We determine the Waring ranks of all sextic binary forms using a Geometric Invariant Theory approach. In particular, we shed new light on a claim by E. B. Elliott at the end of the 19th century concerning the binary sextics with Waring rank 3.
We show that the Waring rank of the $3 times 3$ determinant, previously known to be between $14$ and $18$, is at least $15$. We use syzygies of the apolar ideal, which have not been used in this way before. Additionally, we show that the cactus rank of the $3 times 3$ permanent is at least $14$.
We discuss the Waring rank of binary forms of degree 4 and 5, without multiple factors, and point out unexpected relations to the harmonic cross-ratio, j-invariants and the golden ratio. These computations of ranks for binary forms are used to show t
hat the combinatorics of a line arrangement in the complex projective plane does not determine the Waring rank of the defining equation even in very simple situations.
The wall-and-chamber structure of the dependence of the reductive GIT quotient on the choice of linearisation is well known. In this article, we first give a brief survey of recent results in non-reductive GIT, which apply when the unipotent radical
is graded. We then examine the dependence of these non-reductive quotients on the linearisation and an additional parameter, the choice of one-parameter subgroup grading the unipotent radical, and arrive at a picture similar to the reductive one.
We investigate an extension of a lower bound on the Waring (cactus) rank of homogeneous forms due to Ranestad and Schreyer. We show that for particular classes of homogeneous forms, for which a generalization of this method applies, the lower bound e
xtends to the level of border (cactus) rank. The approach is based on recent results on tensor asymptotic rank.
The Green--Griffiths--Lang and Kobayashi hyperbolicity conjectures for generic hypersurfaces of polynomial degree are proved using intersection theory for non-reductive geometric invariant theoretic quotients and recent work of Riedl and Yang.