number of solutions in homogeneous system

Question

What is the maximum possible number of solutions of homogeneous system $N \times N$ ($N$ variables, $N$ equations) of degree $2$, where in each equation we have linear terms in $x_i$ and quadratic terms in $x_i x_j$, for $i \neq j$, but not quadratic terms in $x_{i}^{2}$?

Answer 1

The system of equations equation $ \sum x_{2i-(2k-1)} x_{2i} = 0$ where $k$ ranges from $1$ to $n$, has infinitely many solutions, namely when all the even (resp odd) indices are 0, and the odd (resp even) can be anything you want. (There are possibly other solutions, which I do not care for.) Hence, you can have infinitely many solutions.

Do you want the finite maximum?