The Wonderful Geometry of the Vandermonde map

Jose AcevedoGrigoriy BlekhermanSebastian DebusCordian Riener

Mar 2023

摘要

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

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