Monatshefte für Mathematik

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

Integer-valued polynomials over matrices and divided differences

  • Giulio Peruginelli


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 \(D\).


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

Mathematic Subject Classification

13B25 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.


  1. 1.
    Bhargava, M.: On P-orderings, rings of integer-valued polynomials and ultrametric analysis. J. Am. Math. Soc. 22(4), 963–993 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Cahen, J.-P., Chabert, J.-L.: Integer-valued polynomials, vol. 48. American Mathematical Society Surveys and Monographs, Providence (1997)Google Scholar
  3. 3.
    Evrard, S., Fares, Y., Johnson, K.: Integer valued polynomials on lower triangular integer matrices. Monats. für Math. 170(2), 147–160 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Frisch, S.: Polynomial separation of points in algebras. In: Chapman, S. (ed.) Arithmetical Properties of Commutative Rings and Modules (Chapel Hill Conf.), pp. 249–254. Dekker, New York (2005)Google Scholar
  5. 5.
    Frisch, S.: Integer-valued polynomials on algebras. J. Algebra 373, 414–425 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Gilmer, R.: Contracted ideals with respect to integral extensions. Duke Math. J. 34, 561–571 (1967)Google Scholar
  7. 7.
    Alan Loper, K., Werner, Nicholas J.: Generalized rings of integer-valued polynomials. J. Number Theory 132(11), 2481–2490 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    McCoy, N.H.: Concerning matrices with elements in a commutative ring. Bull. Am. Math. Soc. 45(4), 280–284 (1939)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Peruginelli, G.: Integral-valued polynomials over sets of algebraic integers of bounded degree. submitted,

Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

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

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