有限域映射的多项式代表：精确的维数二分法
Polynomial representatives of finite-field maps: a sharp dimensional dichotomy
AI总结：
该研究明确有限域映射的多项式代表的维数二分规律，给出代数独立与相关代表的存在条件及构造方法，证明小维度下置换代表的代数独立性。
AI中文摘要：
设$k=\boldsymbol{F}_q$，有限集映射的多项式代表是一组在有理点网格上诱导出该映射的多项式元组。我们证明有限集映射与其代表的几何之间存在精确区分：当$n=1$或$n=2$时，$\boldsymbol{F}_q^n$上任意置换的每个多项式代表都具有代数独立的坐标；当$n\boldsymbol{\text{≥}}3$时，$\boldsymbol{F}_q^n\to\boldsymbol{F}_q^n$的每个集合映射都同时存在代数独立和代数相关的代表，后者可满足$F_2^q-F_2=(F_1^q-F_1)F_3$。更一般地，当$n\boldsymbol{\text{≤}}m$时，$\boldsymbol{F}_q^m\to\boldsymbol{F}_q^n$的每个映射都存在代数独立的代表；当$n\boldsymbol{\text{≥}}3$时，每个此类映射都存在代数相关的代表。相关的依赖构造结合了Artin–Schreier插值定理（通过满足$A^q-A\boldsymbol{|}B^q-B$的多项式$A,B$生成指定值）与三坐标悬挂结构。对于$\boldsymbol{F}_q^3$上的恒等映射，其概象可恰好取为$V^q-V=(U^q-U)W$，这是一个光滑几何整有理曲面。我们还建立了强制代数独立性的低次数与扩域准则，且精确穷举计算证明$\boldsymbol{F}_2^3$上置换的每个2-约化代表都具有代数独立的坐标。
英文摘要：
Let $k=\mathbb{F}_q$. A polynomial representative of a finite-set map is a tuple of polynomials inducing that map on the rational-point grid. We prove a sharp distinction between a finite-set map and the geometry of its representatives. If $n=1$ or $n=2$, every polynomial representative of a permutation of $k^n$ has algebraically independent coordinates. If $n\geq3$, every set map $k^n\to k^n$ has both an algebraically independent and an algebraically dependent representative; the latter may be chosen to satisfy \[ F_2^q-F_2=(F_1^q-F_1)F_3. \] More generally, every map $k^m\to k^n$ has an algebraically independent representative exactly when $n\leq m$, while every such map has a dependent representative when $n\geq3$. The dependent construction combines an Artin--Schreier interpolation theorem, producing prescribed values by polynomials $A,B$ with $A^q-A\mid B^q-B$, with a three-coordinate suspension. For the identity on $k^3$, the scheme-theoretic image may be chosen to be exactly \[ V^q-V=(U^q-U)W, \] a smooth geometrically integral rational surface. We also establish low-degree and extension-field criteria forcing algebraic independence. An exact exhaustive computation additionally proves that every $2$-reduced representative of a permutation of $\mathbb{F}_2^3$ has algebraically independent coordinates.
