J. Korean Math. Soc. 2018; 55(6): 1305-1320
Online first article March 21, 2018 Printed November 1, 2018
https://doi.org/10.4134/JKMS.j170684
Copyright © The Korean Mathematical Society.
Krishnan Rajkumar, Arikatla Satyanarayana Reddy, Devendra Prasad Semwal
Jawaharlal Nehru University, Shiv Nadar University, Shiv Nadar University
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: fixed divisor, generalized factorial, Dedekind domain
MSC numbers: Primary 13F20, 11Rxx
2021; 58(6): 1311-1325
2016; 53(6): 1225-1236
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