Bounded deviations
Let I and H be nonnilpotent ideals of a Noetherian ring R such that \(\sqrt{I}=\sqrt{H}\) and v(I,H,n) (respectively w(I,H,n)) the largest (respectively smallest) integer such that \(I^ n\subseteq H^{v(I,H,n)}\) (respectively \(I^ n\supseteq H^{w(I,H,n)})\). By \textit{H. Samuel} [Ann. Math., II. Ser. 56, 11-21 (1952; Zbl 0049.023)] there exist the integers \(\ell (I,H)=\lim_{n\to \infty}v(I,H,n)/n \), L(I,H):\(=\lim_{n\to \infty}w(I,H,n)/n \) and the deviations d(I,H,n):\(=n\cdot \ell (I,H)-v(I,H,n)\), D(I,H,n):\(=w(I,H,n)-n\cdot L(I,H)\) are nonnegative. In Mem. Coll. Sci., Univ. Kyoto, Ser. A 30, 165- 175 (1957; Zbl 0095.023), \textit{M. Nagata} showed that d(I,H,n), D(I,H,n) are bounded as n varies, let us say e(I,H) (respectively E(I,H)) is the least upper bound of d(I,H,n) (respectively D(I,H,n)). Now suppose that I is a regular ideal and let \({\mathcal P}(I)\) be the set of ideals projectively equivalent to I. The aim of this paper is to find when each of the following sets is bounded: \(\{\) e(I,H)\(| H\in {\mathcal P}(I)\}\), \(\{\) e(H,I)\(| H\in {\mathcal P}(I)\}\), \(\{\) E(I,H)\(| H\in {\mathcal P}(I)\}\), \(\{\) E(H,I)\(| H\in {\mathcal P}(I)\}\). For example the second and the third sets above are bounded when R is an analytically unramified local ring.
scientific article; zbMATH DE number 4204503