Basic r-symmetric tropical polynomials generate the semifield of S_n-invariant tropical rational functions, and arbitrary permutation groups admit a quadratic degree bound that is optimal for alternating groups.
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The basic tropical polynomials generate the semifield of $r$-symmetric tropical rational functions
Basic r-symmetric tropical polynomials generate the semifield of S_n-invariant tropical rational functions, and arbitrary permutation groups admit a quadratic degree bound that is optimal for alternating groups.