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Quasi-quadratic modules in valuation rings and valued fields
We define quasi-quadratic modules in a commutative ring generalizing the notion of quadratic modules.
The main result is a structure theorem of...
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Free Bianalytic Maps between Spectrahedra and Spectraballs in a Generic Setting
Given a tuple E = (E1, . . . , Eg) of d × d matrices, the collection...
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A note on λ-compact spaces
We extend the well-known and important fact that “a topological space X is compact if and only if every ideal in C ( X ) is fixed” , to more general...
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The tracial moment problem and trace-optimization of polynomials
The main topic addressed in this paper is trace-optimization of polynomials in noncommuting (nc) variables: given an nc polynomial f , what is the...
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The matricial relaxation of a linear matrix inequality
Given linear matrix inequalities (LMIs) L 1 and L 2 it is natural to ask:
(Q 1 ) when does one dominate the other, that is, does
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Tracial Nullstellensätze
We survey some of the latest developments on the geometry of polynomials in noncommuting variables, focusing on various Nullstellensätze both in the...
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Positivstellensätze for real function algebras
We look for algebraic certificates of positivity for functions which are not necessarily polynomial functions. Similar questions were examined...
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Sums of squares on reducible real curves
We ask whether every polynomial function that is non-negative on a real algebraic curve can be expressed as a sum of squares in the coordinate ring....
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Noncommutative Real Algebraic Geometry Some Basic Concepts and First Ideas
We propose and discuss how basic notions (quadratic modules, positive elements, semialgebraic sets, Archimedean orderings) and results...
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Sums of Squares, Moment Matrices and Optimization Over Polynomials
We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NP-hard in...
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A note on the representation of positive polynomials with structured sparsity
We consider real polynomials in finitely many variables. Let the variables consist of finitely many blocks that are allowed to overlap in a certain...
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Maximal Quadratic Modules on ∗-rings
We generalize the notion of and results on maximal proper quadratic modules from commutative unital rings to ∗-rings and discuss the relation of this...
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Embedding Ordered Valued Domains into Division Rings
We study some classes of ordered domains that are embeddable in division rings. We prove the ordered version of the Cohn–Lichtman embedding theorem...
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Real Closed Graded Fields
We investigate homogeneous orderings on G -graded rings where G is an arbitrary ordered abelian group. For this we introduce the notion of real closed...
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Positivity and Convexity in Rings of Fractions
Given a commutative ring A equipped with a preordering A + (in the most general sense, see below), we look for a fractional ring extension (= “ring of...
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On hermitian polynomial optimization
We compare three levels of algebraic certificates for evaluating the maximum modulus of a complex analytic polynomial, on a compact semi-algebraic...
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Central Extensions of *-ordered Skew Fields
A *-ordering of a skew field D induces an ordering of the field K of its central symmetric elements. Let F be an ordered field extension of K . We...