Chapter

Commutative Algebra

pp 133-151

Date:

The Probability That Int n (D) Is Free

  • Jesse ElliottAffiliated withDepartment of Mathematics, California State University Email author 

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Abstract

Let D be a Dedekind domain with quotient field K. The ring of integer-valued polynomials on D is the subring Int(D) = { fK[X]: f(D) ⊆ D} of the polynomial ring K[X]. The Pólya-Ostrowski group PO(D) of D is a subgroup of the class group of D generated by the well-known factorial ideals n! D of D. A regular basis of Int(D) is a D-module basis consisting of one polynomial of each degree. It is well known that Int(D) has a regular basis if and only if the group PO(D) is trivial, if and only if the D-module Int n (D) = { f ∈ Int(D): degfn} is free for all n. In this paper we provide evidence for and prove special cases of the conjecture that, if PO(D) is finite, then the natural density of the set of nonnegative integers n such that Int n (D) is free exists, is rational, and is at least 1∕ | PO(D) | . Moreover, we compute this density or determine a conjectural value for several examples of Galois number fields of degrees 2, 3, 4, 5, and 6 over \(\mathbb{Q}\).

Keywords

Integer-valued polynomial Integral domain Pólya-Ostrowski group Class group Number field

Keywords

MSC 13F20 13G05 11R04 11R29