Let D be an integral domain, $D^w$ be the $w$-integral closure of D, X be an indeterminate over D, and $N_v={f{in}D[X]{mid}c(f)_v=D}$. In this paper, we introduce the concept of $t$-locally APVD. We show that D is a $t$-locally APVD and a UMT-domain if and only if D is a $t$-locally APVD and $D^w$ is a $PvMD$, if and only if D[X] is a $t$-locally APVD, if and only if $D[X]_{N_v}$ is a locally APVD.