Real Algebraic and Analytic Geometry |

Noncommutative polynomials nonnegative on a variety intersect a convex set.

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Submission: 2013, August 1.

*Abstract:
By a result of Helton and McCullough, open bounded convex free
semialgebraic sets are exactly open (matricial) solution sets D_L of a
linear matrix inequality (LMI) L(X)>0. This paper gives a precise
algebraic certificate for a polynomial being nonnegative on a convex
semialgebraic set intersect a variety, a so-called "Perfect"
Positivstellensatz.
For example, given a generic convex free semialgebraic set D_L we
determine all "(strong sense) defining polynomials" p for D_L. This
follows from our general result for a given linear pencil L and a finite
set I of rows of polynomials. A matrix polynomial p is positive where L
is positive and I vanishes if and only if p has a weighted sum of
squares representation module the "L-real radical" of I. In such a
representation the degrees of the polynomials appearing depend in a very
tame way only on the degree of p and the degrees of the elements of I.
Further, this paper gives an efficient algorithm for computing the
L-real radical of I.
Our Positivstellensatz has a number of additional consequences which are
presented.*

Mathematics Subject Classification (2010): 13J30, 14A22, 46L07, 16S10, 14P10, 47Lxx, 16Z05, 90C22.

Keywords and Phrases: free real algebraic geometry, linear matrix inequality (LMI), spectrahedron, free algebra, complete positivity, symbolic computation, semidefinite programming (SDP).

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