Non-Commuting Plynomials

[Sid Andal]

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?

Thanks,

SWA

[Waldek Hebisch]

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).

--
Waldek Hebisch

[Ralf Hemmecke]

To me the above structure doesn't quite look like something that fits
into the Ore context.

I do not immediately see, how the grading would be done for that example
that corresponds to the following paper.

https://scholar.google.de/citations?view_op=view_citation&hl=de&user=iKkds9kAAAAJ&citation_for_view=iKkds9kAAAAJ:qjMakFHDy7sC
https://ul.qucosa.de/api/qucosa%3A34526/attachment/ATT-0/

Ralf

[Waldek Hebisch]

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

[Kurt Pagani]

I would use a function/relation/equations on the TERMS=Record(k:FMB,c:R) of  XDistributedPolynomial(B,R). This way one gets a (pseudo) quotient algebra (see my example ;).

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