Stable complete coordinates for multisets of points via basic $r$-symmetric tropical polynomials

Abstract

A multiset of $n$ unordered points in $\mathbb{R}^r$ -- a point cloud, or, for $r=2$, a persistence barcode of birth-death pairs -- is a point of the orbit space $\mathbb{R}^{nr}/S_n$ for the symmetric group $S_n$ permuting the rows of an $n \times r$ matrix; a separating family of invariants on this space is exactly a complete set of permutation-independent coordinates. We provide one that is explicit, small, and stable, in the max-plus (tropical) setting: for all $n \geq 1$ and $r \geq 1$, the $\binom{n+r}{r}$ basic $r$-symmetric tropical polynomials, of degree at most $n$, separate the orbits of $S_n$ on $\mathbb{R}^{nr}$. This settles in full a problem left open in [Kubo, J. Pure Appl. Algebra 223 (2019) 72-85], where separation was known only for $r=2$ and special cases of $r \geq 3$, and yields a family far smaller and of lower degree than the general separating sets from Derksen's recent theory of tropical invariants for permutation actions ($nr + (nr)!/n!$ invariants of degree $O(n^2 r^2)$). The proof is elementary and constructive: the basic values are identified with a transportation problem, and the multiset is recovered from the dual by an explicit algorithm. We further show the coordinate map is a bi-Lipschitz embedding for all $n$ and $r$, being an injective max filter bank (via the bi-Lipschitz theory of max filtering), with an explicit Lipschitz constant for the forward bound and a fully explicit, dimension-free distortion when $r=1$. Finally we determine when the pairwise values suffice (exactly $n \leq 3$) and show that invariants on at least three columns and of degree less than $n$ are necessary in general, the obstruction being a standard non-uniqueness configuration from discrete tomography.