The convex Positivstellensatz in a free algebra
By William Helton, Igor Klep, Scott McCullough. 2011
62 citations recorded by the source corpus.
Abstract Given a monic linear pencil L in g variables, let P L = ( P L ( n ) ) n ∈ N where P L ( n ) : = { X ∈ S n g ∣ L ( X ) ⪰ 0 } , and S n g is the set of g -tuples of symmetric n × n matrices. Because L is a monic linear pencil, each P L ( n ) is convex with interior, and conversely it is known that convex bounded noncommutative semialgebraic sets with interior are all of the form P L . The main result of this paper establishes a perfect noncommutative Nichtnegativstellensatz on a convex semialgebraic set. Namely, a noncommutative matrix-valued polynomial p is positive semidefinite on P L if and only if it has a weighted sum of squares representation with optimal degree bounds: p = s ∗ s + ∑ j finite f j ∗ L f j , where s , f j are matrices of noncommutative polynomials of degree no greater than deg ( p ) 2 . 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.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/44e5273410d9b9baec93adc2f8fd787c6ff122c7
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In Algebra and Number Theory
- J. William Helton
- Igor Klep
- Scott McCullough
- Nathanial P. Brown Narutaka Ozawa
- Gelu Popescu
- Victor Vinnikov
- Dmitry S. Kaliuzhnyi-Verbovetskyi
In Computational Theory and Mathematics
- Scott McCullough
- Igor Klep
- J. William Helton
- Victor Vinnikov
- Petter Brand'en
- Nathanial P. Brown Narutaka Ozawa
- Gelu Popescu
- Adam W. Marcus
In Geometry and Topology
Citations in the corpus
- The Tracial Hahn-Banach Theorem, Polar Duals, Matrix Convex Sets, and Projections of Free Spectrahedra (2014, background): “ 2r+1 is positive semidenite on Kif and only if p2M x(L) r. Remark 5.2. Several remarks are in order. (1)In case there are no y-variables in L, Theorem5.1reduces to the Convex Positivstellensatz of [HKM12]. (2)If r= 0, i.e., pis linear, then Theorem5.1reduces to Corollary4.15. (3)A Positivstellensat”