Semistar dimension of polynomial rings and Pr\"{u}fer-like domains

Let $D$ be an integral domain and $\star$ a semistar operation stable and of finite type on it. In this paper 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 domains). We also extend Arnold's formula to the setting of semistar operations.