Determinantal Representations and the Hermite Matrix
For researchers in algebraic geometry and convex optimization, this work provides theoretical connections but is incremental, building on known concepts.
This paper connects determinantal representations of real polynomials to sums-of-squares decompositions of the Hermite matrix, showing that definite determinantal representations always exist if denominators are allowed.
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.