# unexpected TypeError while creating a polyhedron over ZZ with Normaliz

The following code

Polyhedron( ieqs=[(-1, 1, -1, 0, -4), (2, -1, 1, 0, 4), (0, 1, 1, 2, -2), (2, -1, -1, -2, 2), (0, 0, -1, 1, 1), (1, 0, 1, -1, -1), (0, 1, 0, 0, 1), (1, -1, 0, 0, -1)], base_ring=ZZ, backend='normaliz' )

gives me TypeError: no conversion of this rational to integerin Sage 9.4. It can also be seen in SagMathCell - https://sagecell.sagemath.org/?q=soiohx

The error disappears if base_ring is changed to QQ. What's wrong?

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1

Does it have integer vertices ?

( 2021-09-06 20:51:04 +0200 )edit

Does it have to? I thought it's reduced to the integral hull when defined over ZZ.

( 2021-09-06 20:56:48 +0200 )edit

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Thanks to the pointer from FrédéricC, it appears that Sage expects that the definition of polyhedron over ZZ is strict with all vertices being integral upfront. The behavior I expected can be achieved by first defining a polyhedron over QQ, computing its integral hull, and then changing base ring to ZZ like

P = Polyhedron( ieqs=[(-1, 1, -1, 0, -4), (2, -1, 1, 0, 4), (0, 1, 1, 2, -2), (2, -1, -1, -2, 2), (0, 0, -1, 1, 1), (1, 0, 1, -1, -1), (0, 1, 0, 0, 1), (1, -1, 0, 0, -1)], base_ring=QQ, backend='normaliz' )
Q = P.integral_hull().change_ring(ZZ)

The error message in the case of non-integral vertices could have been more informative, nevertheless.

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