{"id":2650,"date":"2025-07-26T18:04:20","date_gmt":"2025-07-26T18:04:20","guid":{"rendered":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/?page_id=2650"},"modified":"2025-07-26T21:29:04","modified_gmt":"2025-07-26T21:29:04","slug":"on-finiteness-of-some-noncommutative-grobner-bases-over-finite-fields","status":"publish","type":"page","link":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/on-finiteness-of-some-noncommutative-grobner-bases-over-finite-fields\/","title":{"rendered":"On finiteness of some noncommutative Gr\u00f6bner bases over finite fields"},"content":{"rendered":"\n
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Moroccan Journal of Algebra and Geometry with Applications<\/strong><\/p>\n\n\n\n

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On finiteness of some noncommutative Gr\u00f6bner bases over finite fields<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

Yatma Diop<\/strong>\\,^1<\/span><\/i> and Laila Mesmoudi\\,^2<\/span>
\\,^{1,2}<\/span>Department of Mathematics and Computer Sciences, Cheikh Anta Diop University, Dakar, Senegal
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Pages 74-80 | Received 26 July 2024, Accepted 03 December 2024, Published 10 July 2025 <\/span><\/p>\n\n\n\n

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Abstract<\/span><\/strong><\/p>\n\n\n\n

Eisenbud and al. proved that if \\mathbb{K}<\/span> is a field of characteristic 0 and \\gamma:\\mathbb{K}\\langle X_1,\u2026,X_n\\rangle\\longrightarrow\\mathbb{K}[x_1,\u2026x_n]<\/span>, the map from the noncommutative ploynomial ring to the commutative one which sends X_i<\/span> to x_i<\/span> then any noncommutative ideal \\mathcal{J}=\\gamma^{-1}(\\mathcal{I})<\/span> has a finite Gr\u00f6bner basis even after a linear change of variables. By an example they prove that if \\mathbb{K}<\/span> is of characteristic p\\neq 0<\/span> then this result does not always hold. In this work, we consider a coefficient finite field \\mathbb{K}<\/span>. Then we first give a necessary and sufficient condition for any ideal of the form \\gamma^{-1} (\\mathcal{I})<\/span> to have finite Gr\u00f6bner basis. We secondly prove that this condition is satisfied for any 0-dimensionnal \\mathcal{I}<\/span>. We finish by investigating the particular case where \\mathcal{I}<\/span> is a principal ideal.<\/p>\n\n\n\n

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Keywords<\/strong>:<\/span> Finite field, Finiteness, Initial ideal, Linear change of variables, Noncommutative Gr\u00f6bner bases.<\/p>\n\n\n\n

MSC numbers<\/strong>:<\/span> 16Z05,08A62,13B25,13B02.<\/p>\n\n\n\n

Downloads:<\/strong> Full-text PDF<\/a><\/p>\n<\/div>\n\n\n\n

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Moroccan Journal of Algebra and Geometry with Applications Latest articles On finiteness of some noncommutative Gr\u00f6bner bases over finite fields Yatma Diop and Laila Mesmoudi 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 Read More<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-page-builder.php","meta":{"footnotes":""},"class_list":["post-2650","page","type-page","status-publish","hentry","entry"],"_links":{"self":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2650","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/comments?post=2650"}],"version-history":[{"count":6,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2650\/revisions"}],"predecessor-version":[{"id":2704,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2650\/revisions\/2704"}],"wp:attachment":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/media?parent=2650"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}