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.