Super-additive sequences and algebras of polynomials

Abstract:

If $K$ is a field with discrete valuation $\nu$ and $D= \{ a \in K: \nu (a) \geq 0\}$, then an algebra $D[x]\subseteq A\subseteq K[x]$ has associated to it a sequence of fractional ideals $\{{\mathcal I}_n:n=0,1,2,\dots \}$ with ${\mathcal I}_n$ consisting of $0$ and the leading coefficients of elements of $A$ of degree no more than $n$ and a sequence of integers $\{a(n):n=0,1,2,\dots \}$ with $a(n)=-\nu ({\mathcal I}_n)$. Combinatorial properties of this integer sequence reflect algebraic properties of $A$, and these are used to identify the degrees of generators of $A$ and to characterize finitely generated algebras $A$ by a periodicity property of this sequence.