{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T20:25:43Z","timestamp":1787343943951,"version":"build-2736575974"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,9,22]],"date-time":"2025-09-22T00:00:00Z","timestamp":1758499200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,9,22]],"date-time":"2025-09-22T00:00:00Z","timestamp":1758499200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100007465","name":"UiT The Arctic University of Norway","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100007465","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Found Comput Math"],"published-print":{"date-parts":[[2026,8]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    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. \u201cUndecidability of polynomial inequalities in weighted graph homomorphism densities\u201d. In:\n                    <jats:italic>Forum of Mathematics, Sigma<\/jats:italic>\n                    . Vol.\u00a012. 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 Mel\u00e1nov\u00e1, Bernd Sturmfels, and Rosa Winter. \u201cRecovery from Power Sums\u201d. In:\n                    <jats:italic>Experimental Mathematics<\/jats:italic>\n                    (2022), pp.\u00a01\u201310., and generalize results of Man-Duen Choi, Tsit-Yuen Lam, and Bruce Reznick. \u201cEven symmetric sextics\u201d. In:\n                    <jats:italic>Mathematische Zeitschrift<\/jats:italic>\n                    195.4 (1987), pp.\u00a0559\u2013580. on nonnegative even symmetric polynomials. We also show that undecidability does not hold for the normalized power sum map.\n                  <\/jats:p>","DOI":"10.1007\/s10208-025-09718-6","type":"journal-article","created":{"date-parts":[[2025,9,22]],"date-time":"2025-09-22T14:46:18Z","timestamp":1758552378000},"page":"2005-2051","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Wonderful Geometry of the Vandermonde map"],"prefix":"10.1007","volume":"26","author":[{"given":"Jose","family":"Acevedo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Grigoriy","family":"Blekherman","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sebastian","family":"Debus","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Cordian","family":"Riener","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,9,22]]},"reference":[{"issue":"4","key":"9718_CR1","doi-asserted-by":"publisher","first-page":"1097","DOI":"10.1007\/s00454-024-00652-y","volume":"73","author":"J Acevedo","year":"2025","unstructured":"J.\u00a0Acevedo and G.\u00a0Blekherman, Power mean inequalities and sums of squares, Discrete Comput. 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Math."},{"key":"9718_CR4","doi-asserted-by":"crossref","unstructured":"G.\u00a0Blekherman, A.\u00a0Raymond, and F.\u00a0Wei, Undecidability of polynomial inequalities in weighted graph homomorphism densities, Forum Math. Sigma 12 (2024), Paper No. e40, 16.","DOI":"10.1017\/fms.2024.19"},{"issue":"3","key":"9718_CR5","doi-asserted-by":"publisher","first-page":"764","DOI":"10.1007\/s00454-020-00208-w","volume":"65","author":"G Blekherman","year":"2021","unstructured":"G.\u00a0Blekherman and C.\u00a0Riener, Symmetric non-negative forms and sums of squares, Discrete Comput. Geom. 65 (2021), no.\u00a03, 764\u2013799.","journal-title":"Discrete Comput. Geom."},{"key":"9718_CR6","doi-asserted-by":"crossref","unstructured":"J.\u00a0Bochnak, M.\u00a0Coste, and M.-F. Roy, Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol.\u00a036, Springer-Verlag, Berlin, 1998.","DOI":"10.1007\/978-3-662-03718-8"},{"key":"9718_CR7","doi-asserted-by":"crossref","unstructured":"B.\u00a0Bollob\u00e1s, Relations between sets of complete subgraphs, in Proc. Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975), Congressus Numerantium XV, Utilitas Math., Winnipeg, 1976, pp.\u00a079\u201384.","DOI":"10.1017\/S0305004100052063"},{"issue":"4","key":"9718_CR8","doi-asserted-by":"publisher","first-page":"559","DOI":"10.1007\/BF01166704","volume":"195","author":"MD Choi","year":"1987","unstructured":"M.\u00a0D. Choi, T.\u00a0Y. Lam, and B.\u00a0Reznick, Even symmetric sextics, Math. Z. 195 (1987), no.\u00a04, 559\u2013580.","journal-title":"Math. 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Foregger, On the relative extrema of a linear combination of elementary symmetric functions, Linear and Multilinear Algebra 20 (1987), no.\u00a04, 377\u2013385.","journal-title":"Linear and Multilinear Algebra"},{"key":"9718_CR12","doi-asserted-by":"crossref","unstructured":"D.\u00a0Gale, Neighborly and cyclic polytopes, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, RI, 1963, pp.\u00a0225\u2013232.","DOI":"10.1090\/pspum\/007\/0152944"},{"key":"9718_CR13","doi-asserted-by":"crossref","unstructured":"A.\u00a0B. Givental\u2019, Moments of random variables and the equivariant Morse lemma, Uspekhi Mat. 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Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995."}],"container-title":["Foundations of Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-025-09718-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10208-025-09718-6","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-025-09718-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:43:24Z","timestamp":1787341404000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10208-025-09718-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,22]]},"references-count":36,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2026,8]]}},"alternative-id":["9718"],"URL":"https:\/\/doi.org\/10.1007\/s10208-025-09718-6","relation":{},"ISSN":["1615-3375","1615-3383"],"issn-type":[{"value":"1615-3375","type":"print"},{"value":"1615-3383","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,22]]},"assertion":[{"value":"2 May 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 April 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 April 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 September 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}