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LSpice
LSpice
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computational Computational tool for checking the existence of non trivial-trivial rational solutionzero of a cubic formsform

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-trivial rational solutionzero or not  ? I think for real field there is always a non trivial real solution  ,we 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-trivial rational solution zero.

computational tool for checking non trivial rational solution of a cubic forms

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 this cubic homogeneous form $ f $ has any non trivial rational solution 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 solution .

Computational tool for checking the existence of non-trivial rational zero of a cubic form

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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Sky
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computational tool for checking non trivial rational solution of a cubic forms

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 this cubic homogeneous form $ f $ has any non trivial rational solution 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 solution .