Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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FIXED DIVISOR OF A MULTIVARIATE POLYNOMIAL AND GENERALIZED FACTORIALS IN SEVERAL VARIABLES

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

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References

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