### A Factorization Algorithm for G-Algebras and Applications

**2016-01-31**

1602.00296 | math.RA

It has been recently discovered by Bell, Heinle and Levandovskyy that a large
class of algebras, including the ubiquitous $G$-algebras, are finite
factorization domains (FFD for short).
Utilizing this result, we contribute an algorithm to find all distinct
factorizations of a given element $f \in \mathcal{G}$, where $\mathcal{G}$ is
any $G$-algebra, with minor assumptions on the underlying field.
Moreover, the property of being an FFD, in combination with the factorization
algorithm, enables us to propose an analogous description of the factorized
Gr\"obner basis algorithm for $G$-algebras. This algorithm is useful for
various applications, e.g. in analysis of solution spaces of systems of linear
partial functional equations with polynomial coefficients, coming from
$\mathcal{G}$. Additionally, it is possible to include inequality constraints
for ideals in the input.

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