Classification of polynomial translation hypersurfaces of finite type (Q1895234)

The problem of determining the surfaces of finite type in \(E^3\) is a difficult question within the theory of finite type submanifolds. So far, no other examples than spheres, circular cylinders and minimal surfaces are known and it is believed (although still unknown) that they are the only ones (Chen's conjecture). The corresponding problem for Euclidean hypersurfaces in higher dimensions is a direct generalization for which there are only partial answers. In this line, a natural question would be to consider whether a sort of Bernstein problem could be valid in this context. That is: Do complete finite type graphs exist other than hyperplanes in the Euclidean space? In connection with this question, the authors study finite type translation hypersurfaces of the Euclidean space, giving partial answers to the above problem. In particular, they show that the only finite type polynomial translation hypersurfaces of the Euclidean space are the hyperplanes.