Reduced-representative variant

Determine whether the coordinates of the unique q-reduced polynomial representative of every permutation of the finite vector space k^n are algebraically independent over k, for dimensions n≥3 other than the established case (q,n)=(2,3).

Background

The paper disproves the Maubach–Willems conjecture for arbitrary polynomial representatives in dimensions at least three by constructing dependent representatives of finite-set maps, while proving that every representative of a permutation is algebraically independent in dimensions one and two. This motivates restricting attention to the canonical q-reduced representative, which is uniquely determined by the permutation through reduction modulo the finite-grid ideal.

The authors settle the smallest three-dimensional case over the field with two elements by exhaustive computation: every 2-reduced representative of a permutation of F_23 has algebraically independent coordinates. For general q and n≥3, however, the paper does not determine whether reduction can produce a permutation representative whose coordinate kernel is a positive-height prime contained in (U_1q-U_1,...,U_nq-U_n).

References

The answer is affirmative for n=1,2, because Theorems~\ref{thm:dimension-one} and~\ref{thm:dimension-two} apply to every representative, and Proposition~\ref{prop:reduced-f2} gives an affirmative answer for $(q,n)=(2,3)$. Apart from this case, the present paper leaves the question open for $n\geq3$.

— Polynomial representatives of finite-field maps: a sharp dimensional dichotomy  (2608.24612 - Barańczuk et al., 25 Aug 2026) in Open Problem 'Reduced-representative variant,' Section 6.2 ('Remaining problems'), specifically Proposition 6.1 and the paragraph following it

Corollary~\ref{cor:degree-lower} gives

d_{q,n}\geq\left\lceil q{1/(n-1)}\right\rceil.

Proposition~\ref{prop:compact-f2} below gives $d_{2,3}\leq14$, but no minimum is determined here.

— Polynomial representatives of finite-field maps: a sharp dimensional dichotomy  (2608.24612 - Barańczuk et al., 25 Aug 2026) in Open Problem 'Minimum degree,' Section 6.2 ('Remaining problems'), following Corollary 5.4