scholarly journals Algebraic boundaries of Hilbert’s SOS cones

2012 ◽  
Vol 148 (6) ◽  
pp. 1717-1735 ◽  
Author(s):  
Grigoriy Blekherman ◽  
Jonathan Hauenstein ◽  
John Christian Ottem ◽  
Kristian Ranestad ◽  
Bernd Sturmfels
Keyword(s):  
Algebraic Geometry ◽  
K3 Surfaces ◽  
Sums Of Squares ◽  
Numerical Algebraic Geometry ◽  
Hankel Matrices ◽  
Severi Variety ◽  
The Difference ◽  
Rank Constraints ◽  
Extreme Rays

AbstractWe study the geometry underlying the difference between non-negative polynomials and sums of squares (SOS). The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether–Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. The non-SOS extreme rays of the two cones of non-negative forms are parametrized, respectively, by the Severi variety of plane rational sextics and by the variety of quartic symmetroids.

scholarly journals Maximum Likelihood for Matrices with Rank Constraints

2014 ◽  
Vol 5 (1) ◽  
Author(s):  
Jonathan Hauenstein
Keyword(s):  
Algebraic Geometry ◽  
Maximum Likelihood ◽  
Optimization Problem ◽  
Likelihood Estimation ◽  
Numerical Algebraic Geometry ◽  
Likelihood Functions ◽  
Determinantal Varieties ◽  
Discrete Random Variables ◽  
Rank Constraints ◽  
Bounded Rank

Maximum likelihood estimation is a fundamental optimization problem in statistics. Westudy this problem on manifolds of matrices with bounded rank. These represent mixtures of distributionsof two independent discrete random variables. We determine the maximum likelihood degree for a rangeof determinantal varieties, and we apply numerical algebraic geometry to compute all critical points oftheir likelihood functions. This led to the discovery of maximum likelihood duality between matrices ofcomplementary ranks, a result proved subsequently by Draisma and Rodriguez.

scholarly journals Theta Surfaces

2020 ◽  
Author(s):  
Daniele Agostini ◽  
Türkü Özlüm Çelik ◽  
Julia Struwe ◽  
Bernd Sturmfels
Keyword(s):  
Algebraic Geometry ◽  
Theta Functions ◽  
Minkowski Sum ◽  
Zero Set ◽  
Analytic Surface ◽  
Numerical Algebraic Geometry ◽  
Abelian Integrals ◽  
Space Curves ◽  
Quartic Curves ◽  
Special Plane

Abstract A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincaré showed that any analytic surface that is the Minkowski sum of two space curves in two different ways is a theta surface. The four space curves that generate such a double translation structure are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their theta surfaces, and we develop the numerical algebraic geometry of degenerations of theta functions.

scholarly journals Identifiability and numerical algebraic geometry

PLoS ONE ◽  
2019 ◽  
Vol 14 (12) ◽  
pp. e0226299
Author(s):  
Daniel J. Bates ◽  
Jonathan D. Hauenstein ◽  
Nicolette Meshkat
Keyword(s):  
Algebraic Geometry ◽  
Numerical Algebraic Geometry

Numerical irreducible decomposition over a number field

2018 ◽  
Vol 17 (10) ◽  
pp. 1850195
Author(s):  
Timothy M. McCoy ◽  
Chris Peterson ◽  
Andrew J. Sommese
Keyword(s):  
Algebraic Geometry ◽  
Number Field ◽  
Polynomial System ◽  
Field Extension ◽  
General Point ◽  
Numerical Algebraic Geometry ◽  
Irreducible Decomposition ◽  
Algebraic Set ◽  
Field Of Definition ◽  
Witness Set

Let [Formula: see text] be a set of elements in the polynomial ring [Formula: see text], let [Formula: see text] denote the ideal generated by the elements of [Formula: see text], and let [Formula: see text] denote the radical of [Formula: see text]. There is a unique decomposition [Formula: see text] with each [Formula: see text] a prime ideal corresponding to a minimal associated prime of [Formula: see text] over [Formula: see text]. Let [Formula: see text] denote the reduced algebraic set corresponding to the common zeroes of the elements of [Formula: see text]. Techniques from numerical algebraic geometry can be used to determine the numerical irreducible decomposition of [Formula: see text] over [Formula: see text]. This corresponds to producing a witness set for [Formula: see text] for each [Formula: see text] together with the degree and dimension of [Formula: see text] (a point in a witness set for [Formula: see text] can be considered as a numerical approximation for a general point on [Formula: see text]). The purpose of this paper is to show how to extend these results taking into account the field of definition for the polynomial system. In particular, let [Formula: see text] be a number field (i.e. a finite field extension of [Formula: see text]) and let [Formula: see text] be a set of elements in [Formula: see text]. We show how to extend techniques from numerical algebraic geometry to determine the numerical irreducible decomposition of [Formula: see text] over [Formula: see text].

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