Normalization of Quaternionic Polynomials in Coordinate-Free Quaternionic Variables in Conjugate-Alternating Order
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917992782102528 |
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| author | Li, Hongbo Wang, Zhengyang Liu, Yue Huang, Lei Shao, Changpeng |
| author_facet | Li, Hongbo Wang, Zhengyang Liu, Yue Huang, Lei Shao, Changpeng |
| contents | Quaternionic polynomials occur naturally in applications of quaternions in science and engineering, and normalization of quaternionic polynomials is a basic manipulation. Once a Groebner basis is certified for the defining ideal I of the quaternionic polynomial algebra, the normal form of a quaternionic polynomial can be computed by routine top reduction with respect to the Groebner basis. In the literature, a Groebner basis under the conjugate-alternating order of quaternionic variables was conjectured for I in 2013, but no readable and convincing proof was found.
In this paper, we present the first readable certification of the conjectured Groebner basis. The certification is based on several novel techniques for reduction in free associative algebras, which enables to not only make reduction to S-polynomials more efficiently, but also reduce the number of S-polynomials needed for the certification. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14990 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normalization of Quaternionic Polynomials in Coordinate-Free Quaternionic Variables in Conjugate-Alternating Order Li, Hongbo Wang, Zhengyang Liu, Yue Huang, Lei Shao, Changpeng Symbolic Computation I.1.1, F.2.1 Quaternionic polynomials occur naturally in applications of quaternions in science and engineering, and normalization of quaternionic polynomials is a basic manipulation. Once a Groebner basis is certified for the defining ideal I of the quaternionic polynomial algebra, the normal form of a quaternionic polynomial can be computed by routine top reduction with respect to the Groebner basis. In the literature, a Groebner basis under the conjugate-alternating order of quaternionic variables was conjectured for I in 2013, but no readable and convincing proof was found. In this paper, we present the first readable certification of the conjectured Groebner basis. The certification is based on several novel techniques for reduction in free associative algebras, which enables to not only make reduction to S-polynomials more efficiently, but also reduce the number of S-polynomials needed for the certification. |
| title | Normalization of Quaternionic Polynomials in Coordinate-Free Quaternionic Variables in Conjugate-Alternating Order |
| topic | Symbolic Computation I.1.1, F.2.1 |
| url | https://arxiv.org/abs/2504.14990 |