Growth of multiplicities of graded families of ideals
Huy Tai Ha, Pham An Vinh

TL;DR
This paper investigates the asymptotic behavior of the length function of graded families of ideals in Noetherian local rings, establishing bounds on the difference of lengths that grow polynomially with degree related to the ring's dimension.
Contribution
It provides a bound on the difference of lengths of successive ideals in graded families, showing the growth is controlled by a polynomial of degree d-1.
Findings
Existence of a constant b3 controlling the growth
Difference of lengths is bounded by b3 n^{d-1}
Growth rate is polynomial in n
Abstract
Let be a Noetherian local ring of dimension . Let be a graded family of -primary ideals in . We examine how far off from a polynomial can the length function be asymptotically. More specifically, we show that there exists a constant such that for all ,
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