Semistar dimension of polynomial rings and Prufer-like domains

Let $D$ be an integral domain and $star$ a semistar operation stable and of finite type on it. We define the semistar dimension (inequality) formula and discover their relations with $star$-universally catenarian domains and $star$-stably strong S-domains. As an application, we give new characterizations of $star$-quasi-Pr"{u}fer domains and UM$t$ domains in terms of dimension inequality formula (and the notions of universally catenarian domain, stably strong S-domain, strong S-domain, and Jaffard domain). We also extend Arnold's formula to the setting of semistar operations.