Let $M$ be affine variety $M \subset \mathbb A^n$.
$z\in M \Leftrightarrow z = (x, xy, xy^2, ... , xy^{n-1}),where\hspace{2mm} x,y \in \mathbb C, x \neq 0$.
I need to find the equations $f_1, ..., f_k$ such that $M = V(f_1, ..., f_k)$. And find irreducible components.
I solved this only in case $n = 3$. In general case I only found variety with same dimension containing M. Got really stucked. Any help is welcome :)
In a geometric progression, the ratio between successive terms is constant. So, if $z = (a_1,a_2,a_3,\dots,a_n)$ is a point in $\mathbb A^n$ representing a geometric progression, then $\frac{a_2}{a_1}=\frac{a_3}{a_2} = \frac{a_4}{a_3} = \cdots$. If you multiply by an appropriate common denominator then you can turn this into a set of polynomial equations.