TY - JOUR
AU - Ladzoryshyn, N.B.
AU - Petrychkovych, V.M.
AU - Zelisko, H.V.
PY - 2020/12/20
Y2 - 2024/07/12
TI - Matrix Diophantine equations over quadratic rings and their solutions
JF - Carpathian Mathematical Publications
JA - Carpathian Math. Publ.
VL - 12
IS - 2
SE - Scientific articles
DO - 10.15330/cmp.12.2.368-375
UR - https://journals.pnu.edu.ua/index.php/cmp/article/view/4069
SP - 368-375
AB - <p>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.</p>
ER -