# Publications

## To appear

(100) A random copositive matrix is completely positive with positive probability, *SIAM Journal on Applied Algebra and Geometry* (with and Tea Štrekelj)

(99) Matrix Extreme Points and Free Extreme points of Free Spectrahedra, *Optimization Methods and Software* (with Aidan Epperly, , )

## Published

### 2024

(98) State polynomials: positivity, optimization and nonlinear Bell inequalities, (with , and )

(97) Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators, (with , Anirban Chowdhury, Aidan Epperly, )

### 2023

(95) Globally trace-positive noncommutative polynomials and the unbounded tracial moment problem, (with and )

(94) Ranks of linear matrix pencils separate simultaneous similarity orbits, (with , Visu Makam and )

(93) Noncommutative Nullstellensätze and Perfect Games, (with , )

(92) Noncommutative Polynomial Optimization, (with and Abhishek Bhardwaj)

### 2022

(91) Dimension-free entanglement detection in multipartite Werner states, (with , and )

(90) Facial structure of matrix convex sets, (with Tea Štrekelj)

(89) Sparse Noncommutative Polynomial Optimization, (with and )

(88) Factorization of noncommutative polynomials and Nullstellensätze for the free algebra, (with and )

(87) Noncommutative partially convex rational functions, (with , Mark Mancuso, and )

(86) Optimization over trace polynomials, (with and )

### 2021

(85) Noncommutative polynomials describing convex sets, (with , and )

(84) Positive univariate trace polynomials, (with and )

(83) Noncommutative partial convexity via \(\Gamma\)-convexity, (with , Mark Mancuso, and )

(82) A note on group representations, determinantal hypersurfaces and their quantizations, *“Operator Theory, Functional Analysis and Applications”* () edited by M.A. Bastos, L. Castro, A.Y. Karlovich, 393–402, Birkhäuser, 2021 (with )

### 2020

(81) Local theory of free noncommutative functions: germs, meromorphic functions and Hermite interpolation, (with and )

(80) Bianalytic free maps between spectrahedra and spectraballs, (with , and )

(79) Multipartite rational functions, (with and )

(78) Plurisubharmonic noncommutative rational functions, (with , , and )

(77) A Matrix Positivstellensatz with lifting polynomials, (with )

(76) Noncommutative rational functions invariant under the action of a finite solvable group, (with , Gregor Podlogar and )

### 2019

(75) Dilations, Linear Matrix Inequalities, the Matrix Cube Problem and Beta Distributions, *Mem. Amer. Math. Soc.* 257 vol. 1232 (2019), 104pp (with , and
)

(74) There are many more positive maps than completely positive maps, (with , and ). Supplementary material (a Mathematica notebook file) is available as a nb file, and as a pdf file.

(73) Free bianalytic maps between spectrahedra and spectraballs in a generic setting, *“Interpolation and Realization Theory with Applications to Control Theory”* (), edited by V. Bolotnikov, S. ter Horst, A.C.M. Ran, V. Vinnikov, 43–60, Birkhäuser, 2019 (with , and )

### 2018

(72) Bianalytic Maps Between Free Spectrahedra, (with , and )

(71) Geometry of free loci and factorization of noncommutative polynomials, (with and ). A preliminary version was published as Oberwolfach Preprint 2017-23

(70) Skolem-Noether algebras, (with , , )

(69) Positive trace polynomials and the universal Procesi-Schacher conjecture, (with and )

(68) Extreme points of matrix convex sets, free spectrahedra and dilation theory, (with , and ). The post-production version fixes issues identified by Tom Kriel and Ben Passer.

(67) Minimizer extraction in polynomial optimization is robust, (with , )

(66) The real spectrum of a noncommutative ring and the Artin-Lang homomorphism theorem,

### 2017

(65) The tracial Hahn-Banach theorem, polar duals, matrix convex sets, and projections of free spectrahedra, (with and )

(64) Free loci of matrix pencils and domains of noncommutative rational functions, (with ). The post-production version contains an appendix due independently to and .

(63) Free function theory through matrix invariants, (with )

(62) Regular and positive noncommutative rational functions, (with and )

(61) Null- and Positivstellensätze for rationally resolvable ideals, (with and )

(60) Circular Free Spectrahedra, (with , and )

### 2016

(59) Optimization of polynomials in non-commuting variables, , Springer-Verlag, XV, 104. ISBN 978-3-319-33336-6 (with and )

(58) Matrix Convex Hulls of Free Semialgebraic Sets, (with and ). Supplementary material (a Mathematica notebook file) is available here.

(57) Instances of the Kaplansky-Lvov multilinear conjecture for polynomials of degree three, (with ). Supplementary material (a Mathematica notebook file) is available as a nb file, and as a pdf file.

(56) Sweeping words and the length of a generic vector subspace of \(M_n(\mathbb F)\), (with )

(55) Constrained trace-optimization of polynomials in freely noncommuting variables, (with )

(54) On trace-convex noncommutative polynomials, (with and Chris Nelson)

### 2015

(53) Rational sums of hermitian squares of free noncommutative polynomials, *Ars Math. Contemp.* 9 (2015) 253–269
(with and
).

### 2014

(52) A tracial Nullstellensatz, (with )

(51) Noncommutative polynomials nonnegative on a variety intersect a convex set, (with and Chris Nelson)

(50) Addendum to “Connes’ embedding conjecture and sums of hermitian squares”, (with , and )

(49) On real one-sided ideals in a free algebra, (with , , and Chris Nelson)

### 2013

(48) The matricial relaxation of a linear matrix inequality, (with and ). Supplementary material (a zip file containing three Mathematica notebooks and their pdf versions) is available here.

(47) An exact duality theory for semidefinite programming based on sums of squares, (with )

(46) A local-global principle for linear dependence of noncommutative polynomials, (with )

(45) The tracial moment problem and trace-optimization of polynomials, (with , and )

(44) Algorithmic aspects of sums of hermitian squares of noncommutative polynomials, (with , and )

(43) Free convex algebraic geometry, *“Semidefinite Optimization and Convex Algebraic Geometry”* edited by G. Blekherman, P. Parrilo, R. Thomas, 341–405, SIAM, 2013 (with and
)

### 2012

(42) The convex Positivstellensatz in a free algebra, (with and )

(41) The truncated tracial moment problem, *J. Operator Theory* 68 (2012) 141–163 (with )

(40) Free analysis, convexity and LMI domains, *“Mathematical methods in systems, optimization and control”* (Operator Theory: Advances and Applications) edited by H. Dym, M. de Oliveira, M. Putinar, 195–219, Birkhäuser, 2012
(with and
)

(39) Constrained polynomial optimization problems with noncommuting variables, (with and )

(38) Convexity and Semidefinite Programming in dimension-free matrix unknowns, *“Handbook of Semidefinite, Conic and Polynomial Optimization”* edited by M. Anjos and J. B. Lasserre, 377–405, Springer, 2012 (with and
)

### 2011

(37) Trace-positive polynomials,

(36) Proper Analytic Free Maps, (with and ). Supplementary material (a zip file containing a Mathematica notebook and its pdf version) is available here.

(35) Infeasibility certificates for linear matrix inequalities, *Oberwolfach Preprints (OWP)* 28 (2011) (with )

(34) Tracial Nullstellensätze, *“Notions of Positivity and the Geometry of Polynomials”* (J. Borcea memorial volume) edited by P. Brändén, M. Passare and M. Putinar, 79–101, Birkhäuser, 2011 (with )

(33) Finite rings in which commutativity is transitive, (with and )

(32) Analytic mappings between noncommutative pencil balls, (with and )

(31) NCSOStools: a computer algebra system for symbolic and numerical computation with noncommutative polynomials, (with and ). The accompanying MATLAB toolbox is NCSOStools.

### 2010

(30) The Procesi-Schacher conjecture and Hilbert's 17th problem for algebras with involution, (with )

(29) Pure states, positive matrix polynomials and sums of hermitian squares, *Indiana Univ. Math. J.* 59 (2010) 857–874 (with )

(28) A Note on Values of Noncommutative Polynomials, (with )

(27) A note on the nonexistence of sum of squares certificates for the BMV conjecture, (with and )

(26) Semidefinite programming and sums of hermitian squares of noncommutative polynomials, (with )

(25) Sign patterns for chemical reaction networks, (with and Vitaly Katsnelson). Accompanying Chemical Reaction Network Software runs under Mathematica.

(24) Lie algebras with abelian centralizers, (with )

(23) \(*\)-orderable semigroups, (with )

(22) Trace-positive polynomials and the quartic tracial moment problem, (with )

(21) Relaxing LMI Domination Matricially, *49th IEEE Conference on Decision and Control (CDC)*, 2010, 3331–3336 (with and
)

### 2009

(20) Noncommutative ball maps, (with , and Nick Slinglend)

(19) Noncommutative Polynomials, Lie Skew-Ideals and Tracial Nullstellensätze, (with )

(18) Determinant Expansions of Signed Matrices and of Certain Jacobians, (with and ). Accompanying Chemical Reaction Network Software (under Mathematica) can be found here.

(17) The Procesi-Schacher conjecture and Hilbert's 17th problem for algebras with involution, *Oberwolfach Reports* 6 (2009) 1388–1390 (with )

### 2008

(16) Connes’ embedding conjecture and sums of hermitian squares, (with )

(15) Sums of hermitian squares and the BMV conjecture, (with ). Complimentary Mathematica notebook and its pdf output.

(14) The Joly-Becker theorem for \(*\)-orderings, (with )

### 2007

(13) Embedding ordered valued domains into division rings,

(12) A Nichtnegativstellensatz for polynomials in noncommuting variables, (with )

(11) On valuations, places and graded rings associated to \(*\)-orderings,

(10) The Prime Spectrum and the Extended Prime Spectrum of Noncommutative Rings, (with )

(09) Sums of hermitian squares, Connes’ embedding problem and the BMV Conjecture, *Oberwolfach Reports* 4 (2007) 779–782 (with )

### 2006

(08) Embedding \(*\)-ordered domains into skew fields,

(07) Generalized orderings and rings of fractions, (with )

(06) Central extensions of \(*\)-ordered skew fields, (with )

### 2005

(05) On \(*\)-orderable groups, (with )

(04) Central extensions of n-ordered division rings, (with )

### 2004

(03) A Kadison-Dubois representation for associative rings,

(02) The noncommutative graded Positivstellensatz,

(01) n-real valuations and the higher level version of the Krull-Baer theorem, (with )