We know that the determinant of a Hadamard product of two positive semidefinite matrices $|{\bf A}\circ{\bf B}|$ is greater than or equal to $|{\bf A}||{\bf B}|$. Are there any general results on arbitrary matrices or something specific on other classes of matrices?

I am specifically interested in the determinant of $|{\bf V}\circ {\bf R}|$ where the $i$-th row of ${\bf V}$ is $\left[x_1^{d_i} \;x_2^{d_i}\;\ldots \;x_N^{d_i}\right]$ and ${\bf R}$'s elements are all roots of unity.