Topics in the geometry of spaces of symmetric tensors

Abstract

A tensor is a multi-dimensional data array, occurring ubiquitously in mathematics, physics, engineering and, more generally, in all the situations where one needs to organize data by more than two indices. If tensors remain unchanged after index reordering, they are called symmetric tensors, and in real-life situations they can be simply thought of as homogeneous polynomials of a fixed degree and a given number of variables. A key feature of symmetric tensors is their rank, namely the minimal number of data that is required to fully describe them. The work of my thesis is concerned with low-dimensional symmetric tensors. A central role is played by the so-called catalecticant matrices, that store polynomial's coefficients in a suitable order. The first part of my work focuses on the study of some geometric objects, known as reciprocal varieties, which are defined by taking inverses of catalecticant matrices. Points on these varieties are also tensors, and therefore the rank structure is analyzed, together with other relevant geometric properties. The second main topic pertains to alternative notions of rank, which approximate the classical one. Under suitable conditions, I give formulas to compute these invariants via catalecticant matrices of inhomogeneous polynomials.