Noncommutative Polynomials Describing Convex Sets

J. William Helton, Igor Klep, Scott McCullough, Jurij Volčič

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


The free closed semialgebraic set Df determined by a hermitian noncommutative polynomial f∈ M δ(C< x, x>) is the closure of the connected component of { (X, X) ∣ f(X, X) ≻ 0 } containing the origin. When L is a hermitian monic linear pencil, the free closed semialgebraic set DL is the feasible set of the linear matrix inequality L(X, X) ⪰ 0 and is known as a free spectrahedron. Evidently these are convex and it is well known that a free closed semialgebraic set is convex if and only it is a free spectrahedron. The main result of this paper solves the basic problem of determining those f for which Df is convex. The solution leads to an efficient algorithm that not only determines if Df is convex, but if so, produces a minimal hermitian monic pencil L such that Df= DL. Of independent interest is a subalgorithm based on a Nichtsingulärstellensatz presented here: given a linear pencil L~ and a hermitian monic pencil L, it determines if L~ takes invertible values on the interior of DL. Finally, it is shown that if Df is convex for an irreducible hermitian f∈ C< x, x> , then f has degree at most two, and arises as the Schur complement of an L such that Df= DL.

Original languageEnglish
Pages (from-to)575-611
Number of pages37
JournalFoundations of Computational Mathematics
Issue number2
StatePublished - 1 Apr 2021
Externally publishedYes


  • Convex set
  • Free locus
  • Linear matrix inequality (LMI)
  • Noncommutative rational function
  • Semialgebraic set
  • Spectrahedron

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics


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