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MATRIX EQUATIONS AND HILBERT'S TENTH PROBLEM

Bell

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Abstract

We show a reduction of Hilbert's tenth problem to the solvability of the matrix equation over non-commuting integral matrices, where Z is the zero matrix, thus proving that the solvability of the equation is undecidable. This is in contrast to the case whereby the matrix semigroup is commutative in which the solvability of the same equation was shown to be decidable in general. The restricted problem where k = 2 for commutative matrices is known as the "A-B-C Problem" and we show that this problem is decidable even for a pair of non-commutative matrices over an algebraic number field.

Acceptance Date Apr 10, 2008
Publication Date Apr 10, 2008
Journal International Journal of Algebra and Computation
Print ISSN 0218-1967
Publisher World Scientific Publishing
Pages 1231 - 1241
DOI https://doi.org/10.1142/S0218196708004925
Keywords Matrix equations, undecidability, formal power series, diophantine equations
Publisher URL https://www.worldscientific.com/doi/abs/10.1142/S0218196708004925

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