Noncommutative Polynomials Nonnegative on a Variety Intersect a Convex Set
نویسندگان
چکیده
By a result of Helton and McCullough [HM12], 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. Such polynomials must have the form
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The convex Positivstellensatz in a free algebra
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