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A lower bound for the degree of polynomials satisfied by matrices

Published online by Cambridge University Press:  09 April 2009

K. R. Pearson
Affiliation:
La Trobe UniversityBundoora, Victoria, 3083 Australia
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Abstract

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R. Paré and W. Schelter (1978) have extended the Cayley-Hamilton theorem by showing that for each n<1 there is an integer k such that all n x n matrices over any (possibly noncommutative) ring satisfy a monic polynomial of degree k. We give a lower bound for this degree, namely π(n), which is defined as the shortest possible length of a sequence with entries from {1, 2, …, n}.

MSC classification

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1979

References

Adleman, L. (1974), ‘Short permutation strings’, Discrete Math. 10, 197200.CrossRefGoogle Scholar
Harary, Frank, Norman, Robert Z. and Cartwright, Dorwin (1965), Structural models: an introduction to the theory of directed graphs (John Wiley, New York).Google Scholar
Kleitman, D. J. and Kwiatkowski, D. J. (1976), ‘A lower bound on the length of a sequence containing all permutations as subsequences’, J. Comb. Theory 21, 129136.CrossRefGoogle Scholar
Koutas, P. J. and Hu, T. C. (1975), ‘Shortest string containing all permutations’, Discrete Math. 11, 125132.CrossRefGoogle Scholar
Paré, R. and Schelter, W. (1978), ‘Finite extensions are integral’, J. Alg. 53, 477479.CrossRefGoogle Scholar
Robson, J. C. (1979), ‘Polynomials satisfied by matrices’, J. Alg. 55, 509520.CrossRefGoogle Scholar
Wilson, Robin J. (1972), Introduction to graph theory (Oliver and Boyd, Edinburgh).Google Scholar