Suppose $T$ be a system of polynomials homogenous of degree 2 solvable over $\mathbb{R}$ and $\mathbb{Q}_p$ for all primes $p$. So, can we claim that $T$ is solvable over $\mathbb{Q}$? I think as of now there is no local global principle proven for system of polynomials, but could there be one?

everyprojective variety can be written (after a suitable embedding) as an intersection of quadrics. So the answer to your question is no, there is no local-global principle. (But I don't think the question is stupid and you shouldn't have been downvoted.) $\endgroup$1more comment