“Positive” noncommutative polynomials are sums of squares

By Øóö Ý ½ ¡ ¡ ¡ Ò Ì ½ ¡ ¡ ¡ Ì Ò Òò Ööúúúøø Øøøñ Ûûøø Øøø Öö×ùðøøòò, Ññøööü É´½ ¡ ¡ ¡ Ò. 2002

203 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cdf0165938219290036f2f7cd62eadc0fbca1936

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Citations in the corpus

Foundations of Free Noncommutative Function Theory (2012, background): “, [74, 73, 98] — in the context of noncommutative convexity and noncommutative real semialgebraic geometry, where one considers polynomials and rational functions in d noncommuting indeterminates evaluated on d-tuples of matrices over R [49, 56, 59, 58, 54, 57].”
Free holomorphic functions and interpolation (2007, methodology): “In related areas of research, we remark the work of Helton, McCullough, Putinar, and Vinnikov, on symmetric noncommutative polynomials ([13], [14], [15], [16]), and the work of Muhly and Solel on representations of tensor algebras over C correspondences (see [19], [20]).”
Singularities of rational functions and minimal factorizations: The noncommutative and the commutative setting (2009, background): “Most optimization problems of system theory and control are dimensionless in the sense that the natural variables are matrices of an arbitrary size, and their domain is described by size-independent polynomial or rational inequalities (see [15,29,30]).”
The inverse function theorem and the Jacobian conjecture for free analysis (2013, result): “e nonlocal data in many contexts which is in strong contrastto the classical case. For another example, compare between Putinar’s Positivstellensatz [11] to Helton’s noncommutative Positivstellensatz [6]. 1.1. Some examples of domains of invertibility. We brieﬂy give some examples of applications of”
Noncommutative sums of squares (2005, methodology): “The purpose of this note is to give a proof of Helton’s sum-of-squares (SoS) theorem [Helton 2002] based upon a theorem of Carathéodory and a Hahn–Banach separation argument.”
Operator Theory on Noncommutative Domains (2007, unknown): “In a related area of research, we remark the work of Helton, McCullough, Putinar, and Vinnikov, on symmetric noncommutative polynomials ([48], [49], [51], [50], [52]), which contains a new type of engineering application.”
The implicit function theorem and free algebraic sets (2014, unknown): “Nc-functions have been studied for a variety of reasons: by Anderson [4] as a generalization of the Weyl calculus; by Taylor [20] in the context of the functional calculus for non-commuting operators; Popescu [14–17] in the context of extending classical function theory to d -tuples of bounded opera”
Noncommutative Real Algebraic Geometry Some Basic Concepts and First Ideas (2007, unknown): “Then there exist diagonal matrices Dl ∈ Mn(R[t]), matrices X±,l ∈ Mn(R[t]) and polynomials zl ∈ ∑ R[t]2, l = 1, . . . ,m, such that: (i) X+lX−l = X−lX+l = zjI, Dl = X−lAX t −l, zlA = X+lDlX t +l, (ii) For s ∈ Rd, A(s) ≥ 0 if and only if Dl(s) ≥ 0 for all l = 1, . . . ,m. Proof.”
Matrix-J-unitary Non-commutative Rational Formal Power Series (2004, unknown): “, matrix-positivity of non-commutative polynomials (non-commutative Positivstellensatz) [29, 40, 31, 32], matrix-positivity of FPS kernels [34], matrix-convexity [21, 30].”
Pick Interpolation for free holomorphic functions (2013, background): “Nc-functions have been studied for a variety of reasons: by Taylor [29], in the context of the functional calculus for non-commuting operators; Voiculescu [30, 31], in the context of free probability; Popescu [23, 24, 25, 26], in the context of extending classical function theory to d-tuples of boun”