# Open Collections

## BIRS Workshop Lecture Videos ## BIRS Workshop Lecture Videos

### Noncommutative polynomials describing convex sets Klep, Igor

#### Description

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 NichtsingulÃ Â¤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 VolÃ iÃ .