On Tue, Dec 17, 2024 at 09:46:03PM +0100, 'Ralf Hemmecke' via FriCAS - computer
algebra system wrote:
>
>
> On 12/17/24 21:15, Waldek Hebisch wrote:
> > On Tue, Dec 17, 2024 at 07:57:40AM -0800, Sid Andal wrote:
> > > I'm trying to construct polynomials in non-commuting variables in x, y,
> > > and
> > > z
> > > over the integers: Z<x, y, z>, or over some other commutative ring.
> > >
> > > The XPolynomial domain constructor allows to define such polynomials.
> > >
> > > However, additionally, I'd like to be able to construction the quotient,
> > > (Z<x, y, z>/I), where I is the ideal generated, say, by the following
> > > three
> > > commutators:
> > >
> > > [x, y] = x + 2y - z + 1
> > > [x, z] = 3x - y + 5z - 7
> > > [y, z] = - 4x + 8 y - 2 z + 9
> > >
> > > Are there any suitable constructors to help with this?
> >
> > AFAICS what you have above is a multivariate version of Ore algebra,
> > we have SparseMultivariateSkewPolynomial which implements them.
> >
> > We have nothing ready to use for general ideals. If your ideal
> > have a known finite Groebner basis, then it would be reasonably
> > easy to write a new constructor for quotient (in terms of
> > Groebner basis of the ideal).
>
> To me the above structure doesn't quite look like something that fits into
> the Ore context.
Well, with simplest possible grading commutators are of order 2
while left hand sides are of order 1, which allows strightforward
reduction algorithm. Concerning name, people proposed various
definitions of what multivariate Ore algebra should be.
However, I looked more carefully and it seems that
SparseMultivariateSkewPolynomial can not handle this.
--
Waldek Hebisch
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