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 V(f ) ⊂ (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.

Abstract. Recently, the second and third authors developed sums of nonnegative circuit polynomials (SONC) as a new certificate of nonnegativity for real polynomials, which is independent of sums of squares. In this paper we show that the SONC cone is full-dimensional in the cone of nonnegative polynomials. We establish a Positivstellensatz which guarantees that every polynomial which is positive on a given compact, semialgebraic set can be represented by the constraints of the set and SONC polynomials. Based on this Positivstellensatz, we provide a hierarchy of lower bounds converging to the minimum of a polynomial on a given compact set K. Moreover, we show that these new bounds can be computed efficiently via interior point methods using results about relative entropy functions.

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