Lets say I have a multivariate polynomial $P(x_1, \ldots, x_n)$ of degree at most $d = d(n) = n^c$ for a fixed constant $c$. I know that the Kronecker map ($x_i \to x^{d^i}$) preserves zeroness/nonzeroness of a polynomial, that is, $P(x_1, \ldots, x_n) \neq 0$ iff $P(x^{d^1}, \ldots, x^{d^n}) \neq 0$ .

Instead of this univariate map, if I substitute $x_i = \omega_N^{d^i}$ where $\omega_N$ is the $N$-th ($N = d^{n^2}$) root of unity, is the zeroness/nonzeroness still preseved? If yes, what is the best lower bound known on this non-vanishing sum of roots of unity?