基本热带多项式生成r-对称热带有理函数的半域
The basic tropical polynomials generate the semifield of $r$-symmetric tropical rational functions
AI总结：
本文证明基本$r$-对称热带多项式生成所有$r$-对称热带有理函数的半域，将生成元次数界从$nr p_1 \boldsymbol{p}_{n!}$降至$n$，还给出任意置换群不变半域的二次最优生成界。
AI中文摘要：
设对称群$S_n$通过行置换作用于$n×r$实矩阵空间，其轨道为$\boldsymbol{R}^r$中$n$个点的多重集。基本$r$-对称热带多项式构成$\binom{n+r}{r}-1$个次数不超过$n$的非常数不变量，可区分轨道并将轨道空间双李普希茨嵌入。我们证明该族生成所有$r$-对称热带有理函数的半域，解答了《J. Pure Appl. Algebra 223 (2019) 72-85》中提出的问题。Derksen曾指出，任意置换群$G≤S_N$的不变半域由次数不超过$N p_1 \boldsymbol{p}_{|G|}$（$p_i$为第$i$个素数）的元素生成，对行作用而言为$nr p_1 \boldsymbol{p}_{n!}$；本文结果将其替换为次数不超过$n$的生成元。生成表达式是对多重集从其排序列列重新组合的有限最小值，惩罚项来自基本值，通过双李普希茨不等式可抑制错误重组。相同惩罚项将基本坐标映射的像描述为单个热带有理函数的零点集，并给出表达式算法。包含单列值的基本族子族生成当且仅当它们可区分轨道。对任意置换群$G≤S_N$，相同机制在次数不超过$\boldsymbol{max}\boldsymbol{\boldsymbol{N, \binom{N}{2}}}$（与群阶无关的二次界）下生成不变半域；结合Cahill、Iverson、Mixon和Packer的一般性定理，可得到$2N+1$个可区分轨道的不变热带多项式，以及$3N$个生成元，每个任务至少需要$N$个。该二次界是最优的：每个次数小于$\binom{N}{2}$的$A_N$-不变热带多项式都是$S_N$-不变的，因此交错群$A_N$的每个可分族都包含至少一个次数为$\binom{N}{2}$的元素。
英文摘要：
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 $\binom{N}{2}$.
