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## Re: [Axiom-developer] groebner?

**From**: |
Ralf Hemmecke |

**Subject**: |
Re: [Axiom-developer] groebner? |

**Date**: |
Thu, 16 Feb 2006 22:52:40 +0100 |

**User-agent**: |
Thunderbird 1.5 (X11/20051201) |

Hi Ray,
I think that is correct.
However, I guess, you are asking AXIOM to do the wrong thing.
HDMP(X, R) represents the polynomial ring R[X]. In your case R is

`"Polynomial Integer". If you look up "Polynomial" in Hyperdoc, you will
``find that this is a domain that allows any symbol as a variable and
``Integers as coefficients. Roughly speaking, it is Z[Y] (where Y is a
``infinitely set of variables).
`

`If the groebner routine is given something of the form Z[Y][X] where
``Z[Y] is the coefficient domain then the result has to be interpreted in
``Z(Y)[X] (the fraction field of the coefficients ajoint with the
``variables from the set X).
`

`Unfortunately, in the way you coerce your polynomials X is a subset of
``Y. In other words the polynomial eq1H lies in the ring of coefficients
``and is therefore a constant.
`

`Well, computing a GrÃ¶bner basis of a non-zero constant always gives the
``ideal generated by 1. So I would have expected [1] as an output. But [p]
``is also correct, since p is an element of Z(Y) and, therefore,
``invertible. p generates the same ideal as 1 and that is equal to the
``whole ring Z(Y)[X].
`

`You see, be careful with Polynomial(Integer). The interpreter is not as
``smart as you may wish in this case. However, you could help by adding
``the right types.
`
list: List(Symbol) :=[p,Vr,Vt,e]
eq: HDMP(vars, INT) := ((-Vr^3+Vr^2)*Vt+Vr^3-Vr^2)*p
groebner [eq]

`The result is the same as [eq] which is to be expected if you just give
``one polynomial.
`
Ralf
On 02/16/2006 01:40 PM, Raymond E. Rogers wrote:

Could someone tell me if this is wrong; or if it's right why?
-------------------------
list:=[p,Vr,Vt,e]
eq1H:=((-Vr^3+Vr^2)*Vt+Vr^3-Vr^2)*p :: HDMP(list,POLY INT)
Excerpt from an axiom session:
(24) -> eq1H
(24) ->
3 2 3 2
(24) ((- Vr + Vr )Vt + Vr - Vr )p
Type:
HomogeneousDistributedMultivariatePolynomial([p,Vr,Vt,e],Polynomial Integer)
(25) -> groebner [eq1H]
(25) ->
(25) [p]
Type: List
HomogeneousDistributedMultivariatePolynomial([p,Vr,Vt,e],Polynomial Integer)
-----------------------------
Ray