On finiteness of some noncommutative Gröbner bases over finite fields
Yatma Diop\,^1 and Laila Mesmoudi\,^2
\,^{1,2}Department of Mathematics and Computer Sciences, Cheikh Anta Diop University, Dakar, Senegal
Pages 74-80 | Received 26 July 2024, Accepted 03 December 2024, Published 10 July 2025
Abstract
Eisenbud and al. proved that if \mathbb{K} is a field of characteristic 0 and \gamma:\mathbb{K}\langle X_1,…,X_n\rangle\longrightarrow\mathbb{K}[x_1,…x_n], the map from the noncommutative ploynomial ring to the commutative one which sends X_i to x_i then any noncommutative ideal \mathcal{J}=\gamma^{-1}(\mathcal{I}) has a finite Gröbner basis even after a linear change of variables. By an example they prove that if \mathbb{K} is of characteristic p\neq 0 then this result does not always hold. In this work, we consider a coefficient finite field \mathbb{K}. Then we first give a necessary and sufficient condition for any ideal of the form \gamma^{-1} (\mathcal{I}) to have finite Gröbner basis. We secondly prove that this condition is satisfied for any 0-dimensionnal \mathcal{I}. We finish by investigating the particular case where \mathcal{I} is a principal ideal.
Keywords: Finite field, Finiteness, Initial ideal, Linear change of variables, Noncommutative Gröbner bases.
MSC numbers: 16Z05,08A62,13B25,13B02.
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