A Complete Characterization and Converse Result on Catenarity and Strong s -Domains Over Special Pullbacks

Let [Formula: see text] be a two-dimensional, integrally closed, Noetherian, and coequidimensional domain, and let [Formula: see text] be a height-one prime ideal of [Formula: see text]. Given a field [Formula: see text], consider the domain [Formula: see text]. This paper aims to characterize precisely when the polynomial ring [Formula: see text] is a strong [Formula: see text]-domain and when it is catenarian. Our findings extend and refine previous results in the literature, culminating in a converse to the main theorem of [Formula: see text], thereby shedding new light on the interplay between strong [Formula: see text]-properties and catenarity in polynomial extensions.