Existence of orthogonal companions for bivariate local permutation polynomials

Determine whether every bivariate local permutation polynomial over a finite field admits another bivariate local permutation polynomial that forms an orthogonal system with it.

Background

A bivariate local permutation polynomial is a polynomial over a finite field whose specialization in either variable is a permutation polynomial for every fixed value of the other variable. Two such polynomials are companions when the corresponding pair of equations has a unique solution for every prescribed pair of field elements, equivalently when the associated Latin squares are orthogonal.

The paper introduces permutation group polynomials and constructs particular families with explicit companions, but it does not resolve the general existence question for an arbitrary bivariate local permutation polynomial. The unresolved issue is therefore whether the companion property exists universally or only for specially structured families.

References

For a given bivariate local permutation polynomial f1(X1, X2), it is uncertain that whether there exists another local permutation polynomial f2 such that f1 and f2 form an orthogonal system. This uncertainty foster an interesting research question to investigate the existence of such pairs.

— Enumeration of certain permutation group polynomials  (2608.28118 - Hasan et al., 28 Aug 2026) in Section 1, page 2