LMI Representations of Convex Semialgebraic Sets and Determinantal Representations of Algebraic Hypersurfaces: Past, Present, and Future

TL;DR

This paper reviews the development of LMI representations for convex semialgebraic sets and determinantal representations of algebraic hypersurfaces, highlighting past progress, current challenges, and future directions in the field.

Contribution

It provides a comprehensive overview of the history, recent advances, and open problems related to LMI and determinantal representations in convex algebraic geometry.

Findings

Summary of key developments over the past decade

Identification of open questions in LMI representations

Presentation of new results related to algebraic hypersurfaces

Abstract

10 years ago or so Bill Helton introduced me to some mathematical problems arising from semidefinite programming. This paper is a partial account of what was and what is happening with one of these problems, including many open questions and some new results.