41 research outputs found
Sums of hermitian squares and the BMV conjecture
Recently Lieb and Seiringer showed that the Bessis-Moussa-Villani conjecture
from quantum physics can be restated in the following purely algebraic way: The
sum of all words in two positive semidefinite matrices where the number of each
of the two letters is fixed is always a matrix with nonnegative trace. We show
that this statement holds if the words are of length at most 13. This has
previously been known only up to length 7. In our proof, we establish a
connection to sums of hermitian squares of polynomials in noncommuting
variables and to semidefinite programming. As a by-product we obtain an example
of a real polynomial in two noncommuting variables having nonnegative trace on
all symmetric matrices of the same size, yet not being a sum of hermitian
squares and commutators.Comment: 21 pages; minor changes; a companion Mathematica notebook is now
available in the source fil
The matricial relaxation of a linear matrix inequality
Given linear matrix inequalities (LMIs) L_1 and L_2, it is natural to ask:
(Q1) when does one dominate the other, that is, does L_1(X) PsD imply L_2(X)
PsD? (Q2) when do they have the same solution set? Such questions can be
NP-hard. This paper describes a natural relaxation of an LMI, based on
substituting matrices for the variables x_j. With this relaxation, the
domination questions (Q1) and (Q2) have elegant answers, indeed reduce to
constructible semidefinite programs. Assume there is an X such that L_1(X) and
L_2(X) are both PD, and suppose the positivity domain of L_1 is bounded. For
our "matrix variable" relaxation a positive answer to (Q1) is equivalent to the
existence of matrices V_j such that L_2(x)=V_1^* L_1(x) V_1 + ... + V_k^*
L_1(x) V_k. As for (Q2) we show that, up to redundancy, L_1 and L_2 are
unitarily equivalent.
Such algebraic certificates are typically called Positivstellensaetze and the
above are examples of such for linear polynomials. The paper goes on to derive
a cleaner and more powerful Putinar-type Positivstellensatz for polynomials
positive on a bounded set of the form {X | L(X) PsD}.
An observation at the core of the paper is that the relaxed LMI domination
problem is equivalent to a classical problem. Namely, the problem of
determining if a linear map from a subspace of matrices to a matrix algebra is
"completely positive".Comment: v1: 34 pages, v2: 41 pages; supplementary material is available in
the source file, or see http://srag.fmf.uni-lj.si
Ein gekoppeltes Atmosphäre-Ozean-Modell für das Ostsee-Einzugsgebiet
textabstractWe show that ConnesĘĽ embedding conjecture (CEC) is equivalent to a real version of the same (RCEC). Moreover, we show that RCEC is equivalent to a real, purely algebraic statement concerning trace positive polynomials. This purely algebraic reformulation of CEC had previously been given in both a real and a complex version in a paper of the last two authors. The second author discovered a gap in this earlier proof of the equivalence of CEC to the real algebraic reformulation (the proof of the complex algebraic reformulation being correct). In this note, we show that this gap can be filled with help of the theory of real von Neumann algebras
An exact duality theory for semidefinite programming based on sums of squares
Farkas' lemma is a fundamental result from linear programming providing
linear certificates for infeasibility of systems of linear inequalities. In
semidefinite programming, such linear certificates only exist for strongly
infeasible linear matrix inequalities. We provide nonlinear algebraic
certificates for all infeasible linear matrix inequalities in the spirit of
real algebraic geometry: A linear matrix inequality is infeasible if and only
if -1 lies in the quadratic module associated to it. We also present a new
exact duality theory for semidefinite programming, motivated by the real
radical and sums of squares certificates from real algebraic geometry.Comment: arXiv admin note: substantial text overlap with arXiv:1108.593
A local-global principle for linear dependence of noncommutative polynomials
A set of polynomials in noncommuting variables is called locally linearly
dependent if their evaluations at tuples of matrices are always linearly
dependent. By a theorem of Camino, Helton, Skelton and Ye, a finite locally
linearly dependent set of polynomials is linearly dependent. In this short note
an alternative proof based on the theory of polynomial identities is given. The
method of the proof yields generalizations to directional local linear
dependence and evaluations in general algebras over fields of arbitrary
characteristic. A main feature of the proof is that it makes it possible to
deduce bounds on the size of the matrices where the (directional) local linear
dependence needs to be tested in order to establish linear dependence.Comment: 8 page
Addendum to “Connesʼ embedding conjecture and sums of hermitian squares” [Adv. Math. 217 (4) (2008) 1816–1837]
We show that ConnesĘĽ embedding conjecture (CEC) is equivalent to a real version of the same (RCEC). Moreover, we show that RCEC is equivalent to a real, purely algebraic statement concerning trace positive polynomials. This purely algebraic reformulation of CEC had previously been given in both a real and a complex version in a paper of the last two authors. The second author discovered a gap in this earlier proof of the equivalence of CEC to the real algebraic reformulation (the proof of the complex algebraic reformulation being correct). In this note, we show that this gap can be filled with help of the theory of real von Neumann algebras