  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.
