Abstract. Below we discuss the partition of the space of real univariate polynomials according to the number of positive and negative roots and signs of the coefficients. We present several series of non-realizable combinations of signs together with the numbers of positive and negative roots. We provide a detailed information about possible non-realizable combinations as above up to degree 8 as well as a general conjecture about such combinations.Cogito ergo sum (I think, therefore I am) 1

We consider integrals that generalize both the Mellin transforms of rational functions of the form 1/f and the classical Euler integrals. The domains of integration of our so-called Euler-Mellin integrals are naturally related to the coamoeba of f , and the components of the complement of the closure of the coamoeba give rise to a family of these integrals. After performing an explicit meromorphic continuation of Euler-Mellin integrals, we interpret them as A-hypergeometric functions and discuss their linear independence and relation to Mellin-Barnes integrals.

In memory of Mikael Passare, who continues to inspire.Abstract. Given a hypersurface coamoeba of a Laurent polynomial f , it is an open problem to describe the structure of the set of connected components of its complement. In this paper we approach this problem by introducing the lopsided coamoeba. We show that the closed lopsided coamoeba comes naturally equipped with an order map, i.e. a map from the set of connected components of its complement to a translated lattice inside the zonotope of a Gale dual of the point configuration supp(f ). Under a natural assumption, this map is a bijection. Finally we use this map to obtain new results concerning coamoebas of polynomials of small codimension. 1

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We describe the algebraic boundary of the cone of sums of nonnegative circuit polynomials (sonc). This cone is generated by monomials and singular circuit polynomials. We interpret a singular circuit polynomial as compositions of an agiform in the sense of Reznick and a real torus action. Our main result is that the algebraic pieces of the boundary of the sonc cone are parametrized by families of tropical hypersurfaces. Each piece is contained in a rational variety called the positive discriminant, defined as the image of a Horn-Kapranov-type map. We give a complete description of the semi-algebraic stratification of the boundary of the sonc cone in the univariate case.We recover that the sonc cone equals the sage cone, and we prove that that the sonc cone is equal to the nonnegativity cone in codimension at most two. We also derive a combinatorial characterization of support sets for which the sonc cone is equal to the nonnegativity cone, disproving in the process a conjecture by Chandrasekaran, Murray, and Wierman.