Asymptotic Behavior of Characteristic Sequences of Integer-Valued Polynomials

✍ Scribed by Jacques Boulanger; Jean-Luc Chabert

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 226 KB
• Volume: 80
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1999.2432

✦ Synopsis

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 n=0 I n g n where g n is a monic polynomial in D[X] of degree n and I n is a fractional ideal of D. For each maximal ideal m of D, let v m be the corresponding valuation of K; we determine here the asymptotic behavior of the characteristic sequences [v m (I n )] n # N 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.