2014/02/03 by Sadik Iliman, Iliman, Sadik, Timo de Wolff +1 · 4 citations
Computer Science · Mathematics · #11E25 #12D10 #14M25 #14P10 #14T05 #26C10 #52B20 #Advanced Algebra and Logic #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.1402.0462
openalex publication_date 2014/02/03 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
We completely characterize sections of the cones of nonnegative polynomials,\nconvex polynomials and sums of squares with polynomials supported on circuits,\na genuine class of sparse polynomials. In particular, nonnegativity is\ncharacterized by an invariant, which can be immediately derived from the\ninitial polynomial. Furthermore, nonnegativity of such polynomials f\ncoincides with solidness of the amoeba of f, i.e., the Log-absolute-value\nimage of the algebraic variety \V(f) \⊂ (\ℂ^*)n of\nf.\n These results generalize earlier works both in amoeba theory and real\nalgebraic geometry by Fidalgo, Kovacec, Reznick, Theobald and de Wolff and\nsolve an open problem by Reznick. They establish the first direct connection\nbetween amoeba theory and nonnegativity of real polynomials. Additionally,\nthese statements yield a completely new class of nonnegativity certificates\nindependent from sums of squares certificates.\n