Distinct distances on regular varieties over finite fields

In this paper we study some generalized versions of a recent result due to Covert, Koh, and Pi (2015). More precisely, we prove that if a subset $\mathcal{E}$ in a regular variety satisfies $|$\mathcal{E}$|\gg q^{\frac{d-1}{2}+\frac{1}{k-1}}$,then$\Delta_{k, F}($\mathcal{E}$)\supseteq $\mathbb{F}$_q\setminus {0}$forsomecertainfamiliesofpolynomials$F($\mathbf{x}$)\in $\mathbb{F}$_q[x_1, \ldots, x_d]$.