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Suppose we consider a arbitrary cubic homogeneous form $f$ in in four or five variables over the rational field. Is there any computational tool or algorithm to check whether this cubic homogeneous form $ f $ has any non-trivial rational zero or not? I think for real field there is always a non trivial real solution, we don't need any computational tool. But for rational field we can't say it has always a non zero rational solution as for an example $ 5x_1^3 + 12 x_{2}^3 + 9x_3^3 + 10 x_4^3 $ does not admit a non-trivial rational zero.

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  • $\begingroup$ In short, no. It is expected that the Brauer-Manin obstruction is the only one to the Hasse Principle in this case for smooth cubics, but this is an open problem. This obstruction is computable in principle. $\endgroup$ Dec 24, 2021 at 21:30
  • $\begingroup$ Can you give me some reference of Hasse principle and Brauer Manin obstruction, also for which forms ,having non trivial real and finite field solution implies rational solutions. $\endgroup$
    – Sky
    Dec 25, 2021 at 7:50
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    $\begingroup$ There is a vast literature on this and also quite a few mathoverflow posts. You can get quite far with just google. Try Poonen's book on rational points on varieties. In any case, the correct thing to check is $p$-adic solubility, not a solution in a finite field. $\endgroup$ Dec 25, 2021 at 19:36
  • $\begingroup$ If you have some specific example in mind there may be some hope $\endgroup$ Dec 25, 2021 at 19:37

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