MATRIX EQUATIONS AND HILBERT'S TENTH PROBLEM

Bell, Paul

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.

Citation

Bell, P. (2008). MATRIX EQUATIONS AND HILBERT'S TENTH PROBLEM. International Journal of Algebra and Computation, 1231 - 1241. https://doi.org/10.1142/S0218196708004925

• 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
• Public URL: https://keele-repository.worktribe.com/output/423601
• Publisher URL: https://www.worldscientific.com/doi/abs/10.1142/S0218196708004925