On Variants of Inverse Cluster Size Problem & General Magnification
Shubham Jaiswal, M Krithika, P Vanchinathan
TL;DR
This work expands the inverse cluster size program by introducing Primitive Extensions and General Magnification concepts, establishing a primitive variant with existence results for degree $n$ and cluster size $r$ via transitive solvable Galois actions, and developing a broad general magnification framework (SGM and WGM) that tracks how cluster invariants multiply under composition. It provides a concrete criterion linking unique chains to primitiveness and offers case-by-case constructions (including $S_n$-realizations and PSL$_2(F_p)$-based examples) to realize targeted cluster profiles. The results clarify when extensions are primitive or general primitive and demonstrate how generalized magnification interacts with the algebraic structure of Galois groups, yielding explicit methods to realize prescribed cluster sizes in both primitive and general primitive settings. The paper thus furnishes a structured toolkit for engineering field extensions with specified cluster behavior, with potential implications for Galois-realization problems and the understanding of cluster phenomena in arithmetic dynamics.
Abstract
In this article we establish certain variants of the Inverse Cluster Size problem. We introduce the notion of primitive extensions and establish the Primitive variant of the problem. Precisely, we prove the existence of primitive extensions over number fields of any given degree and cluster size less than the degree. We also introduce the notions of Strong and Weak General Magnification and the notion of general primitive extensions. We establish some interesting cases of the General primitive variant of the problem.