An extension of Büchi’s problem for polynomial rings in zero characteristic

Abstract:

We prove a strong form of the “$n$ Squares Problem” over polynomial rings with characteristic zero constant field. In particular we prove : for all $r\ge 2$ there exists an integer $M=M(r)$ depending only on $r$ such that, if $z_1,z_2,...,z_M$ are $M$ distinct elements of $F$ and we have polynomials $f,g,x_1,x_2,\dots ,x_M\in F[t]$, with some $x_i$ non-constant, satisfiying the equations $x_i^r=(z_i+f)^r+g$ for each $i$, then $g$ is the zero polynomial.