2011/08/22 by Tim Netzer, Netzer, Tim, Daniel Plaumann +3
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #FOS: Electrical engineering #FOS: Mathematics #Mathematics and Applications #Matrix Theory and Algorithms #Optimization and Control (math.OC) #Polynomial and algebraic computation #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.1108.4380
openalex publication_date 2011/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the problem of writing real polynomials as determinants of symmetric linear matrix polynomials. This problem of algebraic geometry, whose roots go back to the nineteenth century, has recently received new attention from the viewpoint of convex optimization. We relate the question to sums of squares decompositions of a certain Hermite matrix. If some power of a polynomial admits a definite determinantal representation, then its Hermite matrix is a sum of squares. Conversely, we show how a determinantal representation can sometimes be constructed from a sums-of-squares decomposition of the Hermite matrix. We finally show that definite determinantal representations always exist, if one allows for denominators.