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Signature Gr\"obner bases in free algebras over rings
Authors
Clemens Hofstadler
Thibaut Verron
Publication date
18 July 2023
Publisher
Doi
Cite
View
on
arXiv
Abstract
We generalize signature Gr\"obner bases, previously studied in the free algebra over a field or polynomial rings over a ring, to ideals in the mixed algebra
R
[
x
1
,
.
.
.
,
x
k
]
⟨
y
1
,
…
,
y
n
⟩
R[x_1,...,x_k]\langle y_1,\dots,y_n \rangle
R
[
x
1
​
,
...
,
x
k
​
]
⟨
y
1
​
,
…
,
y
n
​
⟩
where
R
R
R
is a principal ideal domain. We give an algorithm for computing them, combining elements from the theory of commutative and noncommutative (signature) Gr\"obner bases, and prove its correctness. Applications include extensions of the free algebra with commutative variables, e.g., for homogenization purposes or for performing ideal theoretic operations such as intersections, and computations over
Z
\mathbb{Z}
Z
as universal proofs over fields of arbitrary characteristic. By extending the signature cover criterion to our setting, our algorithm also lifts some technical restrictions from previous noncommutative signature-based algorithms, now allowing, e.g., elimination orderings. We provide a prototype implementation for the case when
R
R
R
is a field, and show that our algorithm for the mixed algebra is more efficient than classical approaches using existing algorithms.Comment: 10 page
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oai:arXiv.org:2302.06483
Last time updated on 04/03/2023