Multivarate dependence

[gituliar]

@magv take a look at examples/eec_01.m and examples/eec_02.m files which contain matrices of x, z, eps variables. As usually, x is a free variable, eps -> 0, and z is a new parameter.

Currently, Fuchsia is able to normalize and factorize diagonal blocks of eec_02.m up to some point when eigenvalues become rational. (That requires some change of variables and is not important here.) What is important is that Fuchsia can deal with additional symbolic dependence in z in normalization and factorization! There are still some problems with fuchsification step: try it with eec_01.m. Can you take a look what is going on there and tell whether it is possible to extend fuchsification to deal with eec_01.m.

[magv]

I'm not sure what you're looking for here:

sage: m = import_matrix_from_file("examples/eec_01.m")
sage: fuchsify(m.subs({z:123}), x)
FuchsiaError: matrix cannot be reduced to Fuchsian form

So, the matrix is irreducible with z=123 (or any other integer I've tried), so it follows that it's irreducible with symbolic z as well.

Also note that residue eigenvalues of this matrix at x=1/z are [(2*eps - 1)/z^2, 0]. So even if this matrix was fuchsifiable, it would not be normalizable (at least by Fuchsia), because we can't perform 1/z^2 normalization steps without knowing z. We've talked about this problem before.

[gituliar]

That makes sense. The matrix eec_01.m was not a complete system, I fixed that issue and managed to find the epsilon form, see eec_01_eps.m.

It means that in principle Fuchsia is able to solve multivariate problems: above is the first example taken from the real physical problem, i.e., not artificially constructed. For me it looks like this direction is worth to invest in. What is your feeling?

[magv]

Yes, it does seem to work correctly for this matrix. Possibly because it's a triangular matrix and 'z' never appears on the diagonal of any residue. It will work for a number of similar matrices too. And yet, you've seen the counter-examples, and you know that Fuchsia is not designed to detect them. It will fail unexpectedly when given one of them, or worse: fail silently and give incorrect results. I'm sure you agree that the latter is 100% unacceptable. Basically speaking, if you want to advertise limited support for symbolic constants in matrices, you need to audit every step of every algorithm, determine if those steps are possible (logically) and will work (in Sage) given the assumption of symbolic constants, add checks if breakage is possible, and raise errors to inform the user. This is doable, of course. You just need to be very thorough.

[magv]

I think it's long past due to post an update to this issue. In short, Fuchsia does work fine with multivariate matrices, and in fact this capability has been successfully applied to research problems.

The reservation I had previously are matrices containing terms like 1/(x-a)/(x-b), because depending on whether a equals b or not, this expression may be a double pole or two single poles, and thus require completely different reductions. In such cases Fuchsia will silently assume that a <> b, and reduce accordingly. The reservations I had are resolved by seeing that in practice if you have ambiguous terms like that, then you already had to deal with them long before Fuchsia comes into play. For example, a diagram with a pair of equal masses already requires different treatment than a diagram with distinct masses. In this regard, having Fuchsia assume a <> b does not pose a limitation in practice.