Polynomial representatives of finite-field maps: a sharp dimensional dichotomy

Abstract

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.