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?

  • 3
    $\begingroup$ No. Look at Hasse principle. $\endgroup$
    – abx
    Commented Aug 27, 2021 at 10:15
  • $\begingroup$ Please look at my updated question. In case of homogenous polynomials of degree 2 is the answer still no? $\endgroup$ Commented Aug 27, 2021 at 10:52
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    $\begingroup$ 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.) $\endgroup$
    – Gro-Tsen
    Commented Aug 27, 2021 at 10:57
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    $\begingroup$ 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. $\endgroup$
    – Will Sawin
    Commented Aug 27, 2021 at 14:17
  • $\begingroup$ Thanks. @WillSawin $\endgroup$ Commented Aug 27, 2021 at 15:40


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