Singular Values of Trilinear Forms

Let T : H1 H2 H3 ! C be a trilinear form where H1, H2, H3 are separable Hilbertspaces. In the hypothesis that at least two of the three spaces are nite dimensional weshow that the norm square = kTk2 is a root of a certain algebraic equation, usually ofvery high degree, which we baptize the milleniar equation, because it is an analogue ofthe secular equation in the bilinear case. More generally, as indicated in the title, we canconsider singular values of a trilinear form and their squares too satisfy the same equation.We work out the binary case (i.e., all three spaces are two dimensional). Even in thiscase the situation is rather complex so, fault of any genuine results, we content ourselveswith advancing a number of conjectures which we were led to by computer experiments.Finally, we connect the singular values of a trilinear form with the critical values of anassociated family of a one parameter family of bilinear forms. Also here we have to o ermainly only experimental evidence.Classi cation: 47B10.