Let $1 < N \in \mathbb{N}$ and $x, a \in \mathbb{C}^N$ with $a$ fixed; also, let $b \in \mathbb{R}_{\ge 0}^N$ be fixed (this last bit can be weakened to the extent it makes no difference). For $n \in \{1,\dots,N\}$, let $\mathcal{I}(\cdot;n)$ be a bijection on $\{1,\dots,N\}$, e.g. $\mathcal{I}(n';n) = (n-n'-1) \mod N + 1$. I am interested in (at least) two things regarding the system $$x_n \left ( 1 - \sum_{n'} a_{\mathcal{I}(n';n)} b_{n'} \bar x_{\mathcal{I}(n';n)} \right ) \overset{1 \le n \le N}{=} a_n$$
- What is the dimension of the corresponding variety (I think it is positive but am not sure how to show this, let alone compute it)?
- Is this specific system/variety (or a slightly more general one) studied anywhere?
Please explain any answers as if I were a small child, or a golden retriever: the last time I touched any algebraic geometry at all was 20 years ago and even that was at a fairly elementary level. (I can dimly recall that there is such a thing as a Nullstellensatz.)
