That does indeed seem simple. Here is an even shorter version that only
needs one matrix.
R = LaurentPolynomialRing(QQ, "p1, p2, p3, p4, Q0, Q1, Q2, Q3, Q4, w, z")
p1, p2, p3, p4, Q0, Q1, Q2, Q3, Q4, w, z = R.gens()
L = Matrix([[-p1, Q1, 0, 0, 0, -4*Q0/z, 0, 0, 0],
[Q1, p1-p2, Q2, 0, 4*Q0/z, 0, 0, 0, 0],
[0, Q2, p2-p3, Q3, 0, 0, 0, 0, 0],
[0, 0, Q3, p3-2*p4, 0, 0, 0, 0, 2*Q4],
[0, Q0*z, 0, 0, p1, -Q1, 0, 0, 0],
[-Q0*z, 0, 0, 0, -Q1, -p1+p2, -Q2, 0, 0],
[0, 0, 0, 0, 0, -Q2, -p2+p3, -Q3, 0],
[0, 0, 0, 0, 0, 0, -Q3, -p3+2*p4, -Q4],
[0, 0, 0, Q4, 0, 0, 0, -2*Q4, 0]])
print(f"Laurent: {L.det()}")
print(f"SR: {L.change_ring(SR).det().factor()}")
For me, the output is:
Laurent: 0
SR: 4*Q0*Q1*Q2^2*Q3^2*Q4^2*p1*(z + 1)*(z - 1)/z
I thought maybe the problem had something to do with L being singular, but
adding the identity matrix to L doesn't solve the problem:
I = matrix.identity(9)
((L.change_ring(SR) + 42*I).det() - SR((L + 42*I).det())).factor()
Both individual determinants are a mess, but the difference I get is
4*Q0*Q1*Q2^2*Q3^2*Q4^2*(p1 - 42)*(z + 1)*(z - 1)/z
It is surprising to me that this is the same as the above determinant in
SR, except for the 42. And I get a similar answer if I replace 42 with a
different constant, or even put in a variable by defining a polynomial ring
over R.
I think a ticket should be opened.
On Thursday, December 17, 2020 at 3:18:02 PM UTC-7 Michael Orlitzky wrote:
> On 12/16/20 3:27 PM, Linden Disney wrote:
> > Ok I've modified the code to plain sage to make it more useful and I've
> > copied it below. Given that it's hard to compare the determinants of the
> > raw matrices, as they are defined in terms of different variables, I
> > have found the z^2 coefficient in each case and you can see they are
> > different. I've tested this in Sage 9.1 command line and this has worked.
> >
>
> Great, thanks. I was able to simplify this even further... the
> determinants still disagree at the L1/L2 stage, but look a lot nicer.
> Maybe now someone can figure out what's going wrong. With these
> definitions, you can even try (L1 == L2) and it returns True.
>
> p1, p2, p3, p4, Q0, Q1, Q2, Q3, Q4, w, z = SR.var("p1, p2, p3, p4, Q0,
> Q1, Q2, Q3, Q4, w, z")
> L1 = Matrix([[-p1, Q1, 0, 0, 0, -4*Q0/z, 0, 0, 0],
> [Q1, p1-p2, Q2, 0, 4*Q0/z, 0, 0, 0, 0],
> [0, Q2, p2-p3, Q3, 0, 0, 0, 0, 0],
> [0, 0, Q3, p3-2*p4, 0, 0, 0, 0, 2*Q4],
> [0, Q0*z, 0, 0, p1, -Q1, 0, 0, 0],
> [-Q0*z, 0, 0, 0, -Q1, -p1+p2, -Q2, 0, 0],
> [0, 0, 0, 0, 0, -Q2, -p2+p3, -Q3, 0],
> [0, 0, 0, 0, 0, 0, -Q3, -p3+2*p4, -Q4],
> [0, 0, 0, Q4, 0, 0, 0, -2*Q4, 0]])
>
> L1.det()
>
>
> R = LaurentPolynomialRing(QQ, "p1, p2, p3, p4, Q0, Q1, Q2, Q3, Q4, w, z")
> p1, p2, p3, p4, Q0, Q1, Q2, Q3, Q4, w, z = R.gens()
>
> L2 = Matrix([[-p1, Q1, 0, 0, 0, -4*Q0/z, 0, 0, 0],
> [Q1, p1-p2, Q2, 0, 4*Q0/z, 0, 0, 0, 0],
> [0, Q2, p2-p3, Q3, 0, 0, 0, 0, 0],
> [0, 0, Q3, p3-2*p4, 0, 0, 0, 0, 2*Q4],
> [0, Q0*z, 0, 0, p1, -Q1, 0, 0, 0],
> [-Q0*z, 0, 0, 0, -Q1, -p1+p2, -Q2, 0, 0],
> [0, 0, 0, 0, 0, -Q2, -p2+p3, -Q3, 0],
> [0, 0, 0, 0, 0, 0, -Q3, -p3+2*p4, -Q4],
> [0, 0, 0, Q4, 0, 0, 0, -2*Q4, 0]])
>
> L2.det()
>
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