ON-LINE TensorTalk on May 28: Alessandro Oneto

Speaker: Alessandro Oneto (U. Trento)
Title:  On the strength of general homogeneous polynomials
Abstract:
A slice decomposition is an expression of a homogeneous polynomial as a
sum of reducible polynomials with a linear factor. A strength
decomposition is an expression of a homogeneous polynomial as a sum of
reducible polynomials. The slice rank and the strength of a polynomial
are the minimal lengths of such decompositions, respectively. The slice
rank is clearly an upper bound for the strength, but the gap between
these two values can be arbitrary large.
In this talk, after introducing these notions and showing what is
classically known about them (for example, the value of the the slice
rank for general polynomials and the fact that the slice rank is
upper-semicontinuous), I will prove that strength and slice rank
coincide for general homogeneous polynomials in low degrees (at most 7
or equal to 9). In line with a conjecture by Catalisano et al. on
dimensions of secant varieties of varieties of reducible forms, we
conjecture that this holds in any degree.
This is joint work with A. Bik (U. Bern) -- https://arxiv.org/abs/2005.08617