20 may 2020 -- 17:00
Abstract.
WEBINAR
===Tensors and Algebraic Geometry===
organized by E. Angelini, L. Chiantini, G. Ottaviani, M. Mella
Recent advances in border rank and secant varieties of homogeneous varieties.
Wednesday May 20th 17pm (Central European time) https://meet.google.com/nka-dptz-hjx please connect 10 minutes in advance
Jarek Buczyński (Warsaw)
Apolarity, border rank, and multigraded Hilbert scheme
Abstract: The rank of a homogeneous polynomial F is the minimal number of summands r such that F can be expressed as sum of r powers of linear forms. The border rank of F is a minimal r such that F is a limit of polynomials of rank at most r. A classical tool to calculate or estimate the rank is called apolarity lemma. In this talk we introduce an elementary analogue of the apolarity lemma, which is a method to study the border rank. This can be used to describe the border rank of all cases uniformly, including those very special ones that resisted a systematic approach. We work in a general setting, where the base variety is not necessarily a Veronese variety, but an arbitrary smooth toric projective variety, and this includes the cases of border rank of tensors. We also define a border rank version of the variety of sums of powers and analyse how it is useful in studying tensors and polynomials with large symmetries. In particular, it can be applied to provide lower bounds for the border rank of some very interesting tensors, such as the matrix multiplication tensor. A critical ingredient of our work is an irreducible component of a multigraded Hilbert scheme related to the toric variety in question.
The talk is based on a joint work with Weronika Buczyńska, http://arxiv.org/abs/1910.01944