Fixed divisor of a multivariate polynomial and generalized factorials in several variables

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...

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Bibliographic Details
Published inJournal of the Korean Mathematical Society pp. 1305 - 1320
Main Authors Krishnan Rajkumar, Arikatla Satyanarayana Reddy, Devendra Prasad Semwal
Format Journal Article
LanguageEnglish
Published 대한수학회 01.01.2018
Subjects
수학
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Summary: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. KCI Citation Count: 1
ISSN:0304-9914
2234-3008
DOI:10.4134/JKMS.j170684