We are all familiar with conic curve in high school but what is a conic curve in higher dimensions?
Here is my guess: consider the simplest third order PDE: $D^3f=\mathbf 0$
This means $f_{i_1i_2i_3}dx^{i_1}\otimes dx^{i_2}\otimes dx^{i_3}= \bf 0$
When $f$ is a two-variable function, the conic curve is a solution: $f=Ax^2+By^2+Cx+Dy+E+Fxy$.
When $f$ is $n$-variable function, is it true that (the subset of the solution): $f=e+\sum_{i,j\in{1,...n}}a_{i,j}x_{i}x_j+\sum_nc_nx$ is a reasonable generalization of conic curve?