Dear Maxima mailing list,
I'm a bit confused about the behaviour of Maxima's elem command for
rewriting a symmetric polynomial as a polynomials in the elementary
symmetric polynomials. In particular, the answer it gives changes
when you increase the number of underlying variables, although I think
it should not.
For instance, define the polynomial s_k with:
s(k):=elem([k],sum(concat(t,i)^k,i,1,k),makelist(concat(t,i),i,1,k));
Then expand(s(2)) gives 2*e1^2-4*e2, which is twice what I expect. In
general s(k) seems to be k times what I expect.
And if I change the definition of s(k) to use twice as many variables ti with:
s(k):=elem([3*k],sum(concat(t,i)^k,i,1,3*k),makelist(concat(t,i),i,1,3*k));
then s(k) seems to be 3k times what I expect it to be.
The calculation I am actually interested in is more complicated than
this, and involves multi_elem, so the output differs from my
expectations in more subtle ways that I don't understand.
Am I using elem correctly, and if so does anyone know how to explain
what is going on?
With many thanks,
Carl
(not a subscriber, so please cc me when responding)
(PS, compare this with Mathematica's behaviour:
<< Algebra`SymmetricPolynomials`
s[k_] :=
First[SymmetricReduction[Sum[Power[Subscript[t,i],k], {i,1,k}],
Table[Subscript[t,i], {i, 1, k}], Table[Subscript[e,j], {j, 1, k}]]]
s[2] gives e_1^2-2e_2. And the behaviour does not change if you use
twice as many variables t_i:
s[k_] :=
First[SymmetricReduction[Sum[Power[Subscript[t,i],k], {i,1,2k}],
Table[Subscript[t,i], {i, 1, 2k}], Table[Subscript[e,j], {j, 1, 2k}]]]
)