The Wonderful Geometry of the Vandermonde map

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We study the geometry of the image of the nonnegative orthant under the power-sum and elementary symmetric polynomials maps. After analyzing the image in a finite number of 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 Grigoriy Blekherman, Annie Raymond, and Fan Wei. “Undecidability of polynomial inequalities in weighted graph homomorphism densities”. In: Forum of Mathematics, Sigma . Vol. 12. Cambridge University Press. 2024, e40.). We verify the experimental observation that the image has the combinatorial geometry of a cyclic polytope, as noted by Hana Melánová, Bernd Sturmfels, and Rosa Winter. “Recovery from Power Sums”. In: Experimental Mathematics (2022), pp. 1–10., and generalize results of Man-Duen Choi, Tsit-Yuen Lam, and Bruce Reznick. “Even symmetric sextics”. In: Mathematische Zeitschrift 195.4 (1987), pp. 559–580. on nonnegative even symmetric polynomials. We also show that undecidability does not hold for the normalized power sum map.