Hi Michael,
I do not know about preimplemented algorithms outside Sage itself, but it
would not be so difficult to define this map directly. For example, as a
map to the free algebra:
sage: T.<u,v,w> = FreeAlgebra(QQ)
sage: R.<x,y,z> = PolynomialRing(QQ)
sage: def _polarize(f):
....: M = T.indices()
....: D = {}
....: for e, c in zip(f.exponents(), f.coefficients()):
....: c2 = c / multinomial(e)
....: for p in Permutations(M(list(enumerate(e))).to_list()):
....: D[M.prod(p)] = c2
....: return T._from_dict(D)
sage: from sage.categories.morphism import SetMorphism
sage: polarize = SetMorphism(Hom(R, T, Modules(R.base_ring())), _polarize)
sage: polarize(x^2*y + x*y*z)
1/3*u^2*v + 1/3*u*v*u + 1/6*u*v*w + 1/6*u*w*v + 1/3*v*u^2 + 1/6*v*u*w + 1/6*
v*w*u + 1/6*w*u*v + 1/6*w*v*u
Now this is a one-sided inverse for symmetrization:
sage: symmetrize = T.module_morphism(on_basis=lambda a: R.prod(R.gen(ai)^ni
for ai, ni in a._element_list),
....: codomain=R)
sage: f = R.random_element(6, 15)
sage: symmetrize(polarize(f)) == f
True
The main problem with this is the combinatorial complexity, as the output
quickly becomes huge for larger input. So it might be better if you can
avoid computing this representation explicitly in your use case. After all,
all this does is to divide the coefficients by some multinomial
coefficient. Nevertheless, I still think this functionality would be useful
for small cases and nice to have in Sage.
Alternatively, if you are more interested in viewing this as a multilinear
form, then TensorFreeModule may be more suitable. This representation keeps
track of symmetries in order to store coefficients without redundancy. Here
is a quick example, but there may be room for improvement:
def polarize_form(f):
n = f.parent().ngens()
V = FiniteRankFreeModule(f.parent().base_ring(), n, name='V')
B = V.basis('x')
from sage.tensor.modules.tensor_free_module import TensorFreeModule
T = TensorFreeModule(V, (0, n))
t = T([], name='t', sym=range(n))
for e, c in zip(f.exponents(), f.coefficients()):
idx = [j for j, ej in e.sparse_iter() for _ in range(ej)]
t.set_comp(B)[idx] = c / multinomial(e)
return t
sage: t = polarize_form(x^2*y + x*y*z)
sage: t.symmetries()
symmetry: (0, 1, 2); no antisymmetry
sage: t.display()
t = 1/3 x_0*x_0*x_1 + 1/3 x_0*x_1*x_0 + 1/6 x_0*x_1*x_2 + 1/6 x_0*x_2*x_1 +
1/3 x_1*x_0*x_0 + 1/6 x_1*x_0*x_2 + 1/6 x_1*x_2*x_0 + 1/6 x_2*x_0*x_1 + 1/6
x_2*x_1*x_0
sage: V = t.base_module()
sage: t(V([0,0,1]), V([1,7,3]), V([1,0,0]))
7/6
Best,
Markus
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