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A complex frame is a collection of vectors that span C M and define measurements , called intensity measurements, on vectors in C M. In purely mathematical terms, the problem of phase retrieval is to recover a complex vector from its intensity measurements, namely the modulus of its inner product with these frame vectors. We show that any vector is uniquely… (More)

A smooth quartic curve in the complex projective plane has 36 inequivalent representations as a symmetric determinant of linear forms and 63 representations as a sum of three squares. These correspond to Cayley octads and Steiner complexes respectively. We present exact algorithms for computing these objects from the 28 bitangents. This expresses Vinnikov… (More)

Helton and Vinnikov showed that every rigidly convex curve in the real plane bounds a spectrahedron. This leads to the computational problem of explicitly producing a symmetric (positive definite) linear determinantal representation for a given curve. We study three approaches to this problem: an algebraic approach via solving polynomial equations, a… (More)

—We present a complex frame of eleven vectors in 4-space and prove that it defines injective measurements. That is, any rank-one 4 × 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… (More)

The central curve of a linear program is an algebraic curve specified by linear and quadratic constraints arising from complementary slackness. It is the union of the various central paths for minimizing or maximizing the cost function over any region in the associated hyperplane arrangement. We determine the degree, arithmetic genus and defining prime… (More)