2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/110249In this paper we give a solution of the following problem: under what conditions on infinite compact sets $K_1,K_2\subset \C$ and polynomials $f_1,$ $f_2$ the preimages $f_1^{-1}\{K_1\}$ and $f_2^{-1}\{K_2\}$ coincide. Besides, we investigate some related questions. In particular, we show that polynomials sharing an invariant compact set distinct from a point have equal Julia sets.21 pages, last version accepted by GAFADynamical SystemsComplex Variables37F99; 30C99; 41A10; 41A52text