I wonder, if it is faster to compute the complete polynomial directly or go
the way via the free algebra? Anyway, I have to deal with pretty much
variables, hence a nice fast algorithm would be nice.
Best Michael
Am Dienstag, 5. Mai 2020 10:24:39 UTC+2 schrieb Michael Jung:
>
> 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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