Let $A_{i, j}, B_{i, j}, C, D \in \mathbb{Q}$, and consider the following pair of equations $$ A_{1, 1} x_1 y_1 + A_{1, 2} x_1 y_2 + A_{2, 1} x_2 y_1 + A_{2, 2} x_2 y_2 = C $$ $$ B_{1, 1} x_1 y_1 + B_{1, 2} x_1 y_2 + B_{2, 1} x_2 y_1 + B_{2, 2} x_2 y_2 = D. $$ I was interested in figuring out if this system of equations satisfies the local to global principle. In other words, suppose I can find a solution $(\mathbf{x}, \mathbf{y})$ of the system in $\mathbb{R}^4$ and in $\mathbb{Q}^4_p$ for all primes $p$, then there exists a solution $(\mathbf{x}, \mathbf{y}) \in \mathbb{Q}^4$.

I would appreciate any comments or suggestions or counterexamples. Thank you very much!

CandDare not necessarily zero... the projective closure is a dP4 with four singular points at infinity. $\endgroup$4more comments