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BIRS Workshop Lecture Videos

Noncommutative polynomials describing convex sets Klep, Igor


The semialgebraic set $D_f$ determined by a noncommutative polynomial $f$ is the closure of the connected component of $\{(X,X^*): f(X,X^*)\succ0\}$ containing the origin. When $L$ is a linear pencil, the semialgebraic set $D_L$ is the feasible set of the linear matrix inequality $L(X,X^*)\succeq 0$ and is known as a free spectrahedron. Evidently these are convex and by a theorem of Helton \& McCullough, a free semialgebraic set is convex if and only it is a free spectrahedron. \\\\ In this talk we solve the basic problem of determining those $f$ for which $D_f$ is convex. The solution leads to an effective algorithm that not only determines if $D_f$ is convex, but if so, produces a minimal linear pencil $L$ such that $D_f=D_L$. Of independent interest is a subalgorithm based on a Nichtsingulà ¤rstellensatz: given linear pencils $L,L'$, it determines if $L'$ takes invertible values on the interior of $D_L.$ Finally, it is shown that if $D_f$ is convex for an irreducible noncommutative polynomial, then $f$ has degree at most two, and arises as the Schur complement of an $L$ such that $D_f=D_L$. This is based on joint work with Bill Helton, Scott McCullough and Jurij Volà ià .

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