#8693: QuadraticForm::basis_of_short_vectors may not return an actual basis.
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Reporter: afiori | Owner: justin
Type: defect | Status: new
Priority: major | Milestone:
Component: quadratic forms | Keywords: quadratic forms, basis,
automorphisms
Author: | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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QuadraticForm::basis_of_short_vectors does not actually ensure the list of
vectors it returns is a basis, it only assures that it spans a full rank
sub-lattice.
In particular in the following example (E8):
{{{
Q = QuadraticForm( matrix( [[2,0,0,0,0,0,0,1],
[0,2,1,1,1,1,1,1],
[0,1,2,1,1,1,1,1],
[0,1,1,2,1,1,1,1],
[0,1,1,1,2,1,1,1],
[0,1,1,1,1,2,1,1],
[0,1,1,1,1,1,2,0],
[1,1,1,1,1,1,0,2]] ))
B = Q.basis_of_short_vectors()
matrix(B).det()
}}}
The result is -2, which indicates we did not get a basis.
Note that the above means that sage likely returns incorrect results about
the automorphism groups of a number of interesting lattices.
I am attaching some sample code which (once properly merged {and tested})
could be used to correct the issue.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8693>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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