Luke Oeding

Eigenvectors of tensors and algorithms for Waring decomposition

A Waring decomposition of a (homogeneous) polynomial f is a minimal sum of powers of linear forms expressing f. Under certain conditions, such a decomposition is unique. We discuss some algorithms to compute the Waring decomposition, which are linked to the equations of certain secant varieties and to eigenvectors of tensors. In particular we explicitly decompose a cubic polynomial in three variables as the sum of five cubes (Sylvester Pentahedral Theorem).

Toward a Salmon Conjecture

Methods from numerical algebraic geometry are applied in combination with techniques from classical representation theory to show that the variety of 3×3×4 tensors of border rank 4 is cut out by polynomials of degree 6 and 9. Combined with results of Landsberg and Manivel, this furnishes a computational solution of an open problem in algebraic statistics, namely, the set-theoretic version of Allman’s salmon conjecture for 4×4×4 tensors of border rank 4. A proof without numerical computation was given recently by Friedland and Gross.

The ideal of the trifocal variety

Techniques from representation theory, symbolic com- putational algebra, and numerical algebraic geometry are used to nd the minimal generators of the ideal of the trifocal variety. An eective test for determining whether a given tensor is a trifocal tensor is also given.

Hyperdeterminants of polynomials

Abstract The hyperdeterminant of a polynomial (interpreted as a symmetric tensor) factors into several irreducible factors with multiplicities. Using geometric techniques these factors are identified along with their degrees and their multiplicities. The analogous decomposition for the μ -discriminant of polynomial is found.

Secant Cumulants and Toric Geometry

We study the secant line variety of the Segre product of projective spaces using special cumulant coordinates adapted for secant varieties. We show that the secant variety is covered by open normal toric varieties. We prove that in cumulant coordinates its ideal is generated by binomial quadrics. We present new results on the local structure of the secant variety. In particular, we show that it has rational singularities and we give a description of the singular locus. We also classify all secant varieties that are Gorenstein. Moreover, generalizing [SZ12], we obtain analogous results for the tangential variety.

Set-theoretic defining equations of the tangential variety of the Segre variety

Abstract We prove a set-theoretic version of the Landsberg–Weyman Conjecture on the defining equations of the tangential variety of a Segre product of projective spaces. We introduce and study the concept of exclusive rank. For the proof of this conjecture, we use a connection to the author’s previous work and re-express the tangential variety as the variety of principal minors of symmetric matrices that have exclusive rank no more than 1. We discuss applications to semiseparable matrices, tensor rank versus border rank, context-specific independence models and factor analysis models.

Set-theoretic defining equations of the variety of principal minors of symmetric matrices

The variety of principal minors of

Four Lectures on Secant Varieties

n\times n

Equations for the Fifth Secant Variety of Segre Products of Projective Spaces

symmetric matrices, denoted

Computations and equations for Segre-Grassmann hypersurfaces

Z_{n}