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arXiv · 1803.05780

Fixed Divisor of a Multivariate Polynomial and Generalized Factorials in Several Variables

Devendra Prasad·Krishnan Rajkumar·A. Satyanarayana Reddy
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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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Devendra Prasad, Krishnan Rajkumar, A. Satyanarayana Reddy. 2018-03-15. Fixed Divisor of a Multivariate Polynomial and Generalized Factorials in Several Variables. https://arxiv.org/abs/1803.05780

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