It's a classical fact that there are 3264 plane conics tangent to 5 general conics, over $\mathbb{C}$ [1]. It was also shown that the 3264 can be defined over the reals [2] or [3]. More precisely, they showed that there is a configuration of 5 real conics that admit 3264 real conics tangent at real points.
I was wondering if analogous results are known over $\mathbb{Q}$ or finite extensions of it. More generally, over other non-algebraically closed fields of $\text{char}\ne 2$
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asked Mar 10, 2017 at 20:59
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1$\begingroup$ The result over $\mathbf{C}$ is geometric -- it holds over any algebraically closed field of char 0. In particular there must exist such a configuration over some finite extension of $\mathbf{Q}$. $\endgroup$– François BrunaultCommented Mar 11, 2017 at 8:24
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$\begingroup$ This may be relevant: Vakil, Ravi Schubert induction. Ann. of Math. (2) 164 (2006), no. 2, 489–512. $\endgroup$– Alexandre EremenkoCommented Mar 12, 2017 at 19:26
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