The Wonderful Geometry of the Vandermonde map

We study the geometry of the image of the nonnegative orthant under the power-sum map and the elementary symmetric polynomials map. After analyzing the image in finitely many variables, we concentrate on the limit as the number of variables approaches infinity. We explain how the geometry of the limit plays a crucial role in undecidability results in nonnegativity of symmetric polynomials, deciding validity of trace inequalities in linear algebra, and extremal combinatorics - recently observed by Blekherman, Raymond, and F. Wei. We verify the experimental observation that the image has the combinatorial geometry of a cyclic polytope made by Mel\'anov\'a, Sturmfels, and Winter, and generalize results of Choi, Lam, and Reznick on nonnegative even symmetric polynomials. We also show that undecidability does not hold for the normalized power sum map.