume $a + b + c = 0$ in the expression $(a + bw + cw^2)^3$ and would like to group the monomials by the powers of w instead of the monomials in a,b,c.

Something like this will do the job:

K.<w,a,b,c> = QQ[]
J = K.ideal( [cyclotomic_polynomial(3,w), a+b+c] )
f = ((a+b*w+c*w^2)^3).reduce(J)

Then you can extract the coefficients of powers of w as f.coefficient({w:0}) and f.coefficient({w:1}).

That's great! The only downside to this approach is the false symmetry between w,a,b,c in the poly ring definition. This doesn't correspond to reality and brings additional problems. For instance, I want to express various combinations of the w-coefficients as elementary symmetric polynomials. Is there a better way than this?

ABC.<a,b,c> = QQ[]
fABC = ABC(f.coefficient({w:0}))
SymK = SymmetricFunctions(QQ).e()
SymK.from_polynomial(fABC)

Plugging in f to SymK directly doesn't work because f.coefficient({w:0}) is of course not symmetric in K.<w,a,b,c>.

Instead of ABC(f.coefficient({w:0})) you can use f.coefficient({w:0}).specialization({w:0}). In fact for w^0 you can have simply f.specialization({w:0}) but for w^1 it has to be f.coefficient({w:1}).specialization({w:0}).

For an alternative approach, see my answer in https://ask.sagemath.org/question/75522/