A linear map between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations are positive. In this article quantitative bounds on the fraction of positive maps that are completely positive are proved. A main tool are 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 which are not completely positive is given.
COBISS.SI-ID: 111111
For a square-free bivariate polynomial $p$ of degree $n$ we introduce a simple and fast numerical algorithm for the construction of $n \times n$ matrices $A$, $B$, and $C$ such that $\det(A+xB+yC)=p(x,y)$. This is the minimal size needed to represent a bivariate polynomial of degree $n$. Combined with a square-free factorization one can now compute $n \times n$ matrices for any bivariate polynomial of degree $n$. The existence of such symmetric matrices was established by Dixon in 1902, but, up to now, no simple numerical construction has been found, even if the matrices can be nonsymmetric. Such representations may be used to efficiently numerically solve a system of two bivariate polynomials of small degree via the eigenvalues of a two-parameter eigenvalue problem. The new representation speeds up the computation considerably.
COBISS.SI-ID: 18087513
For bivariate polynomials of degree $n \le 5$ we give fast numerical constructions of determinantal representations with $n \times n$ matrices. Unlike some other existing constructions, our approach returns matrices of the smallest possible size $n \times n$ for all (not just generic) polynomials of degreen and does not require any symbolic computation. We can apply these linearizations to numerically compute the roots of a system of two bivariate polynomials by using numerical methods for the two-parameter eigenvalue problems.
COBISS.SI-ID: 18280537