Davenport's Problem With A Difference

Consider two polynomials with rational coefficients which which have the same ranges modulo p for all but finitely many primes p. Davenport asked whether such polynomials must be tied by a linear change of variables. There is a large body of work on this problem, most notably by Fried and Muller, showing that, under additional hypothesis, the answer is affirmative. In general, the answer is far from straightforward.

The goal of this project is to use the recent work of Dr Tomasic in difference algebra and geometry to extend these considerations to the study of difference polynomials over algebraic closures of finite fields equipped with powers of the Frobenius automorphism.