发表机构
University of New Hampshire; Siena College（新罕布什尔大学; 锡耶纳学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究无限维序列空间上置换不变全纯函数，证明在链连通域上对称性导致刚性（常值性），并给出边界反例与向量值推论。
AI 中文摘要
我们研究无限维序列空间的置换不变子集上的全纯函数，这些函数在坐标的所有有限置换下保持不变。在基于零序列空间$\czero$的域上，我们证明置换对称性迫使一阶微分沿每个常数尾部消失：坐标导数在该处一致，且一个平均论证消去了它们的公共值。这导出了常数序列域上的刚性——即常值性——更一般地，在微分在有限支撑方向上消失的任何$\cc$-链连通域上亦然。该方法有一个尖锐的边界：一个精确的障碍阻止了基本论证触及非恒定点的有限多个例外坐标，因此一般最终常数域上的无条件刚性仍是开放的。另一方面，一个显式例子——$\ell^\infty$上的任意Banach极限——表明链连通性和有界性假设不能被舍弃：仅拓扑连通性就允许非常数的对称整函数存在。我们记录了向量值推论，并讨论了其对零集映射的全纯截面的动机性应用，指出了仍然开放的提升问题。
英文摘要
We study holomorphic functions on permutation-invariant subsets of an infinite-dimensional sequence space that are invariant under all finite permutations of coordinates. Working on domains modeled on the space $\czero$ of null sequences, we show that permutation symmetry forces the first differential to vanish along every constant tail: the coordinate derivatives agree there and an averaging argument annihilates their common value. This yields rigidity---constancy---on domains of constant sequences, and more generally on any $\cc$-chain-connected domain on which the differential vanishes on the finitely supported directions. The method has a sharp boundary: a precise obstruction prevents the elementary argument from reaching the finitely many exceptional coordinates of a non-constant point, so unconditional rigidity on general eventually-constant domains remains open. In the other direction, an explicit example---any Banach limit on $\ell^\infty$---shows that the chain-connectedness and boundedness hypotheses cannot be dropped: mere topological connectedness admits non-constant symmetric entire functions. We record the vector-valued corollary and discuss the motivating application to holomorphic cross-sections of the zero-set map, isolating the lifting problem that remains open.
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