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I look for a solver or reference rather than an answer about how to solve the diophantine matrix equation. One states it below.

$\mathbf{X}\mathbf{N} + \mathbf{Y}\mathbf{D} = \mathbf{I}$

All entries of X, N, Y and D belongs to polynomials. Therefore, it is cumbersome to achieve.

I thank in advance.

Bruno Lobo
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1 Answers1

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Solving that system is of course equivalent to solving $$\begin{pmatrix}X & Y\end{pmatrix} \begin{pmatrix}N \\ D\end{pmatrix} = I$$

For examples of algorithms commonly used in computer algebra systems to solve such a system over the integers, look at:

J. D. Dixon. Exact solution of linear equations using p-adic expansion. Numerische Mathematik, 40:137–141, 1982.

T.W.J. Chou and G.E. Collins., Algorithms for the solution of systems of linear Diophantine equations, SIAM J. Computing, 11(4):687--708, 1982.

len
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  • I thank for the answer, but X, Y, N and D in my application os over the polynomial matrix ring. Specifically, I am more interested in any open source tool to obtain such matrices. – Bruno Lobo Jun 29 '19 at 19:24
  • What do you mean by “polynomial matrix ring”? Can you state your problem more clearly and give some context? – len Jun 30 '19 at 03:28