# Question:Possibly a bug in Groebner[NormalForm] ?

## Question:Possibly a bug in Groebner[NormalForm] ?

Maple 14

Hello everyone,

Groebner[NormalForm] fails to produce quotients for the members of Ideal when MonomialOrder is constructed with Ore_algebra.

e.g. Groebner[NormalForm](F[1], G, T, 'Q');
results in "Error, (in Groebner:-NormalForm) numeric exception: division by zero"
if F[1] in <G> and T is MonomialOrder

But it works if we dont ask for quotients or use ShortMonomialOrder

Any ideas if it is intended behaviour or a bug?

Steps to reproduce:
with(Groebner):
with(Ore_algebra):
F := [x+y+z, x*y+y*z+z*x, x*y*z-1]:
A := poly_algebra(x, y, z):
T := MonomialOrder(A, tdeg(x, y, z)):
G := Groebner[Basis](F, T):
# NormalForm works if we dont ask for quotients
Groebner[NormalForm](F, G, T);
# And fails if we do for the members of the Ideal
# in case if Monomial order is specified by Ore_algebra
Groebner[NormalForm](F[1], G, T, 'Q');
Groebner[NormalForm](F[2]+F[1], G, T, 'Q');
Groebner[NormalForm](3*F[3], G, T, 'Q');

Output:
Error, (in Groebner:-NormalForm) numeric exception: division by zero
Error, (in Groebner:-NormalForm) numeric exception: division by zero
Error, (in Groebner:-NormalForm) numeric exception: division by zero

﻿