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.
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
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