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Monatshefte für Mathematik

, Volume 173, Issue 4, pp 559–571 | Cite as

Integer-valued polynomials over matrices and divided differences

  • Giulio Peruginelli
Article

Abstract

Let \(D\) be an integrally closed domain with quotient field \(K\) and \(n\) a positive integer. We give a characterization of the polynomials in \(K[X]\) which are integer-valued over the set of matrices \(M_n(D)\) in terms of their divided differences. A necessary and sufficient condition on \(f\in K[X]\) to be integer-valued over \(M_n(D)\) is that, for each \(k\) less than \(n\), the \(k\)th divided difference of \(f\) is integral-valued on every subset of the roots of any monic polynomial over \(D\) of degree \(n\). If in addition \(D\) has zero Jacobson radical then it is sufficient to check the above conditions on subsets of the roots of monic irreducible polynomials of degree \(n\), that is, conjugate integral elements of degree \(n\) over \(D\).

Keywords

Integer-valued polynomial Divided differences Matrix Integral element Polynomial closure Pullback 

Mathematic Subject Classification

13B25 13F20 11C20 

Notes

Acknowledgments

I wish to thank Keith Johnson for the useful suggestions. I also thank the referee for the several suggestions he/she gave which improved the overall quality of the paper. The author was supported by the Austrian Science Foundation (FWF), Project Number P23245-N18.

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Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Institut für Analysis und Computational Number TheoryTechnische Universität GrazAustria

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