Real Algebraic and Analytic Geometry
Submission: 2013, August 1.
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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