Showing 1–50 of 56 results for author: Helton, J W

The discrete Laplace asymptotic method and its application to the 3XOR satisfiability problem

Authors: Jared A. Hughes, J. William Helton, Peter Schlosser

Abstract: A standard way to calculate the asymptotic behavior of integrals of the form \int_Wg(x)e^{-nh(x)}dx is the (continuous) Laplace asymptotic method. However, also discrete sums like \sum_{x\in W\capΛ_n}g_n(x)e^{-nh_n(x)} have similar behavior, when Λ_n is a discrete grid which becomes infinitely fine, and the functions g_n and h_n converge to g and h respectively. We go even further, and also derive… ▽ More A standard way to calculate the asymptotic behavior of integrals of the form \int_Wg(x)e^{-nh(x)}dx is the (continuous) Laplace asymptotic method. However, also discrete sums like \sum_{x\in W\capΛ_n}g_n(x)e^{-nh_n(x)} have similar behavior, when Λ_n is a discrete grid which becomes infinitely fine, and the functions g_n and h_n converge to g and h respectively. We go even further, and also derive the asymptotic formula for sums of the form \sum_{x\in W\capΛ_n}S_n(x), where the summand S_n asymptotically behaves as g_ne^{-nh_n}. The motivation, and also an immediate application, will be filling in all details in the classical breakthrough paper of Dubois and Mandler from 2002, which gives the solvability (phase transition) threshold of the 3XOR-SAT problem using the second moment method. Various analytical arguments there were lightly described, but the appendix of this paper combines recent results to fill all of them in. We would expect our theorems on asymptotics to apply to other (especially combinatorial) problems as well. For example, they seem effective on 3XOR-GAME problems. △ Less

Submitted 19 September, 2025; originally announced September 2025.

MSC Class: 41A60; 68Q87

The Satisfiability Threshold for K-XOR Games

Authors: Jared A. Hughes, J. William Helton

Abstract: A $K$-XORGAME system corresponds to a $K$-XORSAT system with the additional restriction that the variables divide uniformly into $K$ blocks. This forms a system of $m$ equations with $K n$ unknowns over $\mathbb{Z}_2$, and a perfect strategy corresponds to a solution to these equations. Equivalently, such equations correspond to colorings of a $K$-uniform $K$-partite hypergraph. This paper proves… ▽ More A $K$-XORGAME system corresponds to a $K$-XORSAT system with the additional restriction that the variables divide uniformly into $K$ blocks. This forms a system of $m$ equations with $K n$ unknowns over $\mathbb{Z}_2$, and a perfect strategy corresponds to a solution to these equations. Equivalently, such equations correspond to colorings of a $K$-uniform $K$-partite hypergraph. This paper proves that the satisfiability threshold of $m/n$ for $K$-XORGAME problems exists and equals the satisfiability threshold for $K$-XORSAT. △ Less

Submitted 2 May, 2025; originally announced May 2025.

MSC Class: 05A16 (Primary)

Uniform Convergence of an Asymptotic Approximation to Associated Stirling Numbers

Authors: E. Rodney Canfield, J. William Helton, Jared A. Hughes

Abstract: Let $S_r(p,q)$ be the $r$-associated Stirling numbers of the second kind, the number of ways to partition a set of size $p$ into $q$ subsets of size at least $r$. For $r=1$, these are the standard Stirling numbers of the second kind, and for $r=2$, these are also known as the Ward Numbers. This paper concerns asymptotic expansions of these Stirling numbers; such expansions have been known for many… ▽ More Let $S_r(p,q)$ be the $r$-associated Stirling numbers of the second kind, the number of ways to partition a set of size $p$ into $q$ subsets of size at least $r$. For $r=1$, these are the standard Stirling numbers of the second kind, and for $r=2$, these are also known as the Ward Numbers. This paper concerns asymptotic expansions of these Stirling numbers; such expansions have been known for many years. However, while uniform convergence of these expansions was conjectured in Hennecart's 1994 paper, it has not been fully proved. A recent paper (Connamacher and Dobrosotskaya, 2020) went a long way, by proving uniform convergence on a large set. In this paper we build on that paper and prove convergence "everywhere." △ Less

Submitted 2 September, 2024; originally announced September 2024.

Comments: 15 pages, 2 figures

MSC Class: 05A16 (Primary) 05A18 (Secondary)

Matrix Extreme Points and Free extreme points of Free spectrahedra

Authors: Aidan Epperly, Eric Evert, J. William Helton, Igor Klep

Abstract: A spectrahedron is a convex set defined by a linear matrix inequality, i.e., the set of all $x \in \mathbb{R}^g$ such that \[ L_A(x) = I + A_1 x_1 + A_2 x_2 + \dots + A_g x_g \succeq 0 \] for some symmetric matrices $A_1,\ldots,A_g$. This can be extended to matrix spaces by taking $X$ to be a tuple of real symmetric matrices of any size and using the Kronecker product… ▽ More A spectrahedron is a convex set defined by a linear matrix inequality, i.e., the set of all $x \in \mathbb{R}^g$ such that \[ L_A(x) = I + A_1 x_1 + A_2 x_2 + \dots + A_g x_g \succeq 0 \] for some symmetric matrices $A_1,\ldots,A_g$. This can be extended to matrix spaces by taking $X$ to be a tuple of real symmetric matrices of any size and using the Kronecker product $$L_A(X) = I_n \otimes I_d + A_1 \otimes X_1 + A_2 \otimes X_2 + \dots + A_g \otimes X_g.$$ The solution set of $L_A (X) \succeq 0$ is called a \textit{free spectrahedron}. Free spectrahedra are important in systems engineering, operator algebras, and the theory of matrix convex sets. Matrix and free extreme points of free spectrahedra are of particular interest. While many authors have studied matrix and free extreme points of free spectrahedra, it has until now been unknown if these two types of extreme points are actually different. The results of this paper fall into three categories: theoretical, algorithmic, and experimental. Firstly, we prove the existence of matrix extreme points of free spectrahedra that are not free extreme. This is done by producing exact examples of matrix extreme points that are not free extreme. We also show that if the $A_i$ are $2 \times 2$ matrices, then matrix and free extreme points coincide. Secondly, we detail methods for constructing matrix extreme points of free spectrahedra that are not free extreme, both exactly and numerically. We also show how a recent result due to Kriel (Complex Anal.~Oper.~Theory 2019) can be used to efficiently test whether a point is matrix extreme. Thirdly, we provide evidence that a substantial number of matrix extreme points of free spectrahedra are not free extreme. Numerical work in another direction shows how to effectively write a given tuple in a free spectrahedron as a matrix convex combination of its free extreme points. △ Less

Submitted 29 November, 2022; originally announced December 2022.

Comments: 66 pages

MSC Class: Primary 47L07; 13J30. Secondary 46L07; 90C22

Journal ref: Optim. Methods Softw. 39 (2024) 1263--1308

Synchronous Values of Games

Authors: J. William Helton, Hamoon Mousavi, Seyed Sajjad Nezhadi, Vern I. Paulsen, Travis B. Russell

Abstract: We study synchronous values of games, especially synchronous games. It is known that a synchronous game has a perfect strategy if and only if it has a perfect synchronous strategy. However, we give examples of synchronous games, in particular graph colouring games, with synchronous value that is strictly smaller than their ordinary value. Thus, the optimal strategy for a synchronous game need not… ▽ More We study synchronous values of games, especially synchronous games. It is known that a synchronous game has a perfect strategy if and only if it has a perfect synchronous strategy. However, we give examples of synchronous games, in particular graph colouring games, with synchronous value that is strictly smaller than their ordinary value. Thus, the optimal strategy for a synchronous game need not be synchronous. We derive a formula for the synchronous value of an XOR game as an optimization problem over a spectrahedron involving a matrix related to the cost matrix. We give an example of a game such that the synchronous value of repeated products of the game is strictly increasing. We show that the synchronous quantum bias of the XOR of two XOR games is not multiplicative. Finally, we derive geometric and algebraic conditions that a set of projections that yields the synchronous value of a game must satisfy. △ Less

Submitted 22 August, 2023; v1 submitted 29 September, 2021; originally announced September 2021.

3XOR Games with Perfect Commuting Operator Strategies Have Perfect Tensor Product Strategies and are Decidable in Polynomial Time

Authors: Adam Bene Watts, J. William Helton

Abstract: We consider 3XOR games with perfect commuting operator strategies. Given any 3XOR game, we show existence of a perfect commuting operator strategy for the game can be decided in polynomial time. Previously this problem was not known to be decidable. Our proof leads to a construction, showing a 3XOR game has a perfect commuting operator strategy iff it has a perfect tensor product strategy using a… ▽ More We consider 3XOR games with perfect commuting operator strategies. Given any 3XOR game, we show existence of a perfect commuting operator strategy for the game can be decided in polynomial time. Previously this problem was not known to be decidable. Our proof leads to a construction, showing a 3XOR game has a perfect commuting operator strategy iff it has a perfect tensor product strategy using a 3 qubit (8 dimensional) GHZ state. This shows that for perfect 3XOR games the advantage of a quantum strategy over a classical strategy (defined by the quantum-classical bias ratio) is bounded. This is in contrast to the general 3XOR case where the optimal quantum strategies can require high dimensional states and there is no bound on the quantum advantage. To prove these results, we first show equivalence between deciding the value of an XOR game and solving an instance of the subgroup membership problem on a class of right angled Coxeter groups. We then show, in a proof that consumes most of this paper, that the instances of this problem corresponding to 3XOR games can be solved in polynomial time. △ Less

Submitted 8 August, 2023; v1 submitted 30 October, 2020; originally announced October 2020.

Comments: 61 pages, 9 figures; corrected typos

Journal ref: Communications in Mathematical Physics volume 400, pages 731-791 (2023)

Empirical properties of optima in free semidefinite programs

Authors: Eric Evert, Yi Fu, J. William Helton, John Yin

Abstract: Semidefinite programming is based on optimization of linear functionals over convex sets defined by linear matrix inequalities, namely, inequalities of the form $$L_A(X)=I-A_1X_1-\dots-A_g X_g\succeq0.$$ Here the $X_j$ are real numbers and the set of solutions is called a spectrahedron. These inequalities make sense when the $X_i$ are symmetric matrices of any size, $n\times n$, and enter the form… ▽ More Semidefinite programming is based on optimization of linear functionals over convex sets defined by linear matrix inequalities, namely, inequalities of the form $$L_A(X)=I-A_1X_1-\dots-A_g X_g\succeq0.$$ Here the $X_j$ are real numbers and the set of solutions is called a spectrahedron. These inequalities make sense when the $X_i$ are symmetric matrices of any size, $n\times n$, and enter the formula though tensor product $A_i\otimes X_i$: The solution set of $L_A(X)\succeq0$ is called a free spectrahedron since it contains matrices of all sizes and the defining ``linear pencil" is ``free" of the sizes of the matrices. In this article, we report on empirically observed properties of optimizers obtained from optimizing linear functionals over free spectrahedra restricted to matrices $X_i$ of fixed size $n\times n$. The optimizers we find are always classical extreme points. Surprisingly, in many reasonable parameter ranges, over 99.9\% are also free extreme points. Moreover, the dimension of the active constraint, $\ker(L_A(X^\ell))$, is about twice what we expected. Another distinctive pattern regards reducibility of optimizing tuples $(X_1^\ell,\dots,X_g^\ell)$. We give an algorithm for representing elements of a free spectrahedron as matrix convex combinations of free extreme points; these representations satisfy a very low bound on the number of free extreme points neede △ Less

Submitted 23 February, 2022; v1 submitted 3 June, 2020; originally announced June 2020.

Comments: 46 pages body. Includes table of contents and index

MSC Class: Primary 46L07. Secondary 90C22; 49K10

Journal ref: Experimental Mathematics 2021. pp. (1-25)

Plurisubharmonic Noncommutative Rational Functions

Authors: Harry Dym, J. William Helton, Igor Klep, Scott McCullough, Jurij Volčič

Abstract: A noncommutative (nc) function in $x_1,\dots,x_g,x_1^*,\dots,x_g$ is called plurisubharmonic (plush) if its nc complex Hessian takes only positive semidefinite values on an nc neighborhood of 0. The main result of this paper shows that an nc rational function is plush if and only if it is a composite of a convex rational function with an analytic (no $x_j^*$) rational function. The proof is entire… ▽ More A noncommutative (nc) function in $x_1,\dots,x_g,x_1^*,\dots,x_g$ is called plurisubharmonic (plush) if its nc complex Hessian takes only positive semidefinite values on an nc neighborhood of 0. The main result of this paper shows that an nc rational function is plush if and only if it is a composite of a convex rational function with an analytic (no $x_j^*$) rational function. The proof is entirely constructive. Further, a simple computable necessary and sufficient condition for an nc rational function to be plush is given in terms of its minimal realization. △ Less

Submitted 5 August, 2019; originally announced August 2019.

Journal ref: J. Math. Anal. Appl. 492 (2020) 124421, 23pp

Factorization of noncommutative polynomials and Nullstellensätze for the free algebra

Authors: J. William Helton, Igor Klep, Jurij Volčič

Abstract: This article gives a class of Nullstellensätze for noncommutative polynomials. The singularity set of a noncommutative polynomial $f=f(x_1,\dots,x_g)$ is $Z(f)=(Z_n(f))_n$, where $Z_n(f)=\{X \in M_n^g: \det f(X) = 0\}.$ The first main theorem of this article shows that the irreducible factors of $f$ are in a natural bijective correspondence with irreducible components of $Z_n(f)$ for every suffici… ▽ More This article gives a class of Nullstellensätze for noncommutative polynomials. The singularity set of a noncommutative polynomial $f=f(x_1,\dots,x_g)$ is $Z(f)=(Z_n(f))_n$, where $Z_n(f)=\{X \in M_n^g: \det f(X) = 0\}.$ The first main theorem of this article shows that the irreducible factors of $f$ are in a natural bijective correspondence with irreducible components of $Z_n(f)$ for every sufficiently large $n$. With each polynomial $h$ in $x$ and $x^*$ one also associates its real singularity set $Z^{re}(h)=\{X: \det h(X,X^*)=0\}$. A polynomial $f$ which depends on $x$ alone (no $x^*$ variables) will be called analytic. The main Nullstellensatz proved here is as follows: for analytic $f$ but for $h$ dependent on possibly both $x$ and $x^*$, the containment $Z(f) \subseteq Z^{re}(h)$ is equivalent to each factor of $f$ being "stably associated" to a factor of $h$ or of $h^*$. For perspective, classical Hilbert type Nullstellensätze typically apply only to analytic polynomials $f,h $, while real Nullstellensätze typically require adjusting the functions by sums of squares of polynomials (sos). Since the above "algebraic certificate" does not involve a sos, it seems justified to think of this as the natural determinantal Hilbert Nullstellensatz. An earlier paper of the authors (Adv. Math. 331 (2018) 589-626) obtained such a theorem for special classes of analytic polynomials $f$ and $h$. This paper requires few hypotheses and hopefully brings this type of Nullstellensatz to near final form. Finally, the paper gives a Nullstellensatz for zeros $V(f)=\{X: f(X,X^*)=0\}$ of a hermitian polynomial $f$, leading to a strong Positivstellensatz for quadratic free semialgebraic sets by the use of a slack variable. △ Less

Submitted 1 May, 2020; v1 submitted 9 July, 2019; originally announced July 2019.

Journal ref: Int. Math. Res. Not. IMRN Issue 11 (2022) 343--372

Efficient evaluation of noncommutative polynomials using tensor and noncommutative Waring decompositions

Authors: Eric Evert, J. William Helton, Shiyuan Huang, Jiawang Nie

Abstract: This paper analyses a Waring type decomposition of a noncommuting (NC) polynomial $p$ with respect to the goal of evaluating $p$ efficiently on tuples of matrices. Such a decomposition can reduce the number of matrix multiplications needed to evaluate a noncommutative polynomial and is valuable when a single polynomial must be evaluated on many matrix tuples. In pursuit of this goal we examine a… ▽ More This paper analyses a Waring type decomposition of a noncommuting (NC) polynomial $p$ with respect to the goal of evaluating $p$ efficiently on tuples of matrices. Such a decomposition can reduce the number of matrix multiplications needed to evaluate a noncommutative polynomial and is valuable when a single polynomial must be evaluated on many matrix tuples. In pursuit of this goal we examine a noncommutative analog of the classical Waring problem and various related decompositions. For example, we consider a "Waring decomposition" in which each product of linear terms is actually a power of a single linear NC polynomial or more generally a power of a homogeneous NC polynomial. We describe how NC polynomials compare to commutative ones with regard to these decompositions, describe a method for computing the NC decompositions and compare the effect of various decompositions on the speed of evaluation of generic NC polynomials. △ Less

Submitted 8 November, 2021; v1 submitted 14 March, 2019; originally announced March 2019.

Comments: 31 pages, includes table of contents

MSC Class: Primary 11P05; 46L52. Secondary 15A69; 47A56

Journal ref: Numer. Funct. Anal. Optim Vol. 42 (2021) pp. 39-68

Noncommutative polynomials describing convex sets

Authors: J. W. Helton, I. Klep, S. McCullough, J. Volčič

Abstract: The free closed semialgebraic set $D_f$ determined by a hermitian noncommutative polynomial $f$ is the closure of the connected component of $\{(X,X^*)\mid f(X,X^*)>0\}$ containing the origin. When $L$ is a hermitian monic linear pencil, the free closed semialgebraic set $D_L$ is the feasible set of the linear matrix inequality $L(X,X^*)\geq 0$ and is known as a free spectrahedron. Evidently these… ▽ More The free closed semialgebraic set $D_f$ determined by a hermitian noncommutative polynomial $f$ is the closure of the connected component of $\{(X,X^*)\mid f(X,X^*)>0\}$ containing the origin. When $L$ is a hermitian monic linear pencil, the free closed semialgebraic set $D_L$ is the feasible set of the linear matrix inequality $L(X,X^*)\geq 0$ and is known as a free spectrahedron. Evidently these are convex and it is well-known that a free closed semialgebraic set is convex if and only it is a free spectrahedron. The main result of this paper solves the basic problem of determining those $f$ for which $D_f$ is convex. The solution leads to an efficient algorithm that not only determines if $D_f$ is convex, but if so, produces a minimal hermitian monic pencil $L$ such that $D_f=D_L$. Of independent interest is a subalgorithm based on a Nichtsingulärstellensatz presented here: given a linear pencil $L'$ and a hermitian monic pencil $L$, it determines if $L'$ takes invertible values on the interior of $D_L$. Finally, it is shown that if $D_f$ is convex for an irreducible hermitian polynomial $f$, then $f$ has degree at most two, and arises as the Schur complement of an $L$ such that $D_f=D_L$. △ Less

Submitted 29 May, 2020; v1 submitted 20 August, 2018; originally announced August 2018.

Comments: v2: 37 pages, algorithm is now deterministic; v1: 36 pages, includes table of contents and index

MSC Class: 13J30; 47A56; 52A05; (Primary); 14P10; 15A22; 16W10 (Secondary)

Journal ref: Found. Comput. Math. 21 (2021) 575--611

Arveson extreme points span free spectrahedra

Authors: Eric Evert, J. William Helton

Abstract: Let $ SM_n(\mathbb{R})^g$ denote $g$-tuples of $n \times n$ real symmetric matrices. Given tuples $X=(X_1, \dots, X_g) \in SM_{n_1}(\mathbb{R})^g$ and $Y=(Y_1, \dots, Y_g) \in SM_{n_2}(\mathbb{R})^g$, a matrix convex combination of $X$ and $Y$ is a sum of the form \[ V_1^* XV_1+V_2^* Y V_2 \quad \quad \quad V_1^* V_1+V_2^* V_2=I_n \] where $V_1:\mathbb{R}^n \to \mathbb{R}^{n_1}$ and… ▽ More Let $ SM_n(\mathbb{R})^g$ denote $g$-tuples of $n \times n$ real symmetric matrices. Given tuples $X=(X_1, \dots, X_g) \in SM_{n_1}(\mathbb{R})^g$ and $Y=(Y_1, \dots, Y_g) \in SM_{n_2}(\mathbb{R})^g$, a matrix convex combination of $X$ and $Y$ is a sum of the form \[ V_1^* XV_1+V_2^* Y V_2 \quad \quad \quad V_1^* V_1+V_2^* V_2=I_n \] where $V_1:\mathbb{R}^n \to \mathbb{R}^{n_1}$ and $V_2:\mathbb{R}^n \to \mathbb{R}^{n_2}$ are contractions. Matrix convex sets are sets which are closed under matrix convex combinations. A key feature of matrix convex combinations is that the $g$-tuples $X, Y$, and $V_1^* XV_1+V_2^* Y V_2$ do not need to have the same size. As a result, matrix convex sets are a dimension free analog of convex sets. While in the classical setting there is only one notion of an extreme point, there are three main notions of extreme points for matrix convex sets: ordinary, matrix, and absolute extreme points. Absolute extreme points are closely related to the classical Arveson boundary. A central goal in the theory of matrix convex sets is to determine if one of these types of extreme points for a matrix convex set minimally recovers the set through matrix convex combinations. This article shows that every real compact matrix convex set which is defined by a linear matrix inequality is the matrix convex hull of its absolute extreme points, and that the absolute extreme points are the minimal set with this property. Furthermore, we give an algorithm which expresses a tuple as a matrix convex combination of absolute extreme points with optimal bounds. Similar results hold when working over the field of complex numbers rather than the reals. △ Less

Submitted 11 June, 2019; v1 submitted 23 June, 2018; originally announced June 2018.

Comments: 28 pages; includes a table of contents and an index

MSC Class: Primary 47L07; Secondary 46L07; 90C22

Journal ref: Math. Ann. Vol 375 (2019) pp. 629-653

Bianalytic free maps between spectrahedra and spectraballs

Authors: J. William Helton, Igor Klep, Scott McCullough, Jurij Volčič

Abstract: Linear matrix inequalities (LMIs) are ubiquitous in real algebraic geometry, semidefinite programming, control theory and signal processing. LMIs with (dimension free) matrix unknowns are central to the theories of completely positive maps and operator algebras, operator systems and spaces, and serve as the paradigm for matrix convex sets. The matricial feasibility set of an LMI is called a free s… ▽ More Linear matrix inequalities (LMIs) are ubiquitous in real algebraic geometry, semidefinite programming, control theory and signal processing. LMIs with (dimension free) matrix unknowns are central to the theories of completely positive maps and operator algebras, operator systems and spaces, and serve as the paradigm for matrix convex sets. The matricial feasibility set of an LMI is called a free spectrahedron. In this article, the bianalytic maps between a very general class of ball-like free spectrahedra (examples of which include row or column contractions, and tuples of contractions) and arbitrary free spectrahedra are characterized and seen to have an elegant algebraic form. They are all highly structured rational maps. In the case that both the domain and codomain are ball-like, these bianalytic maps are explicitly determined and the article gives necessary and sufficient conditions for the existence of such a map with a specified value and derivative at a point. In particular, this leads to a classification of automorphism groups of ball-like free spectrahedra. The results depend on a novel free Nullstellensatz, established only after new tools in free analysis are developed and applied to obtain fine detail, geometric in nature locally and algebraic in nature globally, about the boundary of ball-like free spectrahedra. △ Less

Submitted 29 December, 2019; v1 submitted 25 April, 2018; originally announced April 2018.

Comments: v3: 51 pages. Proof of the Nullstellensatz 1.7 has been streamlined, and now depends on the spanning of boundary hair 4.2; the erroneous second part of 1.7 has been removed. v2: a major rewrite. 44 pages, includes a table of contents. New is a definitive classification of bianalytic maps between spectraballs

MSC Class: 47L25; 32H02; 13J30 (Primary); 14P10; 52A05; 46L07 (Secondary)

Journal ref: J. Funct. Anal. 278 (2020) 108472, 61pp

Free bianalytic maps between spectrahedra and spectraballs in a generic setting

Authors: Meric Augat, J. William Helton, Igor Klep, Scott McCullough

Abstract: Given a tuple $E=(E_1,\dots,E_g)$ of $d\times d$ matrices, the collection of those tuples of matrices $X=(X_1,\dots,X_g)$ (of the same size) such that $\| \sum E_j\otimes X_j\|\le 1$ is called a spectraball $\mathcal B_E$. Likewise, given a tuple $B=(B_1,\dots,B_g)$ of $e\times e$ matrices the collection of tuples of matrices $X=(X_1,\dots,X_g)$ (of the same size) such that… ▽ More Given a tuple $E=(E_1,\dots,E_g)$ of $d\times d$ matrices, the collection of those tuples of matrices $X=(X_1,\dots,X_g)$ (of the same size) such that $\| \sum E_j\otimes X_j\|\le 1$ is called a spectraball $\mathcal B_E$. Likewise, given a tuple $B=(B_1,\dots,B_g)$ of $e\times e$ matrices the collection of tuples of matrices $X=(X_1,\dots,X_g)$ (of the same size) such that $I + \sum B_j\otimes X_j +\sum B_j^* \otimes X_j^*\succeq 0$ is a free spectrahedron $\mathcal D_B$. Assuming $E$ and $B$ are irreducible, plus an additional mild hypothesis, there is a free bianalytic map $p:\mathcal B_E\to \mathcal D_B$ normalized by $p(0)=0$ and $p'(0)=I$ if and only if $\mathcal B_E=\mathcal B_B$ and $B$ spans an algebra. Moreover $p$ is unique, rational and has an elegant algebraic representation. △ Less

Submitted 22 January, 2019; v1 submitted 26 November, 2017; originally announced November 2017.

Comments: 19 pages

MSC Class: 47L25; 32H02; 13J30 (Primary); 14P10; 52A05; 46L07 (Secondary)

Journal ref: "Interpolation and Realization Theory with Applications to Control Theory" (Operator Theory: Advances and Applications vol. 272), 43-60, Birkhäuser, 2019

Geometry of free loci and factorization of noncommutative polynomials

Authors: J. William Helton, Igor Klep, Jurij Volčič

Abstract: The free singularity locus of a noncommutative polynomial f is defined to be the sequence $Z_n(f)=\{X\in M_n^g : \det f(X)=0\}$ of hypersurfaces. The main theorem of this article shows that f is irreducible if and only if $Z_n(f)$ is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Apart from its consequences to factorization in a free algebra, the pa… ▽ More The free singularity locus of a noncommutative polynomial f is defined to be the sequence $Z_n(f)=\{X\in M_n^g : \det f(X)=0\}$ of hypersurfaces. The main theorem of this article shows that f is irreducible if and only if $Z_n(f)$ is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Apart from its consequences to factorization in a free algebra, the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry. △ Less

Submitted 2 April, 2018; v1 submitted 17 August, 2017; originally announced August 2017.

Comments: v2: 32 pages, includes a table of contents

MSC Class: 13J30; 15A22; 47A56 (Primary); 14P10; 16U30; 16R30 (Secondary)

Journal ref: Adv. Math. 331 (2018) 589--626

Extreme points of matrix convex sets, free spectrahedra and dilation theory

Authors: Eric Evert, J. William Helton, Igor Klep, Scott McCullough

Abstract: For matrix convex sets a unified geometric interpretation of notions of extreme points and of Arveson boundary points is given. These notions include, in increasing order of strength, the core notions of "Euclidean" extreme points, "matrix" extreme points, and "absolute" extreme points. A seemingly different notion, the "Arveson boundary", has by contrast a dilation theoretic flavor. An Arveson bo… ▽ More For matrix convex sets a unified geometric interpretation of notions of extreme points and of Arveson boundary points is given. These notions include, in increasing order of strength, the core notions of "Euclidean" extreme points, "matrix" extreme points, and "absolute" extreme points. A seemingly different notion, the "Arveson boundary", has by contrast a dilation theoretic flavor. An Arveson boundary point is an analog of a (not necessarily irreducible) boundary representation for an operator system. This article provides and explores dilation theoretic formulations for the above notions of extreme points. The scalar solution set of a linear matrix inequality (LMI) is known as a spectrahedron. The matricial solution set of an LMI is a free spectrahedron. Spectrahedra (resp. free spectrahedra) lie between general convex sets (resp. matrix convex sets) and convex polyhedra (resp. free polyhedra). As applications of our theorems on extreme points, it is shown the polar dual of a matrix convex set K is generated, as a matrix convex set, by finitely many Arveson boundary points if and only if K is a free spectrahedron; and if the polar dual of a free spectrahedron K is again a free spectrahedron, then at the scalar level K is a polyhedron. △ Less

Submitted 4 June, 2019; v1 submitted 30 November, 2016; originally announced December 2016.

Comments: This version corrects dropped hypotheses in Theorem 1.1 (1) and Proposition 6.1. A detailed explanation of the corrections is found in a new section, Section 8. 37 pages, includes table of contents

MSC Class: Primary 47L07; 13J30; Secondary 46L07; 90C22

Journal ref: J. Geom. Anal. 28 (2018) 1373-1408

Circular Free Spectrahedra

Authors: Eric Evert, J. William Helton, Igor Klep, Scott McCullough

Abstract: 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 L… ▽ More 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. △ Less

Submitted 19 April, 2016; originally announced April 2016.

Comments: 29 pages; includes a table of contents and an index

MSC Class: 47L07; 52A05 (Primary); 46N10; 46L07; 32F17 (Secondary)

Journal ref: J. Math. Anal. Appl. 445 (2017) 1047-1070

Bianalytic Maps Between Free Spectrahedra

Authors: Meric Augat, J. William Helton, Igor Klep, Scott McCullough

Abstract: Linear matrix inequalities (LMIs) $I_d + \sum_{j=1}^g A_jx_j + \sum_{j=1}^g A_j^*x_j^*\succeq0$ play a role in many areas of applications and the set of solutions to one is called a spectrahedron. LMIs in (dimension--free) matrix variables model most problems in linear systems engineering, and their solution sets D_A are called free spectrahedra. These are exactly the free semialgebraic convex set… ▽ More Linear matrix inequalities (LMIs) $I_d + \sum_{j=1}^g A_jx_j + \sum_{j=1}^g A_j^*x_j^*\succeq0$ play a role in many areas of applications and the set of solutions to one is called a spectrahedron. LMIs in (dimension--free) matrix variables model most problems in linear systems engineering, and their solution sets D_A are called free spectrahedra. These are exactly the free semialgebraic convex sets. This paper studies free analytic maps between free spectrahedra and, under certain irreducibility assumptions, classifies all those that are bianalytic. The foundation of such maps turns out to be a very small class of birational maps we call convexotonic. The convexotonic maps in g variables sit in correspondence with g-dimensional algebras. If two bounded free spectrahedra D_A and D_B meeting our irreducibility assumptions are free bianalytic with map denoted p, then p must (after possibly an affine linear transform) extend to a convexotonic map corresponding to a g-dimensional algebra spanned by (U-I)A_1,...,(U-I)A_g for some unitary U. Furthermore, B and UA are unitarily equivalent. The article also establishes a Positivstellensatz for free analytic functions whose real part is positive semidefinite on a free spectrahedron and proves a representation for a free analytic map from D_A to D_B (not necessarily bianalytic). Another result shows that a function analytic on any radial expansion of a free spectrahedron is approximable by polynomials uniformly on the spectrahedron. These theorems are needed for classifying free bianalytic maps. △ Less

Submitted 6 December, 2018; v1 submitted 17 April, 2016; originally announced April 2016.

Comments: v4 (post-production version): 61 pages, fixed Proposition 6.3; includes table of contents. v3 (final version): 58 pages. Reworked introduction

MSC Class: 47L25; 32H02; 52A05 (Primary); 14P10; 32E30; 46L07 (Secondary)

Journal ref: Math. Ann. 371 (2018) 883--959

Non-commutative polynomials with convex level slices

Authors: Harry Dym, J. William Helton, Scott McCullough

Abstract: Let a and x denote tuples of (jointly) freely noncommuting variables. A square matrix valued polynomial p in these variables is naturally evaluated at a tuple (A,X) of symmetric matrices with the result p(A,X) a square matrix. The polynomial p is symmetric if it takes symmetric values. Under natural irreducibility assumptions and other mild hypothesis, the article gives an algebraic certificate fo… ▽ More Let a and x denote tuples of (jointly) freely noncommuting variables. A square matrix valued polynomial p in these variables is naturally evaluated at a tuple (A,X) of symmetric matrices with the result p(A,X) a square matrix. The polynomial p is symmetric if it takes symmetric values. Under natural irreducibility assumptions and other mild hypothesis, the article gives an algebraic certificate for symmetric polynomials p with the property that for sufficiently many tuples A, the set of those tuples X such that p(A,X) is positive definite is convex. In particular, p has degree at most two in x. The case of noncommutative quasi-convex polynomials is of particular interest. The problem analysed here occurs in linear system engineering problems. There the A tuple corresponds to the parameters describing a system one wishes to control while the X tuple corresponds to the parameters one seeks in designing the controller. In this setting convexity is typically desired for numerical reasons and to guarantee that local optima are in fact global. Further motivation comes from the theories of matrix convexity and operator systems. △ Less

Submitted 20 June, 2017; v1 submitted 9 December, 2015; originally announced December 2015.

MSC Class: 47A20; 47A63; 46L07 (Primary); 14P10; 13J30 (Secondary)

Applications of Realizations (aka Linearizations) to Free Probability

Authors: J. William Helton, Tobias Mai, Roland Speicher

Abstract: We show how the combination of new "linearization" ideas in free probability theory with the powerful "realization" machinery -- developed over the last 50 years in fields including systems engineering and automata theory -- allows solving the problem of determining the eigenvalue distribution (or even the Brown measure, in the non-selfadjoint case) of noncommutative rational functions of random m… ▽ More We show how the combination of new "linearization" ideas in free probability theory with the powerful "realization" machinery -- developed over the last 50 years in fields including systems engineering and automata theory -- allows solving the problem of determining the eigenvalue distribution (or even the Brown measure, in the non-selfadjoint case) of noncommutative rational functions of random matrices when their size tends to infinity. Along the way we extend evaluations of noncommutative rational expressions from matrices to stably finite algebras, e.g. type II$_1$ von Neumann algebras, with a precise control of the domains of the rational expressions. The paper provides sufficient background information, with the intention that it should be accessible both to functional analysts and to algebraists. △ Less

Submitted 29 September, 2017; v1 submitted 17 November, 2015; originally announced November 2015.

Comments: We have undertaken a major revision, mainly for the sake of clarity and readability

Convex entire noncommutative functions are polynomials of degree two or less

Authors: J. William Helton, J. E. Pascoe, Ryan Tully-Doyle, Victor Vinnikov

Abstract: This paper concerns matrix "convex" functions of (free) noncommuting variables, $x = (x_1, \ldots, x_g)$. Helton and McCullough showed that a polynomial in $x$ which is matrix convex is of degree two or less. We prove a more general result: that a function of $x$ that is matrix convex near $0$ and also that is "analytic" in some neighborhood of the set of all self-adjoint matrix tuples is in fact… ▽ More This paper concerns matrix "convex" functions of (free) noncommuting variables, $x = (x_1, \ldots, x_g)$. Helton and McCullough showed that a polynomial in $x$ which is matrix convex is of degree two or less. We prove a more general result: that a function of $x$ that is matrix convex near $0$ and also that is "analytic" in some neighborhood of the set of all self-adjoint matrix tuples is in fact a polynomial of degree two or less. More generally, we prove that a function $F$ in two classes of noncommuting variables, $a = (a_1, \ldots, a_{\tilde{g}})$ and $x = (x_1, \ldots, x_g)$ that is "analytic" and matrix convex in $x$ on a "noncommutative open set" in $a$ is a polynomial of degree two or less. △ Less

Submitted 23 January, 2015; originally announced January 2015.

Comments: 17 pages

MSC Class: 46L52; 47A56 (Primary); 32A99; 47Lxx (Secondary)

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

Authors: J. William Helton, Igor Klep, Scott A. McCullough, Markus Schweighofer

Abstract: 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 s… ▽ More 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. △ Less

Submitted 7 May, 2016; v1 submitted 3 December, 2014; originally announced December 2014.

Comments: v4: 93 pages including a table of contents and index, added paragraph at the end of the introduction, several minor changes; v3: 90 pages, new section on including balls into spectrahedra, changes to the introduction; v2: title change, introduction rewritten; v1: 77 pages including a table of contents

MSC Class: 47A20; 46L07; 13J30 (Primary); 60E05; 33B15; 90C22 (Secondary)

Journal ref: Mem. Amer. Math. Soc. 257 vol. 1232 (2019), 104pp

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

Authors: J. William Helton, Igor Klep, Scott McCullough

Abstract: 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… ▽ More 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. △ Less

Submitted 28 February, 2016; v1 submitted 30 July, 2014; originally announced July 2014.

Comments: v2: 56 pages, reworked abstract and intro to emphasize the convex duality aspects; v1: 60 pages; includes an index and table of contents

MSC Class: Primary 14P10; 47L25; 90C22; Secondary 13J30; 46L07

Journal ref: J. Eur. Math. Soc. 19 (2017) 1845-1897

Matrix Convex Hulls of Free Semialgebraic Sets

Authors: J. William Helton, Igor Klep, Scott McCullough

Abstract: 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 w… ▽ More 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. △ Less

Submitted 29 December, 2013; v1 submitted 20 November, 2013; originally announced November 2013.

Comments: 41 pages; includes table of contents; supplementary material (a Mathematica notebook) can be found at http://www.math.auckland.ac.nz/~igorklep/publ.html

MSC Class: Primary 46L07; 14P10; 90C22; Secondary 13J30; 46L89

Journal ref: Trans. Amer. Math. Soc. 268 (2016) 3105-3139

Noncommutative polynomials nonnegative on a variety intersect a convex set

Authors: J. William Helton, Igor Klep, Christopher S. Nelson

Abstract: 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 semialg… ▽ More 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. △ Less

Submitted 14 March, 2014; v1 submitted 31 July, 2013; originally announced August 2013.

Comments: 69 pages, includes a table of contents

MSC Class: Primary: 13J30; 14A22; 46L07; Secondary: 16S10; 47Lxx; 16Z05; 90C22

Journal ref: J. Funct. Anal. 266 (2014) 6684-6752

Free Semidefinite Representation of Matrix Power Functions

Authors: J. William Helton, Jiawang Nie, Jeremy S. Semko

Abstract: Consider the matrix power function X^p defined over the cone of positive definite matrices S^{n}_{++}. It is known that X^p is convex over S^{n}_{++} if p is in [-1,0] or [1,2] and X^p is concave over S^{n}_{++} if p is in [0,1]. We show that the hypograph of X^p admits a free semidefinite representation if p in [0,1] is rational, and the epigraph of X^p admits a free semidefinite representation i… ▽ More Consider the matrix power function X^p defined over the cone of positive definite matrices S^{n}_{++}. It is known that X^p is convex over S^{n}_{++} if p is in [-1,0] or [1,2] and X^p is concave over S^{n}_{++} if p is in [0,1]. We show that the hypograph of X^p admits a free semidefinite representation if p in [0,1] is rational, and the epigraph of X^p admits a free semidefinite representation if p in [-1,0] or [1,2] is rational. △ Less

Submitted 9 October, 2014; v1 submitted 18 May, 2013; originally announced May 2013.

Comments: 12 pages, 1 figures

MSC Class: 90C22; 15-XX (Primary); 47H07 (Secondary)

Free Convex Algebraic Geometry

Authors: J. William Helton, Igor Klep, Scott McCullough

Abstract: 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<x> 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 en… ▽ More 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<x> 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. △ Less

Submitted 15 April, 2013; originally announced April 2013.

Comments: 70 pages, survey

MSC Class: 47A63; 46L89; 14P10 (Primary); 15A22; 13J30 (Secondary)

Journal ref: In: "Semidefinite Optimization and Convex Algebraic Geometry" edited by G. Blekherman, P. Parrilo, R. Thomas, pp. 341-405, SIAM, 2013

Real Nullstellensatze and *-ideals in *-algebras

Authors: Jakob Cimpric, J. William Helton, Scott McCullough, Christopher Nelson

Abstract: Let F denote either the real or complex field. An ideal I in the free *-algebra F<x,x*> in g freely noncommuting variables and their formal adjoints is a *-ideal if I = I*. When a real *-ideal has finite codimension, it satisfies a strong Nullstellensatz. Without the finite codimension assumption, there are examples of such ideals which do not satisfy, very liberally interpreted, any Nullstellensa… ▽ More Let F denote either the real or complex field. An ideal I in the free *-algebra F<x,x*> in g freely noncommuting variables and their formal adjoints is a *-ideal if I = I*. When a real *-ideal has finite codimension, it satisfies a strong Nullstellensatz. Without the finite codimension assumption, there are examples of such ideals which do not satisfy, very liberally interpreted, any Nullstellensatz. A polynomial p in F<x,x*> is analytic if it is a polynomial in the variables {x} only; that is if p in F<x>. As shown in this article, *-ideals generated by analytic polynomials do satisfy a natural Nullstellensatz and those generated by homogeneous analytic polynomials have a particularly simple description. The article also connects the results here for *-ideals to the literature on Nullstellensatz for left ideals in *-algebras generally and in F<x,x*> in particular. It also develops the concomitant general theory of *-ideals in general *-algebras. △ Less

Submitted 20 February, 2013; v1 submitted 19 February, 2013; originally announced February 2013.

MSC Class: 16W10; 16S10; 16Z05; 14P99; 14A22; 47Lxx; 13J30

Journal ref: Electronic Journal of Linear Algebra, 30 (2015), 19-50

Non-Commutative Representations of Families of k^2 Commutative Polynomials in 2k^2 Commuting Variables

Authors: Harry Dym, J. W. Helton, Caleb Meier

Abstract: Given a collection P of k^2 commutative polynomials in 2k^2 commutative variables, the objective is to find a condensed representation of these polynomials in terms of a single non-commutative polynomial p(X,Y) in two k x k matrix variables X and Y. Algorithms that will generically determine whether the given family P has a non-commutative representation and that will produce such a representation… ▽ More Given a collection P of k^2 commutative polynomials in 2k^2 commutative variables, the objective is to find a condensed representation of these polynomials in terms of a single non-commutative polynomial p(X,Y) in two k x k matrix variables X and Y. Algorithms that will generically determine whether the given family P has a non-commutative representation and that will produce such a representation are developed. These algorithms will determine a non-commutative representation for families P that admit a a non-commutative representation in an open, dense subset of the vector space of non-commutative polynomials in two variables. △ Less

Submitted 4 December, 2012; originally announced December 2012.

Free convex sets defined by rational expressions have LMI representations

Authors: J. William Helton, Scott McCullough

Abstract: Suppose p is a symmetric matrix whose entries are polynomials in freely noncommutating variables and p(0) is positive definite. Let D(p) denote the component of zero of the set of those g-tuples X of symmetric matrices (of the same size) such that p(X) is positive definite. By a previous result of the authors, if D(p) is convex and bounded, then D(p) can be described as the set of all solutions to… ▽ More Suppose p is a symmetric matrix whose entries are polynomials in freely noncommutating variables and p(0) is positive definite. Let D(p) denote the component of zero of the set of those g-tuples X of symmetric matrices (of the same size) such that p(X) is positive definite. By a previous result of the authors, if D(p) is convex and bounded, then D(p) can be described as the set of all solutions to a linear matrix inequality (LMI). This article extends that result from matrices of polynomials to matrices of rational functions in free variables. As a refinement of a theorem of Kaliuzhnyi-Verbovetskyi and Vinnikov, it is also shown that a minimal symmetric descriptor realization r for a symmetric free matrix-valued rational function R in g freely noncommuting variables precisely encodes the singularities of the rational function. This singularities result is an important ingredient in the proof of the LMI representation theorem stated above. △ Less

Submitted 21 November, 2012; v1 submitted 15 September, 2012; originally announced September 2012.

On real one-sided ideals in a free algebra

Authors: Jakob Cimprič, J. William Helton, Igor Klep, Scott McCullough, Christopher Nelson

Abstract: 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<x,x^*> of nonco… ▽ More 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<x,x^*> 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. △ Less

Submitted 14 April, 2013; v1 submitted 23 August, 2012; originally announced August 2012.

Comments: v1: 31 pages; v2: 32 pages

MSC Class: Primary 14P10; 08B20; Secondary 90C22; 16W10; 13J30

Journal ref: J. Pure Appl. Algebra 218 (2014), 269-284

Free analysis, convexity and LMI domains

Authors: J. William Helton, Igor Klep, Scott McCullough

Abstract: 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 f… ▽ More 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. △ Less

Submitted 11 June, 2012; originally announced June 2012.

Comments: 25 pages; largely a survey

MSC Class: 46L52; 47A56; 32A05; 46G20 (Primary) 47A63; 32A10; 14P10 (Secondary)

Journal ref: "Mathematical methods in systems, optimization and control" (Operator Theory: Advances and Applications) edited by H. Dym, M. de Oliveira, M. Putinar, pp. 195-219, Birkhauser, 2012

Non-commutative varieties with curvature having bounded signature

Authors: Harry Dym, J. William Helton, Scott McCullough

Abstract: The signature(s) of the curvature of the zero set V of a free (non-commutative) polynomial is defined as the number of positive and negative eigenvalues of the non-commutative second fundamental form on V determined by p. With some natural hypotheses, the degree of p is bounded in terms of the signature. In particular, if one of the signatures is zero, then the degree of p is at most two. The signature(s) of the curvature of the zero set V of a free (non-commutative) polynomial is defined as the number of positive and negative eigenvalues of the non-commutative second fundamental form on V determined by p. With some natural hypotheses, the degree of p is bounded in terms of the signature. In particular, if one of the signatures is zero, then the degree of p is at most two. △ Less

Submitted 31 January, 2012; originally announced February 2012.

Semidefinite programming in matrix unknowns which are dimension free

Authors: J. William Helton, Igor Klep, Scott McCullough

Abstract: One of the main applications of semidefinite programming lies in linear systems and control theory. Many problems in this subject, certainly the textbook classics, have matrices as variables, and the formulas naturally contain non-commutative polynomials in matrices. These polynomials depend only on the system layout and do not change with the size of the matrices involved, hence such problems are… ▽ More One of the main applications of semidefinite programming lies in linear systems and control theory. Many problems in this subject, certainly the textbook classics, have matrices as variables, and the formulas naturally contain non-commutative polynomials in matrices. These polynomials depend only on the system layout and do not change with the size of the matrices involved, hence such problems are called "dimension-free". Analyzing dimension-free problems has led to the development recently of a non-commutative (nc) real algebraic geometry (RAG) which, when combined with convexity, produces dimension-free Semidefinite Programming. This article surveys what is known about convexity in the non-commutative setting and nc SDP and includes a brief survey of nc RAG. Typically, the qualitative properties of the non-commutative case are much cleaner than those of their scalar counterparts - variables in R^g. Indeed we describe how relaxation of scalar variables by matrix variables in several natural situations results in a beautiful structure. △ Less

Submitted 29 December, 2011; originally announced December 2011.

Comments: 25 pages; survey

MSC Class: Primary 90C22; 14P10; 52A05; Secondary 46N10; 46L07; 13J30

Journal ref: Pages 377-405 in the Handbook of Semidefinite, Conic and Polynomial Optimization edited by M. Anjos and J. B. Lasserre, Springer, 2012

A Semidefinite Approach for Truncated K-Moment Problems

Authors: J. William Helton, Jiawang Nie

Abstract: A truncated moment sequence (tms) of degree d is a vector indexed by monomials whose degree is at most d. Let K be a semialgebraic set.The truncated K-moment problem (TKMP) is: when does a tms y admit a positive Borel measure supported? This paper proposes a semidefinite programming (SDP) approach for solving TKMP. When K is compact, we get the following results: whether a tms y of degree d admits… ▽ More A truncated moment sequence (tms) of degree d is a vector indexed by monomials whose degree is at most d. Let K be a semialgebraic set.The truncated K-moment problem (TKMP) is: when does a tms y admit a positive Borel measure supported? This paper proposes a semidefinite programming (SDP) approach for solving TKMP. When K is compact, we get the following results: whether a tms y of degree d admits a K-measure or notcan be checked via solving a sequence of SDP problems; when y admits no K-measure, a certificate will be given; when y admits a K-measure, a representing measure for y would be obtained from solving the SDP under some necessary and some sufficient conditions. Moreover, we also propose a practical SDP method for finding flat extensions, which in our numerical experiments always finds a finitely atomic representing measure for a tms when it admits one. △ Less

Submitted 6 September, 2012; v1 submitted 2 May, 2011; originally announced May 2011.

The possible shapes of numerical ranges

Authors: J. William Helton, Ilya M. Spitkovsky

Abstract: Which convex subsets of the complex plane are the numerical range W(A of some matrix A? This paper gives a precise characterization of these sets. In addition to this we show that for any A there exists a symmetric matrix B of the same size such that W(A)=W(B). Which convex subsets of the complex plane are the numerical range W(A of some matrix A? This paper gives a precise characterization of these sets. In addition to this we show that for any A there exists a symmetric matrix B of the same size such that W(A)=W(B). △ Less

Submitted 23 April, 2011; originally announced April 2011.

Comments: 4 pages

The convex Positivstellensatz in a free algebra

Authors: J. William Helton, Igor Klep, Scott McCullough

Abstract: 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 esta… ▽ More 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. △ Less

Submitted 11 June, 2012; v1 submitted 23 February, 2011; originally announced February 2011.

Comments: 22 pages

MSC Class: 90C22; 13J30 (Primary) 14P10; 47A57; 08B20 (Secondary)

Journal ref: Adv. Math. 231 (2012), 516-534

Noncommutative Plurisubharmonic Polynomials Part I: Global Assumptions

Authors: Jeremy M. Greene, J. William Helton, Victor Vinnikov

Abstract: We consider symmetric polynomials, p, in the noncommutative free variables (x_1, x_2, ..., x_g). We define the noncommutative complex hessian of p and we call a noncommutative symmetric polynomial noncommutative plurisubharmonic if it has a noncommutative complex hessian that is positive semidefinite when evaluated on all tuples of n x n matrices for every size n. In this paper, we show that the s… ▽ More We consider symmetric polynomials, p, in the noncommutative free variables (x_1, x_2, ..., x_g). We define the noncommutative complex hessian of p and we call a noncommutative symmetric polynomial noncommutative plurisubharmonic if it has a noncommutative complex hessian that is positive semidefinite when evaluated on all tuples of n x n matrices for every size n. In this paper, we show that the symmetric noncommutative plurisubharmonic polynomials are precisely the noncommutative convex polynomials with a noncommutative analytic change of variables; i.e., a noncommutative symmetric polynomial, p, is noncommutative plurisubharmonic if and only if it has the form p = \sum f_j^T f_j + \sum k_j k_j^T + F + F^T where the sums are finite and f_j, k_j, F are all noncommutative analytic. We also present a theory of noncommutative integration for noncommutative polynomials and we prove a noncommutative version of the Frobenius theorem. A subsequent paper by Greene proves that if the noncommutative complex hessian of p takes positive semidefinite values on a "noncommutative open set" then the noncommutative complex hessian takes positive semidefinite values on all matrix tuples. Thus, p has the form above. The proof in the subsequent paper draws on most of the theorems in this paper together with a very different technique involving representations of noncommutative quadratic functions. △ Less

Submitted 14 January, 2011; v1 submitted 30 December, 2010; originally announced January 2011.

Comments: 24 pages

MSC Class: 47A56; 46L07; 32H99; 32A99; 46L89

Proper Analytic Free Maps

Authors: J. William Helton, Igor Klep, Scott McCullough

Abstract: This paper concerns analytic free maps. These maps are free analogs of classical analytic functions in several complex variables, and are defined in terms of non-commuting variables amongst which there are no relations - they are free variables. Analytic free maps include vector-valued polynomials in free (non-commuting) variables and form a canonical class of mappings from one non-commutative dom… ▽ More This paper concerns analytic free maps. These maps are free analogs of classical analytic functions in several complex variables, and are defined in terms of non-commuting variables amongst which there are no relations - they are free variables. Analytic free maps include vector-valued polynomials in free (non-commuting) variables and form a canonical class of mappings from one non-commutative domain D in say g variables to another non-commutative domain D' in g' variables. As a natural extension of the usual notion, an analytic free map is proper if it maps the boundary of D into the boundary of D'. Assuming that both domains contain 0, we show that if f:D->D' is a proper analytic free map, and f(0)=0, then f is one-to-one. Moreover, if also g=g', then f is invertible and f^(-1) is also an analytic free map. These conclusions on the map f are the strongest possible without additional assumptions on the domains D and D'. △ Less

Submitted 1 December, 2010; v1 submitted 8 April, 2010; originally announced April 2010.

Comments: 17 pages, final version. To appear in the Journal of Functional Analysis

MSC Class: 46L52; 47A56; 46G20 (Primary). 47A63; 32A10; 14P10 (Secondary)

Journal ref: J. Funct. Anal. 260, No. 5, 1476-1490 (2011)

The matricial relaxation of a linear matrix inequality

Authors: J. William Helton, Igor Klep, Scott McCullough

Abstract: 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… ▽ More 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". △ Less

Submitted 1 March, 2012; v1 submitted 3 March, 2010; originally announced March 2010.

Comments: v1: 34 pages, v2: 41 pages; supplementary material is available in the source file, or see http://srag.fmf.uni-lj.si/

MSC Class: Primary 46L07; 14P10; 90C22; Secondary 11E25; 46L89; 13J30

Journal ref: Math. Program. 138 (2013), 401-445

Non-Commutative Harmonic and Subharmonic Polynomials

Authors: J. William Helton, Daniel P. McAllaster, Joshua A. Hernandez

Abstract: The paper introduces a notion of the Laplace operator of a polynomial p in noncommutative variables x=(x_1,...,x_g). The Laplacian Lap[p,h] of p is a polynomial in x and in a noncommuting variable h. When all variables commute we have Lap[p,h]=h^2Δ_x p where Δ_x p is the usual Laplacian. A symmetric polynomial in symmetric variables will be called harmonic if Lap[p,h]=0 and subharmonic if the po… ▽ More The paper introduces a notion of the Laplace operator of a polynomial p in noncommutative variables x=(x_1,...,x_g). The Laplacian Lap[p,h] of p is a polynomial in x and in a noncommuting variable h. When all variables commute we have Lap[p,h]=h^2Δ_x p where Δ_x p is the usual Laplacian. A symmetric polynomial in symmetric variables will be called harmonic if Lap[p,h]=0 and subharmonic if the polynomial q(x,h):=Lap[p,h] takes positive semidefinite matrix values whenever matrices X_1,..., X_g, H are substituted for the variables x_1,...,x_g, h. In this paper we classify all homogeneous symmetric harmonic and subharmonic polynomials in two symmetric variables. We find there are not many of them: for example, the span of all such subharmonics of any degree higher than 4 has dimension 2 (if odd degree) and 3 (if even degree). Hopefully, the approach here will suggest ways of defining and analyzing other partial differential equations and inequalities. △ Less

Submitted 26 September, 2009; originally announced September 2009.

Journal ref: Integral Equations Operator Theory 61 (2008), no. 1, 77--102

Every free basic convex semi-algebraic set has an LMI representation

Authors: J. William Helton, Scott McCullough

Abstract: The (matricial) solution set of a Linear Matrix Inequality (LMI) is a convex basic non-commutative semi-algebraic set. The main theorem of this paper is a converse, a result which has implications for both semidefinite programming and systems engineering. For p(x) a non-commutative polynomial in free variables x= (x1, ... xg) we can substitute a tuple of symmetric matrices X= (X1, ... Xg) for x an… ▽ More The (matricial) solution set of a Linear Matrix Inequality (LMI) is a convex basic non-commutative semi-algebraic set. The main theorem of this paper is a converse, a result which has implications for both semidefinite programming and systems engineering. For p(x) a non-commutative polynomial in free variables x= (x1, ... xg) we can substitute a tuple of symmetric matrices X= (X1, ... Xg) for x and obtain a matrix p(X). Assume p is symmetric with p(0) invertible, let Ip denote the set {X: p(X) is an invertible matrix}, and let Dp denote the component of Ip containing 0. THEOREM: If the set Dp is uniformly bounded independent of the size of the matrix tuples, then Dp has an LMI representation if and only if it is convex. Linear engineering systems problems are called "dimension free" if they can be stated purely in terms of a signal flow diagram with L2 performance measures, e.g., H-infinity control. Conjecture: A dimension free problem can be made convex if and only it can be made into an LMI. The theorem here settles the core case affirmatively. △ Less

Submitted 30 August, 2011; v1 submitted 29 August, 2009; originally announced August 2009.

MSC Class: 47Axx (Primary); 47A63; 47L07; 47L30; 14P10 (Secondary)

Analytic mappings between noncommutative pencil balls

Authors: J. William Helton, Igor Klep, Scott McCullough

Abstract: In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. In an earlier paper we characterized NC analyti… ▽ More In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. In an earlier paper we characterized NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call "NC ball maps". In this paper we turn to a more general dimension-free ball B_L, called a "pencil ball", associated with a homogeneous linear pencil L(x):= A_1 x_1 + ... + A_m x_m, where A_j are complex matrices. For an m-tuple X of square matrices of the same size, define L(X):=\sum A_j \otimes X_j and let B_L denote the set of all such tuples X satisfying ||L(X)||<1. We study the generalization of NC ball maps to these pencil balls B_L, and call them "pencil ball maps". We show that every B_L has a minimal dimensional (in a certain sense) defining pencil L'. Up to normalization, a pencil ball map is the direct sum of L' with an NC analytic map of the pencil ball into the ball. That is, pencil ball maps are simple, in contrast to the classical result of D'Angelo on such analytic maps in C^m. To prove our main theorem, this paper uses the results of our previous paper mentioned above plus entirely different techniques, namely, those of completely contractive maps. △ Less

Submitted 1 December, 2010; v1 submitted 5 August, 2009; originally announced August 2009.

Comments: 30 pages, final version. To appear in the Journal of Mathematical Analysis and Applications

MSC Class: 47A56; 46L07

Journal ref: J. Math. Anal. Appl. 376 (2011), no. 2, 407-428

Sign patterns for chemical reaction networks

Authors: J. William Helton, Igor Klep, Vitaly Katsnelson

Abstract: Most differential equations found in chemical reaction networks (CRNs) have the form $dx/dt=f(x)= Sv(x)$, where $x$ lies in the nonnegative orthant, where $S$ is a real matrix (the stoichiometric matrix) and $v$ is a column vector consisting of real-valued functions having a special relationship to $S$. Our main interest will be in the Jacobian matrix, $f'(x)$, of $f(x)$, in particular in whethe… ▽ More Most differential equations found in chemical reaction networks (CRNs) have the form $dx/dt=f(x)= Sv(x)$, where $x$ lies in the nonnegative orthant, where $S$ is a real matrix (the stoichiometric matrix) and $v$ is a column vector consisting of real-valued functions having a special relationship to $S$. Our main interest will be in the Jacobian matrix, $f'(x)$, of $f(x)$, in particular in whether or not each entry $f'(x)_{ij}$ has the same sign for all $x$ in the orthant, i.e., the Jacobian respects a sign pattern. In other words species $x_j$ always acts on species $x_i$ in an inhibitory way or its action is always excitatory. In Helton, Klep, Gomez we gave necessary and sufficient conditions on the species-reaction graph naturally associated to $S$ which guarantee that the Jacobian of the associated CRN has a sign pattern. In this paper, given $S$ we give a construction which adds certain rows and columns to $S$, thereby producing a stoichiometric matrix $\widehat S$ corresponding to a new CRN with some added species and reactions. The Jacobian for this CRN based on $\hat S$ has a sign pattern. The equilibria for the $S$ and the $\hat S$ based CRN are in exact one to one correspondence with each equilibrium $e$ for the original CRN gotten from an equilibrium $\hat e$ for the new CRN by removing its added species. In our construction of a new CRN we are allowed to choose rate constants for the added reactions and if we choose them large enough the equilibrium $\hat e$ is locally asymptotically stable if and only if the equilibrium $e$ is locally asymptotically stable. Further properties of the construction are shown, such as those pertaining to conserved quantities and to how the deficiencies of the two CRNs compare. △ Less

Submitted 20 April, 2009; originally announced April 2009.

Comments: 23 pages

MSC Class: 80A30; 15A48; 37N25

Journal ref: J. Math. Chem. 47 (2010), no. 1, 403-429

Classification of All Noncommutative Polynomials Whose Hessian Has Negative Signature One and A Noncommutative Second Fundamental Form

Authors: Harry Dym, Jeremy M. Greene, J. William Helton, Scott A. McCullough

Abstract: Every symmetric polynomial p(x)=p(x_1,...,x_g) (with real coefficients) in g noncommuting variables x_1, ..., x_g can be written as a sum and difference of squares of noncommutative polynomials. Let s(p), the negative signature of p, denote the minimum number of negative squares used in this representation, and let the noncommutative Hessian of p be defined by the formula p''(x)[h] := d^2p(x+th)… ▽ More Every symmetric polynomial p(x)=p(x_1,...,x_g) (with real coefficients) in g noncommuting variables x_1, ..., x_g can be written as a sum and difference of squares of noncommutative polynomials. Let s(p), the negative signature of p, denote the minimum number of negative squares used in this representation, and let the noncommutative Hessian of p be defined by the formula p''(x)[h] := d^2p(x+th)\dt^2|_{t=0}. In this paper we classify all symmetric noncommutative polynomials p(x) such that s(p'') is 0 or 1 . We also introduce the relaxed Hessian of a symmetric polynomial p of degree d via the formula p''_{L,K}(x)[h] := p''(x)[h] + L p'(x)[h]^{T} p'(x)[h] + K R(x)[h] for L, K real numbers and show that if this relaxed Hessian is positive semidefinite in a suitable and relatively innocuous way, then p has degree at most 2. Here R(x)[h] is a simple universal positive polynomial which is quadratic in h. This analysis is motivated by an attempt to develop properties of noncommutative real algebraic varieties pertaining to their curvature, since, as will be shown elsewhere, - < p''_{L,K}(x)[h]v, v > (appropriately restricted) plays the role of of a noncommutative second fundamental form. △ Less

Submitted 11 March, 2009; originally announced March 2009.

Comments: 44 pages

MSC Class: 14A22; 14P10; 47A13; 46L07

Noncommutative ball maps

Authors: J. William Helton, Igor Klep, Scott McCullough, Nick Slinglend

Abstract: In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These types of functions have recently been use… ▽ More In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These types of functions have recently been used in the study of dimension-free linear system engineering problems. In this paper we characterize NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call "NC ball maps". We find that up to normalization, an NC ball map is the direct sum of the identity map with an NC analytic map of the ball into the ball. That is, "NC ball maps" are very simple, in contrast to the classical result of D'Angelo on such analytic maps over C. Another mathematically natural class of maps carries a variant of the noncommutative distinguished boundary to the boundary, but on these our results are limited. We shall be interested in several types of noncommutative balls, conventional ones, but also balls defined by constraints called Linear Matrix Inequalities (LMI). What we do here is a small piece of the bigger puzzle of understanding how LMIs behave with respect to noncommutative change of variables. △ Less

Submitted 28 September, 2008; originally announced September 2008.

Comments: 46 pages

MSC Class: 47A56; 46L07 (Primary) 32H99; 32A99; 46L89 (Secondary)

Journal ref: J. Funct. Anal. 257 (2009), no. 1, 47-87

Non-Commutative Partial Matrix Convexity

Authors: Damon M. Hay, J. William Helton, Adrian Lim, Scott McCullough

Abstract: Let $p$ be a polynomial in the non-commuting variables $(a,x)=(a_1,...,a_{g_a},x_1,...,x_{g_x})$. If $p$ is convex in the variables $x$, then $p$ has degree two in $x$ and moreover, $p$ has the form $p = L + Λ^T Λ,$ where $L$ has degree at most one in $x$ and $Λ$ is a (column) vector which is linear in $x,$ so that $Λ^TΛ$ is a both sum of squares and homogeneous of degree two. Of course the conv… ▽ More Let $p$ be a polynomial in the non-commuting variables $(a,x)=(a_1,...,a_{g_a},x_1,...,x_{g_x})$. If $p$ is convex in the variables $x$, then $p$ has degree two in $x$ and moreover, $p$ has the form $p = L + Λ^T Λ,$ where $L$ has degree at most one in $x$ and $Λ$ is a (column) vector which is linear in $x,$ so that $Λ^TΛ$ is a both sum of squares and homogeneous of degree two. Of course the converse is true also. Further results involving various convexity hypotheses on the $x$ and $a$ variables separately are presented. △ Less

Submitted 3 April, 2008; originally announced April 2008.

Comments: 24 pages

Determinant Expansions of Signed Matrices and of Certain Jacobians

Authors: J. William Helton, Igor Klep, Raul Gomez

Abstract: This paper treats two topics: matrices with sign patterns and Jacobians of certain mappings. The main topic is counting the number of plus and minus coefficients in the determinant expansion of sign patterns and of these Jacobians. The paper is motivated by an approach to chemical networks initiated by Craciun and Feinberg. We also give a graph-theoretic test for determining when the Jacobian of… ▽ More This paper treats two topics: matrices with sign patterns and Jacobians of certain mappings. The main topic is counting the number of plus and minus coefficients in the determinant expansion of sign patterns and of these Jacobians. The paper is motivated by an approach to chemical networks initiated by Craciun and Feinberg. We also give a graph-theoretic test for determining when the Jacobian of a chemical reaction dynamics has a sign pattern. △ Less

Submitted 28 February, 2008; originally announced February 2008.

Comments: 25 pages

MSC Class: 15A48; 80A30

Journal ref: SIAM J. Matrix Anal. Appl. 31 (2009), no. 2, 732-754

Structured Semidefinite Representation of Some Convex Sets

Authors: J. William Helton, Jiawang Nie

Abstract: Linear matrix Inequalities (LMIs) have had a major impact on control but formulating a problem as an LMI is an art. Recently there is the beginnings of a theory of which problems are in fact expressible as LMIs. For optimization purposes it can also be useful to have "lifts" which are expressible as LMIs. We show here that this is a much less restrictive condition and give methods for actually c… ▽ More Linear matrix Inequalities (LMIs) have had a major impact on control but formulating a problem as an LMI is an art. Recently there is the beginnings of a theory of which problems are in fact expressible as LMIs. For optimization purposes it can also be useful to have "lifts" which are expressible as LMIs. We show here that this is a much less restrictive condition and give methods for actually constructing lifts and their LMI representation. △ Less

Submitted 13 February, 2008; originally announced February 2008.

Comments: 6 pages

Homotopy methods for counting reaction network equilibria

Authors: Gheorghe Craciun, J. William Helton, Ruth J. Williams

Abstract: Dynamical system models of complex biochemical reaction networks are usually high-dimensional, nonlinear, and contain many unknown parameters. In some cases the reaction network structure dictates that positive equilibria must be unique for all values of the parameters in the model. In other cases multiple equilibria exist if and only if special relationships between these parameters are satisfi… ▽ More Dynamical system models of complex biochemical reaction networks are usually high-dimensional, nonlinear, and contain many unknown parameters. In some cases the reaction network structure dictates that positive equilibria must be unique for all values of the parameters in the model. In other cases multiple equilibria exist if and only if special relationships between these parameters are satisfied. We describe methods based on homotopy invariance of degree which allow us to determine the number of equilibria for complex biochemical reaction networks and how this number depends on parameters in the model. △ Less

Submitted 8 September, 2008; v1 submitted 9 November, 2007; originally announced November 2007.

Comments: 27 pages

MSC Class: 80A30; 37C25; 65H10; 92C45