Matrix Diophantine equations over quadratic rings and their solutions
The method for solving the matrix Diophantine equations over quadratic rings is developed. On the basic of the standard form of matrices over quadratic rings with respect to $(z,k)$-equivalence previously established by the authors, the matrix Diophantine equation is reduced to equivalent matrix equ...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Vasyl Stefanyk Precarpathian National University
2020-12-01
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| Series: | Karpatsʹkì Matematičnì Publìkacìï |
| Subjects: | |
| Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/4069 |
| _version_ | 1797205836621152256 |
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| author | N.B. Ladzoryshyn V.M. Petrychkovych H.V. Zelisko |
| author_facet | N.B. Ladzoryshyn V.M. Petrychkovych H.V. Zelisko |
| author_sort | N.B. Ladzoryshyn |
| collection | DOAJ |
| description | The method for solving the matrix Diophantine equations over quadratic rings is developed. On the basic of the standard form of matrices over quadratic rings with respect to $(z,k)$-equivalence previously established by the authors, the matrix Diophantine equation is reduced to equivalent matrix equation of same type with triangle coefficients. Solving this matrix equation is reduced to solving a system of linear equations that contains linear Diophantine equations with two variables, their solution methods are well-known. The structure of solutions of matrix equations is also investigated. In particular, solutions with bounded Euclidean norms are established. It is shown that there exists a finite number of such solutions of matrix equations over Euclidean imaginary quadratic rings. An effective method of constructing of such solutions is suggested. |
| first_indexed | 2024-04-24T08:57:27Z |
| format | Article |
| id | doaj.art-54ad7483991e4f4b8f8de9d23f37af63 |
| institution | Directory Open Access Journal |
| issn | 2075-9827 2313-0210 |
| language | English |
| last_indexed | 2024-04-24T08:57:27Z |
| publishDate | 2020-12-01 |
| publisher | Vasyl Stefanyk Precarpathian National University |
| record_format | Article |
| series | Karpatsʹkì Matematičnì Publìkacìï |
| spelling | doaj.art-54ad7483991e4f4b8f8de9d23f37af632024-04-16T07:04:01ZengVasyl Stefanyk Precarpathian National UniversityKarpatsʹkì Matematičnì Publìkacìï2075-98272313-02102020-12-0112236837510.15330/cmp.12.2.368-3753553Matrix Diophantine equations over quadratic rings and their solutionsN.B. Ladzoryshyn0V.M. Petrychkovych1H.V. Zelisko2Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova str., 79060, Lviv, UkrainePidstryhach Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova str., 79060, Lviv, UkraineIvan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, UkraineThe method for solving the matrix Diophantine equations over quadratic rings is developed. On the basic of the standard form of matrices over quadratic rings with respect to $(z,k)$-equivalence previously established by the authors, the matrix Diophantine equation is reduced to equivalent matrix equation of same type with triangle coefficients. Solving this matrix equation is reduced to solving a system of linear equations that contains linear Diophantine equations with two variables, their solution methods are well-known. The structure of solutions of matrix equations is also investigated. In particular, solutions with bounded Euclidean norms are established. It is shown that there exists a finite number of such solutions of matrix equations over Euclidean imaginary quadratic rings. An effective method of constructing of such solutions is suggested.https://journals.pnu.edu.ua/index.php/cmp/article/view/4069quadratic ringmatrix$(z,k)$-equivalence of matricesmatrix diophantine equationsolution of matrix equation |
| spellingShingle | N.B. Ladzoryshyn V.M. Petrychkovych H.V. Zelisko Matrix Diophantine equations over quadratic rings and their solutions Karpatsʹkì Matematičnì Publìkacìï quadratic ring matrix $(z,k)$-equivalence of matrices matrix diophantine equation solution of matrix equation |
| title | Matrix Diophantine equations over quadratic rings and their solutions |
| title_full | Matrix Diophantine equations over quadratic rings and their solutions |
| title_fullStr | Matrix Diophantine equations over quadratic rings and their solutions |
| title_full_unstemmed | Matrix Diophantine equations over quadratic rings and their solutions |
| title_short | Matrix Diophantine equations over quadratic rings and their solutions |
| title_sort | matrix diophantine equations over quadratic rings and their solutions |
| topic | quadratic ring matrix $(z,k)$-equivalence of matrices matrix diophantine equation solution of matrix equation |
| url | https://journals.pnu.edu.ua/index.php/cmp/article/view/4069 |
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