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Generalized factorials and fixed divisors over subsets of a Dedekind domain. (English) Zbl 0931.13004
For a subset $$X$$ of a Dedekind domain $$D$$ and a polynomial $$F\in D[x]$$, the fixed divisor $$d(X,F)$$ of $$F$$ over $$X$$ is the ideal $$(\{F(a),a\in X\})$$. Continuing earlier work [J. Reine Angew. Math. 490, 101–127 (1997; Zbl 0899.13022)], M. Bharagava shows that for a primitive polynomial $$F$$ of degree $$k$$, $$d(X,F)$$ divides the so-called $$k$$-th factorial ideal $$\nu_k(X)$$, a generalization of the ideal of $$\mathbb{Z}$$ generated by $$k!$$; and that for any ideal divisor $$I$$ of $$\nu_k(X)$$, there exists such an $$F$$ satisfying $$d(X,F) = I$$. These results extend classical theorems of Polya and Gunji-McQuillan, respectively. A further result gives the description of $$d(X,F)$$ in terms of the coefficients of $$F$$.

##### MSC:
 13B25 Polynomials over commutative rings 13A15 Ideals and multiplicative ideal theory in commutative rings 05A10 Factorials, binomial coefficients, combinatorial functions 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
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##### References:
 [1] Bhargava, M., P, J. reine angew. math., 490, 101-127, (1997) · Zbl 0899.13022 [2] Gunji, H.; McQuillan, D.L., On polynomials with integer coefficients, J. number theory, 1, 486-493, (1969) · Zbl 0203.35204 [3] Gunji, H.; McQuillan, D.L., On a class of ideals in an algebraic number field, J. number theory, 2, 207-222, (1970) · Zbl 0199.37402 [4] Pólya, G., Über ganzwertige ganze funktionen, Rend. circ. mat. Palermo, 40, 1-16, (1915) · JFM 45.0655.02
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