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An approach to solving Ak=J−I

✍ Scribed by Yaokun Wu; Qiao Li

Publisher
Elsevier Science
Year
2003
Tongue
English
Weight
196 KB
Volume
373
Category
Article
ISSN
0024-3795

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✦ Synopsis

We conjecture that under the permutation similar equivalence relation there are exactly φ(k) solutions A to the matrix equation

, where φ is Euler's totient function, d > 1 is an integer, k > 0 is an odd integer, J is the matrix of all ones, I is the identity matrix, and A is an unknown (0, 1) matrix. We present an approach to verify this conjecture. It establishes a connection between the work of solving the matrix equation A k = J -I and the problems of both determining the structure of near-k-factor factorizations of cyclic groups and characterizing cycle-powers. We also collect some results about the latter two problems in order to give more insight into this approach.

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The objective of this paper is twofold. First, we describe a method to solve large systems of polynomial equations using modular arithmetics. Then, we apply the approach to the study of the problem of linearizability for a quadratic system of ordinary differential equations.