# Local global principle for a system of polynomial equations

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?

• No. Look at Hasse principle.
– abx
Commented Aug 27, 2021 at 10:15
• Please look at my updated question. In case of homogenous polynomials of degree 2 is the answer still no? Commented Aug 27, 2021 at 10:52
• Restricting to homogeneous polynomials of degree $2$ (quadrics) doesn't help, because, as the answers to this question explain, every projective 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.) Commented Aug 27, 2021 at 10:57
• In fact as soon as there are two equations of degree 2, the Hasse principle can fail. (With one, it's of course OK.) Given any elliptic curve E over Q whose Tate-Shafarevich group has 4-torsion, we can produce such a variety by taking a 4-torsion element, looking at the corresponding $E$-torsor, which fails the Hasse principle, and embedding it into $\mathbb P^3$ by a degree $4$ line bundle, where it will be the intersection of two quadrics. Commented Aug 27, 2021 at 14:17
• Thanks. @WillSawin Commented Aug 27, 2021 at 15:40