Will "solve_right" return integer solutions if one exists?

asked 2016-02-03 20:08:49 +0200

done_with_fish
273 ●6 ●16 ●32

If A is an integer matrix and b is an integer vector, then will the command A.solve_right(b) return an integer vector if the equation Ax=b is solvable over the integers?

answered 2016-02-03 22:09:04 +0200

vdelecroix
7397 ●19 ●81 ●164 http://www.labri.fr/pe...

The answer is no

sage: A = matrix(2,3, [3,4,1,5,9,2])
sage: A.solve_right(vector((1,2)))
(1/7, 1/7, 0)
sage: A * vector((0,0,1))
(1, 2)

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Comments

There is the natural question of how to get integer solutions in an efficient manner... but presumably one could create arbitrarily annoying matrices where that would not be so easy to do programmatically?

kcrisman ( 2016-02-04 14:50:07 +0200 )edit

It is not so hard either ;-) Just some arithmetic with Z-modules.

vdelecroix ( 2016-02-04 16:28:39 +0200 )edit

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