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Abstract: Given an element v in the tensor product of n vector spaces over the
field k (say of dimensions d_1,...,d_n) one can ask for the minimum
integer t such that v can be written as the sum of t decomposable
tensors. The number t is called the tensor rank of v.
In the case where n=2 tensor rank is easily described and the maximum
tensor rank of a tensor is the same as the "generic" tensor rank of a
tensor. When n > 2 things become more complicated. The two numbers
"maximal tensor rank" and "generic tensor rank" are no longer equal and
there are only conjectures about these numbers.
In my talk I will explain why there are really TWO distinct problems
concerning tensor rank (when n > 2) that should be considered (in
analogy with the Waring Problem for Forms) and also give some results about
these problems as well as a counterexample to a published conjecture on
these topics.
The approach we use is through the method of Inverse Systems (as explained
recently by Iarrobino et. al.). This is joint work with M. Catalisano
(Genova) and A. Gimigliano (Bologna).
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