REVIEW 2 major objections 4 minor 11 references
The basic tropical polynomials generate the semifield of $r$-symmetric tropical rational functions
T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read The paper proves that the basic $r$-symmetric tropical polynomials generate the semifield of all $S_n$-invariant tropical rational functions on $\mathbb{R}^{n\times r}$, with explicit penalty-based expressions.
desk verdict A strong, mostly self-contained generation theorem for tropical symmetric rational functions, held back only by an unquantified bi-Lipschitz constant imported from the author's own preprint. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing identity is the sorted-column recovery $b_{i e_\alpha}(M)-b_{(i-1)e_\alpha}(M)$ equal to the $i$-th largest entry of column $\alpha$, which makes the map $s(w)$ that reconstructs sorted columns from basic values tropical rational. The re-assembly maps $R_\sigma$ join the $r$ sorted columns through permutations $\sigma_2,\dots,\sigma_r$; for the true orbit some $\sigma$ realizes $[M]$ itself. The recoupling discriminants $D_\sigma(w)=\max_c |w_c-b_c(R_\sigma(s(w)))|$ are tropical rational, vanish on a correct re-assembly, and by the bi-Lipschitz inequality $\|\Phi(M)-\Phi(M')\|_2\ge c_{n,r}\,d([M],[M'])$ are bounded below by a positive multiple of the distance to any wrong re-assembly. That lower bound calibrates the integer multiplier $\kappa_f$ that makes every incorrect term in the minimum dominate $f(M)$, forcing $g\circ\Phi=f$.
What would settle it
For a specific small case such as $(n,r)=(2,2)$, compute the infimum of $\|\Phi(M)-\Phi(M')\|_2/d([M],[M'])$ over pairs of matrices with distinct row-multisets; if the infimum is $0$ — equivalently, if two distinct orbits share all basic values $b_{(1,0)}$, $b_{(0,1)}$, $b_{(1,1)}$ — then the separating and bi-Lipschitz premise fails and Theorem 3.5 collapses. Alternatively, test the paper's explicit $(2,2)$ formula for $f(M)=\max_i(m_{i1}+m_{i2})$ with the doubling loop for $\kappa_f$ on a dense grid of matrices; a matrix for which no tested $\kappa_f$ reproduces $f(M)$ would contradict the generation claim.
Extended reading notes
Core claim
The central claim, Theorem 3.5, is that the basic-coordinate map $\Phi$ that records all values $b_c$ is a generating coordinate system for the invariant semifield: every $S_n$-invariant tropical rational function $f$ on $\mathbb{R}^{n\times r}$ equals $g\circ\Phi$ for some tropical rational $g$ on $\mathbb{R}^K$, $K=\binom{n+r}{r}-1$. The proof is constructive: the formula $g(w)=\min_{\sigma}[f(R_\sigma(s(w)))+\kappa_f D_\sigma(w)]$ ranges over the $(n!)^{r-1}$ couplings $\sigma$ of the sorted columns recovered from $w$, and each term adds a recoupling discriminant $D_\sigma(w)$ — the maximal deviation between the basic values of $w$ and those of the re-assembled matrix — multiplied by an integer $\kappa_f$ chosen large enough that, by the bi-Lipschitz inequality, any wrong re-assembly pays a penalty exceeding the possible decrease of $f$. The correct coupling has zero penalty and reproduces $f(M)$, so the minimum equals $f(M)$. Hence the invariant semifield is finitely generated in degree at most $n$, and the same mechanism gives a quadratic degree bound for all permutation groups that is optimal for the alternating group.
Load-bearing premise
The theorem rests on the bi-Lipschitz lower bound for the basic coordinate map, taken as a black box from earlier work with a constant whose existence is asserted but not quantified; if that constant were zero, or that earlier proof had a gap, the penalty terms could no longer be guaranteed to outweigh wrong re-assemblies and the generating formula would fail.
Editorial extensions
If this is right
- The semifield of $S_n$-invariant tropical rational functions on $\mathbb{R}^{n\times r}$ is generated by $\binom{n+r}{r}-1$ explicit polynomials of degree at most $n$, replacing the primorial degree bound of prior work.
- The image of the basic coordinate map $\Phi$ is exactly the zero set of the tropical rational function $\min_\sigma D_\sigma$, a finite union of polyhedra, giving a geometric form of the relations among generators.
- For every permutation group $G\le S_N$, the same mechanism generates the invariant semifield in degree at most $\max\{N,\binom{N}{2}\}$, independent of $|G|$; there exist $2N+1$ invariant tropical polynomials that separate orbits and $3N$ that generate, with at least $N$ necessary for each task.
- The quadratic degree bound is optimal: every $A_N$-invariant tropical polynomial of degree below $\binom{N}{2}$ is $S_N$-invariant, so any separating family for $A_N$ must contain a member of degree at least $\binom{N}{2}$.
- For subfamilies of the basic family that contain the single-column values, generating the invariant semifield and separating orbits are equivalent.
Reading between the lines
- The penalty construction suggests a general recipe: any separating bi-Lipschitz invariant coordinate system that admits a tropical rational 'sorting' recovery map generates the invariant semifield; a natural next test is whether such recovery maps exist, with low degree, for linear actions beyond coordinate permutations, where the quadratic bound need not be tight.
- The paper leaves the minimal separating subfamilies and the minimal size of $\kappa_f$ open; a computational scan of the $(n,r)=(2,2)$ and $(3,2)$ cases could reveal how far $\kappa_f$ must exceed the Lipschitz-to-bi-Lipschitz ratio in practice and whether minimal generating expressions correspond to the minimal separating subfamilies.
- Corollary 3.7's description of the image as a finite union of polyhedra is a tropical analogue of the relations in the ring of multisymmetric functions; this invites a normal-form or elimination algorithm for the invariant semifield, in the spirit of classical invariant theory, which the paper does not develop.
- The trace-normal-form proof that low-degree alternating invariants are fully symmetric is carried out with explicit templates; the same mechanism could yield degree lower bounds for other permutation groups generated with a normalizing transposition, and the paper's order-20 group example in Remark 4.10 suggests that low-degree blindness can occur even without such normalizers, pointing toward a g
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that, for the action of S_n permuting the rows of n × r real matrices, the basic r-symmetric tropical polynomials b_c (|c| ≤ n) generate the semifield of S_n-invariant tropical rational functions. The main theorem (Theorem 3.5) gives an explicit expression for any invariant tropical rational f as g∘Φ, where Φ is the basic-coordinate map and g is a minimum over the (n!)^{r-1} couplings of the sorted columns, each term being f evaluated on the re-assembled matrix plus a penalty built from the basic values. The proof combines sorted-column recovery, a re-assembly lemma, a recoupling-discriminant lower bound obtained from the bi-Lipschitz embedding of the orbit space, and the Lipschitz property of tropical rational functions. The paper also gives an image description of Φ, a subfamily generation criterion, an expression algorithm, and a generalization to arbitrary permutation groups with a quadratic degree bound, together with an optimality statement for the alternating group.
Significance. If the result is correct, it is a substantial advance in tropical invariant theory: it replaces Derksen's primorial degree bound for the row-permutation action by generators of degree at most n, answering a question explicitly raised in the author's earlier work. The construction is explicit and algorithmic, and the axiomatic formulation in Section 4 isolates the two structural ingredients, yielding a quadratic degree bound for all permutation groups and a sharp A_N rigidity theorem. The paper is carefully written and contains a useful worked example for (n,r)=(2,2). The central caveat is that the main theorem is conditional on a bi-Lipschitz constant imported from an unpublished preprint, as detailed below.
major comments (2)
- [Section 3.2, Eq. (2) and Lemma 3.3] The load-bearing inequality in the proof of Theorem 3.5 is the discriminant lower bound D_σ(Φ(M)) ≥ c'_{n,r} d∞([M],[R_σ(s(Φ(M)))]), which is derived from the bi-Lipschitz estimate ∥Φ(M)-Φ(M')∥_2 ≥ c_{n,r} d([M],[M']) cited as [2, Corollary 3.8]. The paper gives no proof of this inequality and no quantitative value for c_{n,r}; Remark 3.10 explicitly concedes that the existence of the constant in [2] is qualitative. The lower-bound half of the proof of Theorem 3.5, namely g(w) ≥ f(M) for w=Φ(M), relies on this positive constant for every coupling: without c_{n,r}>0, no finite κ_f can be guaranteed to dominate the Lipschitz loss L_f, and formula (3) may fail. Since this is the single external input on which the main theorem rests, the central claim is conditional on [2] being correct and, in particular, on its Corollary 3.8. Please either prove a positive lower bound for c_{n,r} in this paper, give an explicit bound, or state the main theorem as conditional on that preprint.
- [Section 3.3, Proposition 3.12] The doubling-loop algorithm is presented as an alternative that does not require knowledge of c_{n,r}. This does not circumvent the dependence: the proof of Proposition 3.12 shows termination by invoking Theorem 3.5 ('By Theorem 3.5 the test passes as soon as κ_f ≥ L_f/c'_{n,r}'), and Theorem 3.5 itself uses inequality (2). If c_{n,r} were zero, or if the proof of [2, Corollary 3.8] contained a gap, the loop would have no guaranteed halting time. Thus the algorithm's correctness and termination inherit the same external assumption, and it cannot serve as an independent effectivity argument unless termination is proved using a separately established lower bound.
minor comments (4)
- [Section 3.1, Lemma 3.1] The notation s(w)_{iα} := w_{i eα} - w_{(i-1)eα} uses the symbol w_0, which is not a coordinate of w; please define the convention w_{0 eα} := 0 explicitly, both here and in the algorithm of Section 3.3 where σ_α(i)-1 can be zero.
- [Section 2.2] The paper calls the structure a semifield while also using subtraction (⊘=−) as a total operation on functions. Since a semifield is normally understood without additive inverses, please add a clarifying sentence that 'tropical rational functions' are identified as functions and that the algebraic structure considered is the set of such functions under pointwise max, plus, and subtraction.
- [Corollary 3.9(ii)] The proof asserts that each member of F is a max filter but does not give the template construction. Since the claim is correct, please include one sentence explaining that b_c is the max filter whose template has blocks of 1s in disjoint template rows: rows 1 through c_1 in column 1, rows c_1+1 through c_1+c_2 in column 2, and so on.
- [References] Reference [2] is a submitted preprint and [3] is an arXiv preprint; please update both if they are accepted during revision, and state clearly in the text which results of the paper depend on each of them.
Circularity Check
No circular derivation: Theorem 3.5 is an explicit construction, with the only external input being the same-author bi-Lipschitz bound (2); the missing quantitative support is a verification gap, not a circular reduction.
full rationale
The generation theorem is proved by construction, not by assuming its conclusion. For an invariant tropical rational f, equation (3) defines g(w) = min_σ ( f(R_σ(s(w))) + κ_f D_σ(w) ). The proof establishes g∘Φ = f using Lemma 3.1 (sorted columns are differences of single-column basic values), Lemma 3.2 (some coupling re-assembles [M]), Lemma 3.3 (the recoupling discriminant is bounded below by the quotient distance via the bi-Lipschitz inequality (2)), and Lemma 3.4 (an invariant Lipschitz bound). The occurrence of f inside g is a substitution into the tropical-rational coordinate map w ↦ R_σ(s(w)), not a prior assumption that f lies in the generated semifield. The only load-bearing imported step is (2), quoted as [2, Cor. 3.8]; Remark 3.10 explicitly concedes that the existence of c_{n,r} is qualitative, and Section 3.3's doubling loop still relies on Theorem 3.5 for termination. This is a genuine missing-support/verification gap and a same-author citation, but it is not circular: [2] is a separate bi-Lipschitz embedding statement, not the generation theorem, and its assumptions do not contain the target result. The remaining dependencies ([3], [4], [9]) are external. Accordingly the paper has no circular step, but the central argument is conditional on an unquantified and not independently reproved constant, so the score reflects that concern rather than a circular equivalence.
Assumptions & free parameters
free parameters (1)
- kappa_f =
unspecified; minimal doubling-loop value
assumptions (5)
- domain assumption Phi is a bi-Lipschitz embedding with positive constant c_{n,r} ([2, Cor. 3.8]).
- standard math Tropical rational functions are exactly continuous piecewise linear functions with finitely many affine pieces of integer slope.
- domain assumption For any permutation group G, the separating family of [3, Theorem 1.3] has degree at most max{N, binom(N,2)} and is bi-Lipschitz.
- domain assumption Generic max filter banks with 2N templates separate G-orbits and are bi-Lipschitz ([4, Cor. 13], [9, Cor. 1.5]).
- standard math Semialgebraic dimension and invariance of domain facts.
Cite this review
Pith. "Pith review of The basic tropical polynomials generate the semifield of $r$-symmetric tropical rational functions." pith.science (2026). https://pith.science/paper/K76U3ZRQ
@misc{pith2026260806857,
author = {Pith},
title = {Pith review of: The basic tropical polynomials generate the semifield of $r$-symmetric tropical rational functions},
year = {2026},
howpublished = {\url{https://pith.science/paper/K76U3ZRQ}},
note = {Machine review of arXiv:2608.06857}
}
abstract
Let the symmetric group $S_n$ act on the space of $n \times r$ real matrices by permuting rows, so orbits are multisets of $n$ points in $\mathbb{R}^r$. The basic $r$-symmetric tropical polynomials form a family of $\binom{n+r}{r}-1$ nonconstant invariants of degree at most $n$ that separates orbits and embeds the orbit space bi-Lipschitzly. We prove that this family generates the semifield of all $r$-symmetric tropical rational functions, answering a question raised in [J. Pure Appl. Algebra 223 (2019) 72-85]. Derksen showed that the invariant semifield of any permutation group $G \le S_N$ is generated in degree at most $N p_1 \cdots p_{|G|}$ ($p_i$ the $i$th prime), which for the row action is $nr p_1 \cdots p_{n!}$; the present result replaces this by generators of degree at most $n$. The generating expression is a finite minimum over the ways of re-assembling a multiset from its sorted columns, with penalties from the basic values that, via the bi-Lipschitz inequality, dominate a wrong re-assembly. The same penalties describe the image of the basic coordinate map as the zero set of a single tropical rational function and yield an expression algorithm. Subfamilies of the basic family containing the single-column values generate if and only if they separate. For any permutation group $G \le S_N$ the same mechanism generates the invariant semifield in degree at most $\max\{N, \binom{N}{2}\}$, a quadratic bound independent of the group order; combined with a genericity theorem of Cahill, Iverson, Mixon, and Packer, it yields $2N+1$ invariant tropical polynomials that separate orbits and $3N$ that generate, with at least $N$ necessary for each task. The quadratic bound is optimal: every $A_N$-invariant tropical polynomial of degree less than $\binom{N}{2}$ is $S_N$-invariant, so every separating family for the alternating group $A_N$ contains a member of degree at least $\binom{N}{2}$.
Reference graph
Works this paper leans on
-
[2]
Stable complete coordinates for multisets of points via basic $r$-symmetric tropical polynomials
S. Kubo,Stable complete coordinates for multisets of points via basicr-symmetric tropical polynomials, arXiv:2606.30184 (2026). Submitted
work page Pith review arXiv 2026
-
[3]
Derksen,Tropical invariants for permutation group actions, arXiv:2512.13452 (2025)
H. Derksen,Tropical invariants for permutation group actions, arXiv:2512.13452 (2025)
- [4]
-
[9]
R. Balan and E. Tsoukanis,G-invariant representations using coorbits: bi-Lipschitz proper- ties, arXiv:2308.11784 (2023)
arXiv 2023
-
[1]
Kubo,Basicr-symmetric tropical polynomials, J
S. Kubo,Basicr-symmetric tropical polynomials, J. Pure Appl. Algebra 223 (2019), 72–85
work page 2019
-
[5]
Rydh,A minimal set of generators for the ring of multisymmetric functions, Ann
D. Rydh,A minimal set of generators for the ring of multisymmetric functions, Ann. Inst. Fourier 57 (2007), no. 6, 1741–1769
work page 2007
-
[6]
A. Lopatin and F. Reimers,Separating invariants for multisymmetric polynomials, Proc. Amer. Math. Soc. 149 (2021), 497–508
work page 2021
-
[7]
Weyl,The Classical Groups, Princeton University Press, Princeton, 1939
H. Weyl,The Classical Groups, Princeton University Press, Princeton, 1939
work page 1939
Show all 11 references
-
[8]
Ovchinnikov,Max-min representation of piecewise linear functions, Beitr¨ age Algebra Geom
S. Ovchinnikov,Max-min representation of piecewise linear functions, Beitr¨ age Algebra Geom. 43 (2002), 297–302
2002
-
[10]
Vaccarino,The ring of multisymmetric functions, Ann
F. Vaccarino,The ring of multisymmetric functions, Ann. Inst. Fourier 55 (2005), no. 3, 717–731
2005
-
[11]
Coste,An Introduction to Semialgebraic Geometry, lecture notes, Univ
M. Coste,An Introduction to Semialgebraic Geometry, lecture notes, Univ. Rennes, 2002. 13
2002
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