Hi Markus,
these are awesome pieces of code. Thank you! Unfortunately, I need this as
the full polynomial since I have to do different computations with each
variable...
Particularly, I have a homogeneous polynomial of matrices and want it to
polarize.
Best wishes
MIchael
Am Dienstag, 28. April 2020 20:23:01 UTC+2 schrieb Markus Wageringel:
>
> 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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