If we consider a cubic homogeneous polynomial in $ 5 $ variables , $ ax_{1}^{3} + bx_{2}^{3} + cx_{3}^{3} + dx_{4}^{3} + ex_{5}^{3} + \sum_{i < j<k =1}^{5} f_{ijk} x_{i}x_{j}x_{k} $ where a,b,c,d,e,$f_{ijk} $ all non zero rational coefficient .This eqn may be have trivial rational solution or non trivial rational solution.Is there any neccessary and sufficient condition such that it have no non trivial rational solution ?Also give some example such that this eqn has no non trivial solution . I found some eqn like $ 5 x_{1}^{3} + 12 x_{1}^{3}+ 9 x_{1}^{3} + 10 x_{1}^{3} $ has only trivial rational solution but this does not satisfy my desire example . Also you can think the same question in $ 4 $ variables.\ So generally for a odd integer r what is the neccessary and sufficient condition for homogeneous polynomial of degree $ r$ in $ 2r-1 $ variables defined by above has no non trivial rational solution ??
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1$\begingroup$ There is no known necessaey and sufficient condition. There are some known necessary conditions (to do with local obstructions and Brauer-Manin obstructions). $\endgroup$– Will SawinCommented Aug 18, 2021 at 19:05
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$\begingroup$ okk @WillSawin can you give some necessary conditions ? $\endgroup$– SkyCommented Aug 18, 2021 at 19:12
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1$\begingroup$ A necessary condition is that there is a nontrivial solution mod $n$ for each integer $n$ (and a nontrivial real solution, but this is automatic in the odd degree case). $\endgroup$– Will SawinCommented Aug 18, 2021 at 19:51
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