Quasi-Convex Free Polynomials
Sriram Balasubramanian, Scott McCullough

TL;DR
This paper characterizes quasi-convex free polynomials, showing that symmetric polynomials with convex positivity sets are either quadratic convex or sums of squares, revealing a structural dichotomy in non-commutative polynomial convexity.
Contribution
It provides a classification of symmetric free polynomials with convex positivity sets, establishing that they are either quadratic convex or sums of hermitian squares, a significant structural insight.
Findings
Polynomials with convex positivity sets are at most quadratic.
Such polynomials are either convex or sums of hermitian squares.
The result applies to polynomials evaluated on symmetric matrix tuples.
Abstract
Let denote the ring of polynomials in freely non-commuting variables . There is a natural involution * on determined by and and a free polynomial is symmetric if it is invariant under this involution. If is a tuple of symmetric matrices, then the evaluation is naturally defined and further . In particular, if is symmetric, then . The main result of this article says if is symmetric, and for each and each symmetric positive definite matrix the set is convex, then has degree at most two and is itself convex, or is a hermitian sum of squares.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Topics in Algebra · Advanced Differential Equations and Dynamical Systems · Matrix Theory and Algorithms
