Integer-valued polynomials over matrices and divided differences
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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 \(D\).
KeywordsInteger-valued polynomial Divided differences Matrix Integral element Polynomial closure Pullback
Mathematic Subject Classification13B25 13F20 11C20
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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