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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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