Integral divisibility points for multivariate polynomials

Determine, for two polynomials p,q in d4a4[x_1,d4a4,x_n], when there exist integral points (x_1,d4a4,x_n) in d4a4^n such that p(x_1,d4a4,x_n) divides q(x_1,d4a4,x_n), thereby addressing the general multivariable polynomial-divisibility problem.

Background

The paper recasts positive integral Dynkin friezes as positive integral points on affine varieties defined by systems of polynomial equations. Solving the resulting equations requires determining when certain rational functions take integral values, which leads to divisibility conditions between polynomial evaluations.

The authors emphasize that even the underlying arithmetic question over the integers is unresolved in general. They note that finiteness criteria are known in one and two variables through results of Siegel and Corvaja–Zannier, but these results do not settle the general multivariable setting relevant to the Diophantine study of frieze varieties.

References

Even over the integers, the question of determining when there are integral points (x_1, \ldots, x_n) \in \mathbb{Z}n for two polynomials p, q \in \mathbb{Z}[x_1, \ldots, x_n] such that p(x_1, \ldots, x_n) divides q(x_1, \ldots, x_n) is an open problem.

— A positive Siegel theorem: Dynkin friezes and positive Mordell-Schinzel  (2503.08800 - Zhang, 11 Mar 2025) in Section 2.2, subsection “Frieze polynomials and affine varieties”