  We study the problem of computing weighted sum-of-squares (WSOS) certificates
for positive polynomials over a compact semialgebraic set. Building on the
theory of interior-point methods for convex optimization, we introduce the
concept of dual cone certificates, which allows us to interpret vectors from
the dual of the sum-of-squares cone as rigorous nonnegativity certificates of a
WSOS polynomial. Whereas conventional WSOS certificates are alternative
representations of the polynomials they certify, dual certificates are distinct
from the certified polynomials; moreover, each dual certificate certifies a
full-dimensional convex cone of WSOS polynomials. As a result, rational WSOS
certificates can be constructed from numerically computed dual certificates at
little additional cost, without any rounding or projection steps applied to the
numerical certificates. As an additional algorithmic application, we present an
almost entirely numerical hybrid algorithm for computing the optimal WSOS lower
bound of a given polynomial along with a rational dual certificate, with a
polynomial-time computational cost per iteration and linear rate of
convergence.
