# Re: [sage-support] AssertionError when finding roots over the Real Algebraic Field

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### Dima Pasechnik

Sep 22, 2021, 4:19:47 AM9/22/21
to sage-support, sage-devel, sage-nt

On Wed, Sep 22, 2021 at 8:10 AM Tracy Hall <h.t...@gmail.com> wrote:
I ran into an assertion error when trying to return a sorted list whose key was a certain linear combination of eigenvalues of the Laplacian matrix over graphs on nine vertices. Digging into it a bit, the failure happened when comparing an algebraic real number against the same number that was constructed differently (starting with the graph complement). Digging further, the error happens when finding roots of a certain degree 56 polynomial over AA (all the roots are real) but there is no error doing the same thing over QQbar.

Here is a minimal working example:

P.<z> = QQ[]
rootlist = (z^8 - 32*z^7 + 425*z^6 - 3044*z^5 + 12789*z^4 - 32090*z^3 + 46672*z^2 - 35734*z + 10917).roots(AA)
problem = rootlist[-1][0] - rootlist[0][0] - 9

problem.minpoly().roots(AA)

indeed, problem.minpoly().roots(QQbar) produces a list of 56 QQbar elements, more precisely, pairs (t,1)), each t convertible into AA.
One funny discrepancy is that one of the elements of this list is shown as
(-6.390396068452545? + 0.?e-170*I, 1)

sage: rrr=problem.minpoly().roots(QQbar)
sage: rrr[-1]
(-6.390396068452545? + 0.?e-170*I, 1)
sage: AA(rrr[-1][0])
-6.390396068452545?

Not sure whether this is the cause of the bug, though.

Dima

### Dima Pasechnik

Sep 22, 2021, 4:23:13 AM9/22/21
to sage-support, sage-devel, sage-nt
The behaviour  of QQbar is not very consistent there. Only one root is shown with an imaginary part, but
the polynomial has integer coefficients --- it ought to "know" that complex roots come in pairs :-)

Dima