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

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The categories of this paper are: solv-int
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The categories of this paper are: patt-sol
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The categories of this paper are: comp-gas
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The categories of this paper are: q-bio
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The categories of this paper are: chem-ph
6
The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: funct-an
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The categories of this paper are: ao-sci
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
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The categories of this paper are: alg-geom
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The categories of this paper are: cond-mat
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The categories of this paper are: dg-ga