Representation of powers by polynomials over function fields and a problem of Logic

TL;DR

This paper generalizes B"uchi's problem to function fields for any exponent, providing new insights into undecidability and offering the first solution for rings of functions with exponents greater than 3.

Contribution

It presents the first solution to B"uchi's problem for rings of functions at exponents larger than 3, advancing understanding in logic and function field arithmetic.

Findings

Solved a generalized B"uchi's problem for all exponents in function fields.

Discussed implications for undecidability in logic.

Provided the first example of such a solution for exponents > 3.

Abstract

We solve a generalization of B\"uchi's problem in any exponent for function fields, and briefly discuss some consequences on undecidability. This provides the first example where this problem is solved for rings of functions in the case of an exponent larger than 3.