Asymptotic Behavior of Characteristic Sequences of Integer-Valued Polynomials Original Research Article

Let D be the ring of integers of a number field K and let E be an infinite subset of D. The D-module Int(E, D) of integer-valued polynomials on E is isomorphic to circled plus∞n=0 imagengn where gn is a monic polynomial in D[X] of degree n and imagen is a fractional ideal of D. For each maximal ideal image of D, let vimage be the corresponding valuation of K; we determine here the asymptotic behavior of the characteristic sequences {vimage(imagen)}nset membership, variantimage in the case where E is a homogeneous subset of D. In order to do this, we first study some properties of ultrametric matrices; then we prove explicit formulas in the case where D is a Dedekind domain with infinite residue fields; finally, we extend these results to the case of number fields.