  We completely characterize sections of the cones of nonnegative polynomials,
convex polynomials and sums of squares with polynomials supported on circuits,
a genuine class of sparse polynomials. In particular, nonnegativity is
characterized by an invariant, which can be immediately derived from the
initial polynomial. Furthermore, nonnegativity of such polynomials $f$
coincides with solidness of the amoeba of $f$, i.e., the Log-absolute-value
image of the algebraic variety $\mathcal{V}(f) \subset (\mathbb{C}^*)^n$ of
$f$.
  These results generalize earlier works both in amoeba theory and real
algebraic geometry by Fidalgo, Kovacec, Reznick, Theobald and de Wolff and
solve an open problem by Reznick. They establish the first direct connection
between amoeba theory and nonnegativity of real polynomials. Additionally,
these statements yield a completely new class of nonnegativity certificates
independent from sums of squares certificates.
