# Vanishing sums of roots of unity

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?

• Is there something in your question that prevents me from taking $P(x_1) = x_1 - \omega_N$? – Kimball Mar 19 '18 at 22:49
• @Kimball You are right, this is a counterexample to what I claimed. But maybe the right question here is if there exists a small set of such safe points. Essentially I thought If I picked these points randomly from the extension $\mathbb{Q}[\omega_N]$, I should be safe. Does that look suspicious? – Nikhil Mar 20 '18 at 9:10