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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 \(F\).

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
Full Text: DOI
[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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