arXiv subjects

Igor Klep

Publications and source records attributed to Igor Klep.

At least 73 records · Page 4Linked to original sources

arXiv2017

Null- and Positivstellensätze for rationally resolvable ideals

Hilbert's Nullstellensatz characterizes polynomials that vanish on the vanishing set of an ideal in C[x]. In the free algebra C the vanishing set of a two-sided ideal I is defined in a dimension-free way using images in finite-dimensional representations of C /I. In this article Nullstellensätze for a simple but important class of ideals in the free algebra - called tentatively rationally resolvable here - are presented. An ideal is rationally resolvable if its defining relations can be eliminated by expressing some of the X variables using noncommutative rational functions in the remaining variables. Whether such an ideal I satisfies the Nullstellensatz is intimately related to embeddability of C /I into (free) skew fields. These notions are also extended to free algebras with involution. For instance, it is proved that a polynomial vanishes on all tuples of spherical isometries iff it is a member of the two-sided ideal I generated by 1-\sum_j X_j^* X_j. This is then applied to free real algebraic geometry: polynomials positive semidefinite on spherical isometries are sums of Hermitian squares modulo I. Similar results are obtained for nc unitary groups.

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arXiv2016

Regular and positive noncommutative rational functions

Call a noncommutative rational function $r$ regular if it has no singularities, i.e., $r(X)$ is defined for all tuples of self-adjoint matrices $X$. In this article regular noncommutative rational functions $r$ are characterized via the properties of their (minimal size) linear systems realizations $r=c^* L^{-1}b$. It is shown that $r$ is regular if and only if $L=A_0+\sum_jA_j x_j$ is privileged. Roughly speaking, a linear pencil $L$ is privileged if, after a finite sequence of basis changes and restrictions, the real part of $A_0$ is positive definite and the other $A_j$ are skew-adjoint. The second main result is a solution to a noncommutative version of Hilbert's 17th problem: a positive regular noncommutative rational function is a sum of squares.

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arXiv2016

Instances of the Kaplansky-Lvov multilinear conjecture for polynomials of degree three

Given a positive integer d, the Kaplansky-Lvov conjecture states that the set of values of a multilinear noncommutative polynomial f on the matrix algebra M_d(C) is a vector subspace. In this article the technique of using one-wiggle families of Sylvester's clock-and-shift matrices is championed to establish the conjecture for polynomials f of degree three when d is even or d<17.

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arXiv2016

Dilations, Linear Matrix Inequalities, the Matrix Cube Problem and Beta Distributions

An operator C on a Hilbert space H dilates to an operator T on a Hilbert space K if there is an isometry V from H to K such that C=V^*TV. A main result of this paper is, for a positive integer d, the simultaneous dilation, up to a sharp factor $\vartheta(d)$, of all d-by-d symmetric matrices of operator norm at most one to a collection of commuting self-adjoint contraction operators on a Hilbert space. An analytic formula for $\vartheta(d)$ is derived, which as a by-product gives new probabilistic results for the binomial and beta distributions. Dilating to commuting operators has consequences for the theory of linear matrix inequalities (LMIs). Given a tuple A=(A_1,...,A_g) of symmetric matrices of the same size, L(x):=I-\sum A_j x_j is a monic linear pencil. The solution set S_L of the corresponding linear matrix inequality, consisting of those x in R^g for which L(x) is positive semidefinite (PsD), is a spectrahedron. The set D_L of tuples X=(X_1,...,X_g) of symmetric matrices (of the same size) for which L(X):=I-\sum A_j \otimes X_j is PsD, is a free spectrahedron. A result here is: any tuple X of d-by-d symmetric matrices in a bounded free spectrahedron D_L dilates, up to a scale factor, to a tuple T of commuting self-adjoint operators with joint spectrum in the corresponding spectrahedron S_L. From another viewpoint, the scale factor measures the extent that a positive map can fail to be completely positive. Given another monic linear pencil M, the inclusion D_L \subset D_M obviously implies the inclusion S_L \subset S_M and thus can be thought of as its free relaxation. Determining if one free spectrahedron contains another can be done by solving an explicit LMI and is thus computationally tractable. The scale factor for commutative dilation of D_L gives a precise measure of the worst case error inherent in the free relaxation, over all monic linear pencils M of size d.

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arXiv2016

Circular Free Spectrahedra

This paper considers matrix convex sets invariant under several types of rotations. It is known that matrix convex sets that are free semialgebraic are solution sets of Linear Matrix Inequalities (LMIs); they are called free spectrahedra. We classify all free spectrahedra that are circular, that is, closed under multiplication by exp(i t): up to unitary equivalence, the coefficients of a minimal LMI defining a circular free spectrahedron have a common block decomposition in which the only nonzero blocks are on the superdiagonal. A matrix convex set is called free circular if it is closed under left multiplication by unitary matrices. As a consequence of a Hahn-Banach separation theorem for free circular matrix convex sets, we show the coefficients of a minimal LMI defining a free circular free spectrahedron have, up to unitary equivalence, a block decomposition as above with only two blocks. This paper also gives a classification of those noncommutative polynomials invariant under conjugating each coordinate by a different unitary matrix. Up to unitary equivalence such a polynomial must be a direct sum of univariate polynomials.

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arXiv2016

The Tracial Hahn-Banach Theorem, Polar Duals, Matrix Convex Sets, and Projections of Free Spectrahedra

This article investigates matrix convex sets and introduces their tracial analogs which we call contractively tracial convex sets. In both contexts completely positive (cp) maps play a central role: unital cp maps in the case of matrix convex sets and trace preserving cp (CPTP) maps in the case of contractively tracial convex sets. CPTP maps, also known as quantum channels, are fundamental objects in quantum information theory. Free convexity is intimately connected with Linear Matrix Inequalities (LMIs) L(x) = A_0 + A_1 x_1 + ... + A_g x_g > 0 and their matrix convex solution sets { X : L(X) is positive semidefinite }, called free spectrahedra. The Effros-Winkler Hahn-Banach Separation Theorem for matrix convex sets states that matrix convex sets are solution sets of LMIs with operator coefficients. Motivated in part by cp interpolation problems, we develop the foundations of convex analysis and duality in the tracial setting, including tracial analogs of the Effros-Winkler Theorem. The projection of a free spectrahedron in g+h variables to g variables is a matrix convex set called a free spectrahedrop. As a class, free spectrahedrops are more general than free spectrahedra, but at the same time more tractable than general matrix convex sets. Moreover, many matrix convex sets can be approximated from above by free spectrahedrops. Here a number of fundamental results for spectrahedrops and their polar duals are established. For example, the free polar dual of a free spectrahedrop is again a free spectrahedrop. We also give a Positivstellensatz for free polynomials that are positive on a free spectrahedrop.

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arXiv2015

Sweeping words and the length of a generic vector subspace of M_n(F)

The main result of this short note is a generic version of Paz's conjecture on the lengths of generating sets in matrix algebras. Consider a generic g-tuple A=(A_1,..., A_g) of nxn matrices over a field. We show that whenever $g^{2d}\geq n^2$, the set of all words of degree 2d in A spans the full nxn matrix algebra. Our proofs use generic matrices, are combinatorial and depend on the construction of a special kind of directed multigraphs with few edge-disjoint walks.

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arXiv2014

On trace-convex noncommutative polynomials

To each real continuous function f there is an associated trace function on real symmetric matrices Tr f. The classical Klein lemma states that f is convex if and only if Tr f is convex. In this note we present an algebraic strengthening of this lemma for univariate polynomials f: Tr f is convex if and only if the noncommutative second directional derivative of f is a sum of hermitian squares and commutators in a free algebra. We also give a localized version of this result.

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arXiv2014

Free function theory through matrix invariants

In this article we introduce powerful tools and techniques from invariant theory to free analysis. This enables us to study free maps with involution. These maps are free noncommutative analogs of real analytic functions of several variables. With examples we demonstrate that they do not exhibit strong rigidity properties of their involution-free free counterparts. We present a characterization of polynomial free maps via properties of their finite-dimensional slices. This is used to establish power series expansions for analytic free maps about scalar and non-scalar points; the latter are given by series of generalized polynomials for which we obtain an invariant-theoretic characterization. Finally, we give an inverse and implicit function theorem for free maps with involution.

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arXiv2014

Noncommutative polynomials nonnegative on a variety intersect a convex set

By a result of Helton and McCullough, open bounded convex free semialgebraic sets are exactly open (matricial) solution sets D_L of a linear matrix inequality (LMI) L(X)>0. This paper gives a precise algebraic certificate for a polynomial being nonnegative on a convex semialgebraic set intersect a variety, a so-called "Perfect" Positivstellensatz. For example, given a generic convex free semialgebraic set D_L we determine all "(strong sense) defining polynomials" p for D_L. This follows from our general result for a given linear pencil L and a finite set I of rows of polynomials. A matrix polynomial p is positive where L is positive and I vanishes if and only if p has a weighted sum of squares representation module the "L-real radical" of I. In such a representation the degrees of the polynomials appearing depend in a very tame way only on the degree of p and the degrees of the elements of I. Further, this paper gives an efficient algorithm for computing the L-real radical of I. Our Positivstellensatz has a number of additional consequences which are presented.

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arXiv2013

Matrix Convex Hulls of Free Semialgebraic Sets

This article resides in the realm of the noncommutative (free) analog of real algebraic geometry - the study of polynomial inequalities and equations over the real numbers - with a focus on matrix convex sets $C$ and their projections $\hat C$. A free semialgebraic set which is convex as well as bounded and open can be represented as the solution set of a Linear Matrix Inequality (LMI), a result which suggests that convex free semialgebraic sets are rare. Further, Tarski's transfer principle fails in the free setting: The projection of a free convex semialgebraic set need not be free semialgebraic. Both of these results, and the importance of convex approximations in the optimization community, provide impetus and motivation for the study of the free (matrix) convex hull of free semialgebraic sets. This article presents the construction of a sequence $C^{(d)}$ of LMI domains in increasingly many variables whose projections $\hat C^{(d)}$ are successively finer outer approximations of the matrix convex hull of a free semialgebraic set $D_p=\{X: p(X)\succeq0\}$. It is based on free analogs of moments and Hankel matrices. Such an approximation scheme is possibly the best that can be done in general. Indeed, natural noncommutative transcriptions of formulas for certain well known classical (commutative) convex hulls does not produce the convex hulls in the free case. This failure is illustrated on one of the simplest free nonconvex $D_p$. A basic question is which free sets $\hat S$ are the projection of a free semialgebraic set $S$? Techniques and results of this paper bear upon this question which is open even for convex sets.

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arXiv2013

A tracial Nullstellensatz

The main result of this note is a tracial Nullstellensatz for free noncommutative polynomials evaluated at tuples of matrices of all sizes: Suppose f_1,...,f_r,f are free polynomials, and tr(f) vanishes whenever all tr(f_j) vanish. Then either 1 or f is a linear combination of the f_j modulo sums of commutators.

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arXiv2013

Free Convex Algebraic Geometry

This chapter is a tutorial on techniques and results in free convex algebraic geometry and free real algebraic geometry (RAG). The term free refers to the central role played by algebras of noncommutative polynomials R in free (freely noncommuting) variables x=(x_1,...,x_g). The subject pertains to problems where the unknowns are matrices or Hilbert space operators as arise in linear systems engineering and quantum information theory. The subject of free RAG flows in two branches. One, free positivity and inequalities is an analog of classical real algebraic geometry, a theory of polynomial inequalities embodied in algebraic formulas called Positivstellensätze; often free Positivstellensätze have cleaner statements than their commutative counterparts. Free convexity, the second branch of free RAG, arose in an effort to unify a torrent of ad hoc optimization techniques which came on the linear systems engineering scene in the mid 1990's. Mathematically, much as in the commutative case, free convexity is connected with free positivity through the second derivative: A free polynomial is convex if and only if its Hessian is positive. However, free convexity is a very restrictive condition, for example, free convex polynomials have degree 2 or less. This article describes for a beginner techniques involving free convexity. As such it also serves as a point of entry into the larger field of free real algebraic geometry.

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arXiv2013

On real one-sided ideals in a free algebra

In classical and real algebraic geometry there are several notions of the radical of an ideal I. There is the vanishing radical defined as the set of all real polynomials vanishing on the real zero set of I, and the real radical defined as the smallest real ideal containing I. By the real Nullstellensatz they coincide. This paper focuses on extensions of these to the free algebra R of noncommutative real polynomials in x=(x_1,...,x_g) and x^*=(x_1^*,...,x_g^*). We work with a natural notion of the (noncommutative real) zero set V(I) of a left ideal I in the free algebra. The vanishing radical of I is the set of all noncommutative polynomials p which vanish on V(I). In this paper our quest is to find classes of left ideals I which coincide with their vanishing radical. We completely succeed for monomial ideals and homogeneous principal ideals. We also present the case of principal univariate ideals with a degree two generator and find that it is very messy. Also we give an algorithm (running under NCAlgebra) which checks if a left ideal is radical or is not, and illustrate how one uses our implementation of it.

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arXiv2013

Correction of a proof in "Connes' embedding conjecture and sums of hermitian squares"

We show that Connes' 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.

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arXiv2012

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.

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arXiv2012

Free analysis, convexity and LMI domains

This paper concerns free analytic maps on noncommutative domains. These maps are free analogs of classical holomorphic functions in several complex variables, and are defined in terms of noncommuting variables amongst which there are no relations - they are free variables. Free analytic maps include vector-valued polynomials in free (noncommuting) variables and form a canonical class of mappings from one noncommutative domain D in say g variables to another noncommutative domain D' in g' variables. Motivated by determining the possibilities for mapping a nonconvex noncommutative domain to a convex noncommutative domain, this article focuses on rigidity results for free analytic maps. Those obtained to date, parallel and are often stronger than those in several complex variables. For instance, a proper free analytic map between noncommutative domains is one-one and, if g=g', free biholomorphic. Making its debut here is a free version of a theorem of Braun-Kaup-Upmeier: between two freely biholomorphic bounded circular noncommutative domains there exists a linear biholomorphism. An immediate consequence is the following nonconvexification result: if two bounded circular noncommutative domains are freely biholomorphic, then they are either both convex or both not convex. Because of their roles in systems engineering, linear matrix inequalities (LMIs) and noncommutative domains defined by an LMI (LMI domains) are of particular interest. As a refinement of above the nonconvexification result, if a bounded circular noncommutative domain D is freely biholomorphic to a bounded circular LMI domain, then D is itself an LMI domain.

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arXiv2012

The convex Positivstellensatz in a free algebra

Given a monic linear pencil L in g variables let D_L be its positivity domain, i.e., the set of all g-tuples X of symmetric matrices of all sizes making L(X) positive semidefinite. Because L is a monic linear pencil, D_L is convex with interior, and conversely it is known that convex bounded noncommutative semialgebraic sets with interior are all of the form D_L. The main result of this paper establishes a perfect noncommutative Nichtnegativstellensatz on a convex semialgebraic set. Namely, a noncommutative polynomial p is positive semidefinite on D_L if and only if it has a weighted sum of squares representation with optimal degree bounds: p = s^* s + \sum_j f_j^* L f_j, where s, f_j are vectors of noncommutative polynomials of degree no greater than 1/2 deg(p). This noncommutative result contrasts sharply with the commutative setting, where there is no control on the degrees of s, f_j and assuming only p nonnegative, as opposed to p strictly positive, yields a clean Positivstellensatz so seldom that such cases are noteworthy.

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