The cubical Fermat-Pham-Brieskorn function is given by the expression . The mapping , given by , is a continuous bijection. The reciprocal map is also continuous and is given by . Please notice that the reciprocal map is not differentiable. It follows that maps the level surface to the plane . Hence the level surfaces , are topological planes in . Since the reciprocal map is not differentiable, we can not conclude, that all the level surfaces , are differentiable submanifolds in . The level surfaces of the function produce a non differentiable foliation on , which is topologically the standard foliation of by planes,
The function is a deformation of the function . The functions have critical points, which are the vertices of a cube in . The maximal number of critical points, which can have a function that is a ``small'' perturbution of the function , is the sum of the Milnor numbers of its singularities. The function has only a singularity at with Milnor number . The deformation is called a maximal small deformation of the singularity of the function at , since this deformation realizes the maximal possible number of critical points. Please notice that for negative values of the function has no critical points at all.
The movie SURFACE shows level surfaces of the function . One can see a ballet of appearing and disappearing cycles. From the choreography and geometry of this ballet one can determine the toplogical behavior of the corresponding complex map .
The frames of the movie are made with the software SURF, which is freely available at http://surf.sourceforge.net/. The playoff is done with XANIM, with is part of any Linux distribution.