Hi all,
I am new to sage, so please forgive me if this is a trivial question.
I am trying to express certain polynomials, which are symmetric in a subset
of the variables, in terms of elementary symmetric polynomials on the
symmetric subset (with coefficients that are polynomials in the other
variables.
Here is my setup:
_____________________________________________________________________
R.<x1,x2,x3> = PolynomialRing(ZZ,3)
C.<c1,c2> = PolynomialRing(R,2)
Sym = SymmetricFunctions(R)
e = Sym.elementary()
def ElemSym(p):
# checks whether a polynomial is symmetric (coefficients in ZZ[l1,l2,l3])
f = Sym.from_polynomial(p)
return e(f)
_____________________________________________________________________
If one enters some polynomials of the desired form by hand, e.g.,
g = (x1^2 - 2*x2^2)*c1 +c1*c2 + (x1^2 -2*x2^2)*c2
and calls ElemSym(g)
then sage returns
(x1^2-2*x2^2)*e[1] + e[2]
as expected.
Now I have some code to generate the polynomial which I am interested in, I
store it as p:
p = (output of some functions)
( p is ((x1^3 - 2*x1*x2 + x3)*c1^2 - (x1*x2 - x3)*c1 + x3)*c2^2 + x1^3 +
c1^2*x3 - (x1*x2 - x3)*c1 - ((x1*x2 - x3)*c1^2 - (x1^3 - x1*x2 + x3)*c1 +
x1*x2 - x3)*c2 - 2*x1*x2 + x3)
Now the curious thing: p is (naively at least) symmetric in c1 and c2, but
calling ElemSym(p) returns an error:
ValueError: x0 + 2*x1 + x2 is not a symmetric polynomial
but if I copy the polynomial itself and call
ElemSym(((x1^3 - 2*x1*x2 + x3)*c1^2 - (x1*x2 - x3)*c1 + x3)*c2^2 + x1^3 +
c1^2*x3 - (x1*x2 - x3)*c1 - ((x1*x2 - x3)*c1^2 - (x1^3 - x1*x2 + x3)*c1 +
x1*x2 - x3)*c2 - 2*x1*x2 + x3)),
then it works and I get
(x1^3-2*x1*x2+x3)*e[] + (-x1*x2+x3)*e[1] + x3*e[1, 1] +
(x1^3-x1*x2-x3)*e[2] + (-x1*x2+x3)*e[2, 1] + (x1^3-2*x1*x2+x3)*e[2, 2] +
(3*x1*x2-3*x3)*e[3] + (-2*x1^3+4*x1*x2-2*x3)*e[3, 1] +
(2*x1^3-4*x1*x2+2*x3)*e[4]
as expected.
Can somebody help me understand what is going on here?
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