In this paper the minimax problem of hypothesis about $k$-dimentional random vector components independency testing is studied. The alternative hypothesis corresponds to the set of densities on $\mathbb R^k$ which are sufficiently smooth and sufficiently distant in the metric of type $L_p$ from the set of product-densities on $\mathbb R^k$. There are given the, conditions of minimax discernibility and nondiscernibility (in the sense [1,2]) depending on the degree of smoothness, dimention $k$, distance between hypothesis and alternative density sets and value $p$.