A big homogeneous system with complex coefficients.

Question

Consider the following system of $p-1$ unknowns and $p$ equations:
$$\begin{cases} x_1 + x_2 + \ldots +x_{p-1}=0 \\ \frac{x_1}{z}+ \frac{x_2}{z^2}+\ldots +\frac{x_{p-1}}{z^{p-1}}=0 \\ \frac{x_1}{z^2}+ \frac{x_2}{z^4}+\ldots +\frac{x_{p-1}}{z^{2(p-1)}}=0 \\ \frac{x_1}{z^3}+ \frac{x_2}{z^6}+\ldots +\frac{x_{p-1}}{z^{3(p-1)}}=0 \\. \\ .\\.\\ \frac{x_1}{z^{p-1}}+ \frac{x_2}{z^{2(p-1)}}+\ldots +\frac{x_{p-1}}{z^{(p-1)(p-1)}}=0 \end{cases}$$
We know that $z=e^{\frac{2\pi i}{p}}$.
Is it true that $x_i=0$ for every $i\le p-1$?

Answer 1

Yes, it is true.

The matrix $A$ of coefficients has entries $a_{jk} = z^{(j-1)k}$, $j=1\ldots p$, $k=1\ldots p-1$, where $z$ is a primitive $p$'th root of unity. If you add an extra column of $1$'s, you have a $p \times p$ Vandermonde matrix which is known to be nonsingular as all $z^j$, $j = 1 \ldots p$, are distinct. Thus $A$ has full column rank $p-1$, and its null space must be $\{0\}$.