Number of solutions of multivariate quadratic system of equations

Question

Consider a system of quadratic equations in the variables $\{x_1,x_2,...,x_n\}$ of the following form \begin{equation} x_ix_j = \sum_kC_{ij}^kx_k \qquad \forall i,j\in\mathbb{N}, \end{equation} where each $C_{ij}^k$ is an element in some field $F$. Is there anything we can say in general about the number of solutions to this system, given the $C_{ij}^k$s? In particular, I am interested in the structure of these solutions. The above equations seem to imply that the solutions form an algebra. Does this help us in anyway to solve or approximate the equations or bound the number of solutions?