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Matrix Diophantine equations over quadratic rings and their solutions. (English) Zbl 1458.15029

Summary: 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.

MSC:

15A24 Matrix equations and identities
11D09 Quadratic and bilinear Diophantine equations
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References:

[1] N.S. Dzhaliuk, V.M. Petrychkovych, The structure of solutions of the matrix linear unilateral polynomial equation with two variables , Carpathian Mathematical Publications: Vol. 9 No. 1 (2017) · Zbl 1370.15009
[2] N.B. Ladzoryshyn, On equivalence of pairs of matrices, which determinants are primes powers, over quadratic Euclidean rings , Carpathian Mathematical Publications: Vol. 5 No. 1 (2013) · Zbl 1391.15046
[3] N.S. Dzhaliuk, V.M. Petrychkovych, The matrix Diophantine equations \(AX+BY=C\) , Carpathian Mathematical Publications: Vol. 3 No. 2 (2011) · Zbl 1391.15054
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