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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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참고문헌 (6)
참고문헌 신청
D. Adam / 2005 / Simultaneous orderings in function fields / J. Number Theory 112(2):287~297
D. Adam / 2010 / Subsets of Z with simultaneous orderings / Inte-gers 10(A37):437~451
M. Bhargava / 1998 / Generalized factorials and fixed divisors over subsets of a Dedekind domain / J. Number Theory 72(1):67~75
M. Bhargava / 2000 / The factorial function and generalizations / Amer. Math. Monthly 107(9):783~799
S. T. Chapman / 2005 / Irreducible polynomials and full elasticity in rings of integer-valued polynomials / J. Algebra 293(2):595~610
D. Adam / 2010 / Subsets of Z with simultaneous orderings / Inte-gers 10(A37):437~451
M. Bhargava / 1998 / Generalized factorials and fixed divisors over subsets of a Dedekind domain / J. Number Theory 72(1):67~75
M. Bhargava / 2000 / The factorial function and generalizations / Amer. Math. Monthly 107(9):783~799
S. T. Chapman / 2005 / Irreducible polynomials and full elasticity in rings of integer-valued polynomials / J. Algebra 293(2):595~610
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