DOI QR코드

DOI QR Code

FIXED DIVISOR OF A MULTIVARIATE POLYNOMIAL AND GENERALIZED FACTORIALS IN SEVERAL VARIABLES

  • Received : 2017.10.27
  • Accepted : 2018.03.05
  • Published : 2018.11.01

Abstract

We define new generalized factorials in several variables over an arbitrary subset ${\underline{S}}{\subseteq}R^n$, where R is a Dedekind domain and n is a positive integer. We then study the properties of the fixed divisor $d(\underline{S},f)$ of a multivariate polynomial $f{\in}R[x_1,x_2,{\ldots},x_n]$. We generalize the results of Polya, Bhargava, Gunji & McQuillan and strengthen that of Evrard, all of which relate the fixed divisor to generalized factorials of ${\underline{S}}$. We also express $d(\underline{S},f)$ in terms of the images $f({\underline{a}})$ of finitely many elements ${\underline{a}}{\in}R^n$, generalizing a result of Hensel, and in terms of the coefficients of f under explicit bases.

Keywords

References

  1. D. Adam, Simultaneous orderings in function fields, J. Number Theory 112 (2005), no. 2, 287-297. https://doi.org/10.1016/j.jnt.2004.11.015
  2. D. Adam, J.-L. Chabert, and Y. Fares, Subsets of $\mathbb{Z}$ with simultaneous orderings, Integers 10 (2010), A37, 437-451.
  3. M. Bhargava, P-orderings and polynomial functions on arbitrary subsets of Dedekind rings, J. Reine Angew. Math. 490 (1997), 101-127.
  4. M. Bhargava, Generalized factorials and fixed divisors over subsets of a Dedekind domain, J. Number Theory 72 (1998), no. 1, 67-75. https://doi.org/10.1006/jnth.1998.2220
  5. M. Bhargava, The factorial function and generalizations, Amer. Math. Monthly 107 (2000), no. 9, 783-799. https://doi.org/10.1080/00029890.2000.12005273
  6. P.-J. Cahen, Polynomes a valeurs entieres, Canad. J. Math. 24 (1972), 747-754.
  7. P.-J. Cahen and J.-L. Chabert, Integer-Valued Polynomials, Mathematical Surveys and Monographs, 48, American Mathematical Society, Providence, RI, 1997.
  8. J.-L. Chabert, Integer-valued polynomials: looking for regular bases (a survey), in Commutative algebra, 83-111, Springer, New York, 2014.
  9. J.-L. Chabert and P.-J. Cahen, Old problems and new questions around integer-valued polynomials and factorial sequences, in Multiplicative ideal theory in commutative algebra, 89-108, Springer, New York, 2006.
  10. S. T. Chapman and B. A. McClain, Irreducible polynomials and full elasticity in rings of integer-valued polynomials, J. Algebra 293 (2005), no. 2, 595-610. https://doi.org/10.1016/j.jalgebra.2005.01.026
  11. L. E. Dickson, History of the Theory of Numbers. Vol. II, Chelsea Publishing Co., New York, 1966.
  12. S. Evrard, Bhargava's factorials in several variables, J. Algebra 372 (2012), 134-148. https://doi.org/10.1016/j.jalgebra.2012.09.013
  13. H. Gunji and D. L. McQuillan, On polynomials with integer coefficients, J. Number Theory 1 (1969), 486-493. https://doi.org/10.1016/0022-314X(69)90009-2
  14. H. Gunji and D. L. McQuillan, On a class of ideals in an algebraic number field, J. Number Theory 2 (1970), 207-222. https://doi.org/10.1016/0022-314X(70)90021-1
  15. K. Hensel, Ueber den grossten gemeinsamen Theiler aller Zahlen, welche durch eine ganze Function von n Veranderlichen darstellbar sind, J. Reine Angew. Math. 116 (1896), 350-356.
  16. W. Narkiewicz, Polynomial Mappings, Lecture Notes in Mathematics, 1600, Springer-Verlag, Berlin, 1995.
  17. G. Polya, Uber ganzwertige ganze funktionen, Rend. Circ. Mat. Palermo 40 (1915), 1-16.
  18. M. Wood, P-orderings: a metric viewpoint and the non-existence of simultaneous orderings, J. Number Theory 99 (2003), no. 1, 36-56. https://doi.org/10.1016/S0022-314X(02)00056-2