Re: Urgent aid needed
- To: mathgroup at smc.vnet.net
- Subject: [mg16437] Re: [mg16398] Urgent aid needed
- From: Daniel Lichtblau <danl>
- Date: Sat, 13 Mar 1999 02:21:47 -0500
- References: <199903110717.CAA12380@smc.vnet.net.>
- Sender: owner-wri-mathgroup at wolfram.com
c.a.e at wanadoo.fr wrote:
>
> Hi,
> Here's the subject:
> Let e,p,q be elements of commutative unitary ring such as e^3==-1
> Prove that D==p*e^2-q divides P==e^2*p^3-p^3*q-e^2*q^3+e^2*p*q^3.
> Please write to me at c.a.e at wanadoo.fr
> Thank you in advance
The following suffices to prove this for your set-up.
In[9]:= polys = {p*e^2-q, e^3+1};
In[10]:= vars = {e,p,q};
In[11]:= gb = GroebnerBasis[polys, vars];
In[12]:= PolynomialReduce[e^2*p^3-p^3*q-e^2*q^3+e^2*p*q^3, gb, vars]
[[2]]
Out[12]= 0
This shows that e^2*p^3-p^3*q-e^2*q^3+e^2*p*q^3 is in the ideal
generated by {p*e^2-q, e^3+1}, which gives a not-quite-constructive
existence proof.
Forming the explicit quotient in terms of original generators p*e^2-q
and e^3+1 is a more difficult matter (PolynomialReduce[...][[1]] gives
the quotients in terms of the elements of gb). One needs a way to
express elements of the Groebner basis in terms of these original
generators, which means knowing a basis for certain syzygy ideals. There
is a way to compute this stuff in Mathematica, not terribly difficult in
fact, but alas too much trouble for me to describe in short time.
Daniel Lichtblau
Wolfram Research
- References:
- Urgent aid needed
- From: c.a.e@wanadoo.fr
- Urgent aid needed