225 citations · 542 across the 32 of their papers we have counts for
37 papers
The convex algebraic geometry of higher-rank numerical ranges
Jonathan Nino-Cortes, Cynthia Vinzant
The higher-rank numerical range is a convex compact set generalizing the classical numerical range of a square complex matrix, first appearing in the study of quantum error correct…
Tropicalizing Principal Minors of Positive Definite Matrices
Abeer Al Ahmadieh, Felipe Rincón, Cynthia Vinzant +1
We study the tropicalization of the image of the cone of positive definite matrices under the principal minors map. It is a polyhedral subset of the set of -concave functions on…
A semidefinite programming characterization of the Crawford number
Shmuel Friedland, Cynthia Vinzant
We give a semidefinite programming characterization of the Crawford number. We show that the computation of the Crawford number within precision is computable in poly…
Gap distributions of Fourier quasicrystals via Lee-Yang polynomials
Lior Alon, Cynthia Vinzant
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 polynomi…
Lines, Quadrics, and Cremona Transformations in Two-View Geometry
Erin Connelly, Rekha R. Thomas, Cynthia Vinzant
Given points , we characterize rank deficiency of the matrix with rows $x_i^\top \otimes y_i^\t…
Every real-rooted exponential polynomial is the restriction of a Lee-Yang polynomial
Lior Alon, Alex Cohen, Cynthia Vinzant
A Lee-Yang polynomial is a polynomial that has no zeros in the polydisc and its inverse $ (\mathbb{C}\setminus\overline{\mathbb{D}})^{n…
Determinantal representations and the image of the principal minor map
Abeer Al Ahmadieh, Cynthia Vinzant
In this paper we explore determinantal representations of multiaffine polynomials and consequences for the image of various spaces of matrices under the principal minor map. We sho…
Moments, Sums of Squares, and Tropicalization
Grigoriy Blekherman, Felipe Rincón, Rainer Sinn +2
We use tropicalization to study the duals to cones of nonnegative polynomials and sums of squares on a semialgebraic set . The truncated cones of moments of measures supported o…
Characterizing principal minors of symmetric matrices via determinantal multiaffine polynomials
Abeer Al Ahmadieh, Cynthia Vinzant
Here we consider the image of the principal minor map of symmetric matrices over an arbitrary unique factorization domain . By exploiting a connection with symmetric determinant…
Invariant hyperbolic curves: determinantal representations and applications to the numerical range
Faye Pasley Simon, Cynthia Vinzant
Here we study the space of real hyperbolic plane curves that are invariant under actions of the cyclic and dihedral groups and show they have determinantal representations that cer…
Sparse moments of univariate step functions and allele frequency spectra
Zvi Rosen, Georgy Scholten, Cynthia Vinzant
We study the univariate moment problem of piecewise-constant density functions on the interval and its consequences for an inference problem in population genetics. We show…
Log-Concave Polynomials IV: Approximate Exchange, Tight Mixing Times, and Near-Optimal Sampling of Forests
Nima Anari, Kuikui Liu, Shayan Oveis Gharan +2
We prove tight mixing time bounds for natural random walks on bases of matroids, determinantal distributions, and more generally distributions associated with log-concave polynomia…
Positively Hyperbolic Varieties, Tropicalization, and Positroids
Felipe Rincón, Cynthia Vinzant, Josephine Yu
A variety of codimension in complex affine space is called positively hyperbolic if the imaginary part of any point in it does not lie in any positive linear subspace of dimens…
Differential signatures of algebraic curves
Irina A. Kogan, Michael Ruddy, Cynthia Vinzant
In this paper, we adapt the differential signature construction to the equivalence problem for complex plane algebraic curves under the actions of the projective group and its subg…
Log-Concave Polynomials III: Mason's Ultra-Log-Concavity Conjecture for Independent Sets of Matroids
Nima Anari, Kuikui Liu, Shayan Oveis Gharan +1
We give a self-contained proof of the strongest version of Mason's conjecture, namely that for any matroid the sequence of the number of independent sets of given sizes is ultra lo…
Log-Concave Polynomials II: High-Dimensional Walks and an FPRAS for Counting Bases of a Matroid
Nima Anari, Kuikui Liu, Shayan Oveis Gharan +1
We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid…
Log-Concave Polynomials I: Entropy and a Deterministic Approximation Algorithm for Counting Bases of Matroids
Nima Anari, Shayan Oveis Gharan, Cynthia Vinzant
We give a deterministic polynomial time -approximation algorithm for the number of bases of a given matroid of rank and the number of common bases of any two matroids…
Generalized eigenvalue methods for Gaussian quadrature rules
Grigoriy Blekherman, Mario Kummer, Cordian Riener +2
A quadrature rule of a measure on the real line represents a convex combination of finitely many evaluations at points, called nodes, that agrees with integration against f…
Semi-inverted linear spaces and an analogue of the broken circuit complex
Georgy Scholten, Cynthia Vinzant
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 generali…
The Chow form of a reciprocal linear space
Mario Kummer, Cynthia Vinzant
A reciprocal linear space is the image of a linear space under coordinate-wise inversion. These fundamental varieties describe the analytic centers of hyperplane arrangements and a…
Gram Spectrahedra
Lynn Chua, Daniel Plaumann, Rainer Sinn +1
Representations of nonnegative polynomials as sums of squares are central to real algebraic geometry and the subject of active research. The sum-of-squares representations of a giv…
Low-Rank Sum-of-Squares Representations on Varieties of Minimal Degree
Grigoriy Blekherman, Daniel Plaumann, Rainer Sinn +1
A celebrated result by Hilbert says that every real nonnegative ternary quartic is a sum of three squares. We show more generally that every nonnegative quadratic form on a real pr…
Computing complex and real tropical curves using monodromy
Daniel A. Brake, Jonathan D. Hauenstein, Cynthia Vinzant
Tropical varieties capture combinatorial information about how coordinates of points in a classical variety approach zero or infinity. We present algorithms for computing the rays…
Computing Hermitian determinantal representations of hyperbolic curves
Daniel Plaumann, Rainer Sinn, David E. Speyer +1
Every real hyperbolic form in three variables can be realized as the determinant of a linear net of Hermitian matrices containing a positive definite matrix. Such representations a…