# quadratic 4\times 4 system

$\mathbb{Q}$ is the field of rational numbers. $p_1,\cdots,p_r$ are indeterminates and we consider the quotient field $K=\mathbb{Q}(p_1,\cdots,p_r)$ and its algebraic closure $\overline{K}$. Of course, there are no relations that link the $(p_i)_i$. We study a quadratic system of $4$ equations in $4$ unknowns $x_1,x_2,x_3,x_4$ in $\overline{K}$. Eq (1): for every $i$, $\sum_{j\leq k}a_{i,j,k}x_jx_k+\sum_jb_{i,j}x_j+c_i=0.$ where $a_{i,j,k},b_{i,j},c_i$ are given polynomials over $K$ in the indeterminates $p_1,\cdots,p_r$. When I specialize (randomly) $p_1,\cdots,p_r$ into $p_{0,1},\cdots,p_{0,r}$, I solve the associated system, using the library Groebner of Maple, and I obtain almost always $16$ distinct solutions in the $x_1,x_2,x_3,x_4$. What can I conclude about the generic Eq (1) ? In particular, has it $16$ distinct solutions, that is, according to the Bezout's Theorem, the maximal number of isolated solutions ?
More generally, assume that a random specialization of Eq (1) gives less than $16$ solutions, for instance $8$. Can I conclude that the generic'' Eq (1) has a finite number $s$ of solutions in $\overline{K}^4$ and that $s\geq 8$ ?

Thanks.

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