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Symmetric nonnegative functions, the tropical Vandermonde cell and superdominance of power sums

  • Acevedo, Jose [1] ; Blekherman, Grigoriy [2] ; Debus, Sebastian [3] ; Riener, Cordian [4]
    1. [1] Universidad Industrial de Santander

      Universidad Industrial de Santander

      Colombia

    2. [2] Georgia Institute of Technology

      Georgia Institute of Technology

      Estados Unidos

    3. [3] Chemnitz University of Technology

      Chemnitz University of Technology

      Kreisfreie Stadt Chemnitz, Alemania

    4. [4] The Arctic University of Norway

      The Arctic University of Norway

      Noruega

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  • Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 77, Fasc. 2, 2026, págs. 555-600
  • Idioma: inglés
  • DOI: 10.1007/s13348-025-00481-z
  • Texto completo no disponible (Saber más ...)
  • Resumen

    • We study nonnegative and sums of squares symmetric (and even symmetric) functions of fixed degree. We can think of these as limit cones of symmetric nonnegative polynomials and symmetric sums of squares of fixed degree as the number of variables goes to infinity. We compare these cones, including finding explicit examples of nonnegative polynomials which are not sums of squares for any sufficiently large number of variables, and compute the tropicalizations of their dual cones in the even symmetric case. We find that the tropicalization of the dual cones is naturally understood in terms of the overlooked superdominance order on partitions. The power sum symmetric functions obey this same partial order (analogously to how term-normalized power sums obey the dominance order (Cuttler et al. Eur J Comb 32:745–761, 2011)).

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