  We introduce positive Gorenstein ideals. These are Gorenstein ideals in the
graded ring $\RR[x]$ with socle in degree 2d, which when viewed as a linear
functional on $\RR[x]_{2d}$ is nonnegative on squares. Equivalently, positive
Gorenstein ideals are apolar ideals of forms whose differential operator is
nonnegative on squares. Positive Gorenstein ideals arise naturally in the
context of nonnegative polynomials and sums of squares, and they provide a
powerful framework for studying concrete aspects of sums of squares
representations. We present applications of positive Gorenstein ideals in real
algebraic geometry, analysis and optimization. In particular, we present a
simple proof of Hilbert's nearly forgotten result on representations of ternary
nonnegative forms as sums of squares of rational functions. Drawing on our
previous work, our main tools are Cayley-Bacharach duality and elementary
convex geometry.
