Integer-Valued Polynomials: Looking for Regular Bases (A Survey)



This paper reviews recent results about the additive structure of algebras of integer-valued polynomials and, particularly, the question of the existence and the construction of regular bases. Doing this, we will be led to consider questions of combinatorial, arithmetical, algebraic, ultrametric, or dynamical nature.


Integer-valued polynomials Generalized factorials v-Orderings Kempner’s formula Regular basis Pólya fields Divided differences Mahler’s theorem 

2010 MSC

Primary 13F20 Secondary 11S05 11R21 11B65 



The author thanks the anonymous referee for many valuable suggestions.


  1. 1.
    D. Adam, Simultaneous orderings in function fields. J. Number Theory 112, 287–297 (2005)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    D. Adam, Finite differences in finite characteristic. J. Algebra 296, 285–300 (2006)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    D. Adam, Pólya and Newtonian function fields. Manuscripta Math. 126, 231–246 (2008)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    D. Adam, Polynômes à valeurs entières ainsi que leurs dérivées en caractéristique p. Acta Arith. 148, 351–365 (2011)Google Scholar
  5. 5.
    D. Adam, P.-J. Cahen, Newtonian and Schinzel quadratic fields. J. Pure Appl. Algebra 215, 1902–1918 (2011)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    D. Adam, Y. Fares, Integer-valued Euler-Jackson’s finite differences. Monatsh. Math. 161, 15–32 (2010)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    D. Adam, J.-L. Chabert, Y. Fares, Subsets of \(\mathbb{Z}\) with simultaneous orderings. Integers 10, 437–451 (2010)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Y. Amice, Interpolation p-adique. Bull. Soc. Math. France 92, 117–180 (1964)MATHMathSciNetGoogle Scholar
  9. 9.
    M. Bhargava, P-orderings and polynomial functions on arbitrary subsets of Dedekind rings. J. Reine Angew. Math. 490, 101–127 (1997)MATHMathSciNetGoogle Scholar
  10. 10.
    M. Bhargava, Generalized factorials and fixed divisors over subsets of a Dedekind domain. J. Number Theory 72, 67–75 (1998)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    M. Bhargava, The factorial function and generalizations. Am Math. Monthly 107, 783–799 (2000)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    M. Bhargava, On p-orderings, integer-valued polynomials, and ultrametric analysis. J. Am. Math. Soc. 22, 963–993 (2009)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    M. Bhagava, K. Kedlaya, Continuous functions on compact subsets of local fields. Acta Arith. 91, 191–198 (1999)MathSciNetGoogle Scholar
  14. 14.
    M. Bhargava, P.-J. Cahen, J. Yeramian, Finite generation properties for various rings of integer-valued polynomials. J. Algebra 322, 1129–1150 (2009)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    J. Boulanger, J.-L. Chabert, Asymptotic behavior of characteristic sequences of integer-valued polynomials. J. Number Theory 80, 238–259 (2000)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    J. Boulanger, J.-L. Chabert, S. Evrard, G. Gerboud, The characteristic sequence of integer-valued polynomials on a subset. In Advances in Commutative Ring Theory. Lecture Notes in Pure and Applied Mathematics, vol. 205 (Dekker, New York, 1999), pp. 161–174Google Scholar
  17. 17.
    P.-J. Cahen, J.-L. Chabert, Integer-Valued Polynomials. American Mathematical Society Surveys and Monographs, vol. 48 (American Mathematical Society, Providence, 1997)Google Scholar
  18. 18.
    P.-J. Cahen, J.-L. Chabert, On the ultrametric Stone-Weierstrass theorem and Mahler’s expansion. J. Théor. Nombres Bordeaux 14, 43–57 (2002)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    P.-J. Cahen, J.-L. Chabert, K.A. Loper, High dimension Prüfer domains of integer-valued polynomials. J. Korean Math. Soc. 38, 915–935 (2001)MATHMathSciNetGoogle Scholar
  20. 20.
    L. Carlitz, A class of polynomials. Trans. Am. Math. Soc. 43, 167–182 (1938)CrossRefGoogle Scholar
  21. 21.
    J.-L. Chabert, Generalized factorial ideals. Arab. J. Sci. Eng. Sect. C 26, 51–68 (2001)MATHMathSciNetGoogle Scholar
  22. 22.
    J.-L. Chabert, Factorial groups and Pólya groups in Galoisian extensions of \(\mathbb{Q}\). In Commutative Ring Theory and Applications. Lecture Notes in Pure and Applied Mathematics, vol. 231 (Marcel Dekker, New York, 2003), pp. 77–86Google Scholar
  23. 23.
    J.-L. Chabert, Integer-valued polynomials in valued fields with an application to discrete dynamical systems. In Commutative Algebra and Applications (de Gruyter, Berlin, 2009), pp. 103–134Google Scholar
  24. 24.
    J.-L. Chabert, About polynomials whose divided differences are integer valued on prime numbers. Comm. Algebra, (to appear)Google Scholar
  25. 25.
    J.-L. Chabert, A.-H. Fan, Y. Fares, Minimal dynamical systems on a discrete valuation domain. Discrete Contin. Dyn. Syst. 35, 777–795 (2009)CrossRefMathSciNetGoogle Scholar
  26. 26.
    J.-L. Chabert, S. Evrard, Y. Fares, Regular subsets of valued fields and Bhargava’s v-orderings. Math. Zeitschrift 274, 263–290 (2013)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    S. Evrard, Bhargava’s factorials in several variables. J. Algebra 372, 134–148 (2012)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    S. Evrard, Y. Fares, K. Johnson, Integer valued polynomials on lower triangular integer matrices. Monatsh. Math. 170, 147–160 (2013)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    S. Frisch, Polynomial functions on finite commutative rings. In Advances in Commutative Ring Theory. Lecture Notes in Pure and Applied Mathematics, vol. 205 (Dekker, New York, 1999), pp. 323–336Google Scholar
  30. 30.
    S. Frisch, Integer-valued polynomials on algebras. J. Algebra 373, 414–425 (2013)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    K. Johnson, Limits of characteristic sequences of integer-valued polynomials on homogeneous sets. J. Number Theory 129, 2933–2942 (2009)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    K. Johnson, Computing r-removed P-orderings and P-orderings of order h. Actes des rencontres du CIRM 2(2), 147–160 (2010)Google Scholar
  33. 33.
    K. Johnson, Super-additive sequences and algebras of polynomials. Proc. Am. Math. Soc. 139, 3431–3443 (2011)CrossRefMATHGoogle Scholar
  34. 34.
    K. Johnson, D. Patterson, Projective p-orderings and homogeneous integer-valued polynomials. Integers 11, 597–604 (2011)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    K. Johnson, M. Pavlovski, Integer-valued polynomials on the Hurwitz ring of integral quaternions. Comm. Algebra 40, 4171–4176 (2012)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    A.J. Kempner, Polynomials and their residue systems. Trans. Am. Math. Soc. 22, 240–288 (1921)CrossRefMathSciNetGoogle Scholar
  37. 37.
    A. Leriche, Cubic, quatric and sextic Pólya fields. J. Number Theory 133, 59–71 (2013)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    A. Leriche, About the embedding of a number field in a Pólya field. J. Number Theory, (to appear)Google Scholar
  39. 39.
    K. Mahler, An interpolation series for continuous functions of a p-adic variable. J. Reine Angew. Math. 199, 23–34 (1958); 208, 70–72 (1961)Google Scholar
  40. 40.
    A. Mingarelli, Abstract factorials. arXiv:00705.4299v3 [math.NT]. Accessed 10 Jul 2012Google Scholar
  41. 41.
    A. Ostrowski, Über ganzwertige Polynome in algebraischen Zahlkörpern. J. Reine Angew. Math. 149, 117–124 (1919)MATHGoogle Scholar
  42. 42.
    G. Pólya, Über ganzwertige ganze Funktionen. Rend. Circ. Mat. Palermo 40, 1–16 (1915)CrossRefMATHGoogle Scholar
  43. 43.
    G. Pólya, Über ganzwertige Polynome in algebraischen Zahlkörpern. J. Reine. Angew. Math. 149, 97–116 (1919)MATHGoogle Scholar
  44. 44.
    N. Werner, Integer-valued polynomials over quaternions rings. J. Algebra 324, 1754–1769 (2010)CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    N. Werner, Integer-valued polynomials over matrix rings. Comm. Algebra 40, 4717–4726 (2012)CrossRefMATHMathSciNetGoogle Scholar
  46. 46.
    M. Wood, P-orderings: a metric viewpoint and the non-existence of simultaneous orderings. J. Number Theory 99, 36–56 (2003)CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    H. Zantema, Integer valued polynomials over a number field. Manuscr. Math. 40, 155–203 (1982)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.LAMFA CNRS-UMR 7352Université de PicardieAmiensFrance

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