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The Probability That Int n (D) Is Free

In: Commutative Algebra

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  • Jesse Elliott

    (California State University, Department of Mathematics)

Abstract

Let D be a Dedekind domain with quotient field K. The ring of integer-valued polynomials on D is the subring Int(D) = { f ∈ K[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): degf ≤ n} 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 ℚ $$\mathbb{Q}$$ .

Suggested Citation

  • Jesse Elliott, 2014. "The Probability That Int n (D) Is Free," Springer Books, in: Marco Fontana & Sophie Frisch & Sarah Glaz (ed.), Commutative Algebra, edition 127, pages 133-151, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4939-0925-4_8
    DOI: 10.1007/978-1-4939-0925-4_8
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