Re: [Axiom-developer] groebner?

[Ralf Hemmecke]

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 Gröbner 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