Here is a set of $4$ simultaneous equations which are second degree polynomials in $4$ variables, e.g.
\begin{align} &\sum^2_{\substack{M=0 \\ i+j+k+l=M}}\alpha_{ijkl}x^i_1x^j_2x^k_3x^l_4=0, &\sum^2_{\substack{M=0 \\ i+j+k+l=M}}\beta_{ijkl}x^i_1x^j_2x^k_3x^l_4=0,\\ &\sum^2_{\substack{M=0 \\ i+j+k+l=M}}\gamma_{ijkl}x^i_1x^j_2x^k_3x^l_4=0, &\sum^2_{\substack{M=0 \\ i+j+k+l=M}}\delta_{ijkl}x^i_1x^j_2x^k_3x^l_4=0. \end{align} with $\alpha_{ijkl},\beta_{ijkl},\gamma_{ijkl},\delta_{ijkl} \in \mathbb{R}$.
When do we know if such a system has a solution at all? Are there any general theorems which give the existence to such a system? I feel like because of the high dimension relative to the degree of the equations there might be, but I am not sure.
If there are always solutions, when are these solutions real numbers?
If anyone has any suggested reading which could help address a problem like this I'd be interested in hearing about it. Bonus points if the reference discusses the above type of problems in higher dimensions and with higher order polynomials.
Generically by Bezout's theorem one expects $2^4 = 16$ solutions, but this is only true if solutions are counted 1) over an algebraically closed field such as $\mathbb{C}$, 2) projectively (so including solutions "at infinity"), and 3) with multiplicity. So over $\mathbb{R}$ several things can go wrong.
Heuristically this can be understood as an application of the "principle of continuity." The simplest special case which is relatively easy to understand is if the quadratics factor, so each one says that the $x_i$ lie on one of $2$ possible hyperplanes. So all four equations together say that the $x_i$ lie on one of $2^4$ possible $4$-tuples of hyperplanes. Generically we expect these hyperplanes to intersect in a single point, which is different for each $4$-tuple of hyperplanes. And then the "principle of continuity" asserts that the answers to these sorts of questions, when counted appropriately (which means the three conditions above), don't change if we perturb the equations.
I'm not familiar with how to actually compute with these things but apparently one can try to use resultants or Gröbner bases.