The basic tropical polynomials generate the semifield of $r$-symmetric tropical rational functions

What the math gives to ML

The paper gives an explicit finite family of max-plus invariant features for multisets of points: every basic value is a maximum-weight selection subject to disjointness constraints across feature coordinates. Unlike ordinary sum or mean pooling, the complete family separates row-permutation orbits and is bi-Lipschitz with respect to the optimal matching distance, providing both expressivity and stability guarantees. The most direct neural-network transfer is a permutation-invariant set encoder whose pooling layer computes these tropical elementary multisymmetric features exactly by dynamic programming, optionally followed by a small MLP. This is especially promising for fixed-size sets with low feature dimension, where one wants an invariant representation that preserves geometric information rather than collapsing it through moments or sums.