A small frame and a certificate of its injectivity

We present a complex frame of eleven vectors in 4-space and prove that it defines injective measurements. That is, any rank-one $4\times 4$ Hermitian matrix is uniquely determined by its values as a Hermitian form on this collection of eleven vectors. This disproves a recent conjecture of Bandeira, Cahill, Mixon, and Nelson. We use algebraic computations and certificates in order to prove injectivity.Comment: 4 pages, 3 figure

Semi-inverted linear spaces and an analogue of the broken circuit complex

The image of a linear space under inversion of some coordinates is an affine variety whose structure is governed by an underlying hyperplane arrangement. In this paper, we generalize work by Proudfoot and Speyer to show that circuit polynomials form a universal Groebner basis for the ideal of polynomials vanishing on this variety. The proof relies on degenerations to the Stanley-Reisner ideal of a simplicial complex determined by the underlying matroid. If the linear space is real, then the semi-inverted linear space is also an example of a hyperbolic variety, meaning that all of its intersection points with a large family of linear spaces are real.Comment: 16 pages, 1 figure, minor revisions and added connections to the external activity complex of a matroi

Gap distributions of Fourier quasicrystals via Lee-Yang polynomials

Recent work of Kurasov and Sarnak provides a method for constructing one-dimensional Fourier quasicrystals (FQ) from the torus zero sets of a special class of multivariate polynomials called Lee-Yang polynomials. In particular, they provided a non-periodic FQ with unit coefficients and uniformly discrete support, answering an open question posed by Meyer. Their method was later shown to generate all one-dimensional Fourier quasicrystals with $\mathbb{N}$-valued coefficients ($\mathbb{N}$-FQ). In this paper, we characterize which Lee-Yang polynomials give rise to non-periodic $\mathbb{N}$-FQs with unit coefficients and uniformly discrete support, and show that this property is generic among Lee-Yang polynomials. We also show that the infinite sequence of gaps between consecutive atoms of any $\mathbb{N}$-FQ has a well-defined distribution, which, under mild conditions, is absolutely continuous. This generalizes previously known results for the spectra of quantum graphs to arbitrary $\mathbb{N}$-FQs. Two extreme examples are presented: first, a sequence of $\mathbb{N}$-FQs whose gap distributions converge to a Poisson distribution. Second, a sequence of random Lee-Yang polynomials that results in random $\mathbb{N}$-FQs whose empirical gap distributions converge to that of a random unitary matrix (CUE)

Edges of the Barvinok-Novik orbitope

Here we study the k^th symmetric trigonometric moment curve and its convex hull, the Barvinok-Novik orbitope. In 2008, Barvinok and Novik introduce these objects and show that there is some threshold so that for two points on S^1 with arclength below this threshold, the line segment between their lifts on the curve form an edge on the Barvinok-Novik orbitope and for points with arclenth above this threshold, their lifts do not form an edge. They also give a lower bound for this threshold and conjecture that this bound is tight. Results of Smilansky prove tightness for k=2. Here we prove this conjecture for all k.Comment: 10 pages, 3 figures, corrected Lemma 4 and other minor revision