Algebraic geometry and sums of squares

2020 ◽  
pp. 59-82
Author(s):  
Mauricio Velasco
Keyword(s):  
Algebraic Geometry ◽  
Sums Of Squares

scholarly journals There Are Many More Positive Maps Than Completely Positive Maps

2017 ◽  
Vol 2019 (11) ◽  
pp. 3313-3375 ◽  
Author(s):  
Igor Klep ◽  
Scott McCullough ◽  
Klemen Šivic ◽  
Aljaž Zalar
Keyword(s):  
Algebraic Geometry ◽  
Main Tool ◽  
Positive Semidefinite ◽  
Sums Of Squares ◽  
Completely Positive Maps ◽  
Positive Semidefinite Matrices ◽  
Positive Polynomials ◽  
Completely Positive ◽  
Positive Maps ◽  
Matrix Spaces

Abstract A $\ast$-linear map $\Phi$ between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations $I_n\otimes \Phi$ are positive. In this article, quantitative bounds on the fraction of positive maps that are completely positive are proved. A main tool is the real algebraic geometry techniques developed by Blekherman to study the gap between positive polynomials and sums of squares. Finally, an algorithm to produce positive maps that are not completely positive is given.

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.

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