Optimal bounds for Büchi's problem in modular arithmetic II

Abstract

Given a prime $p\ge 5$ and an integer $s\ge 1$, we show that there exists an integer $M$ such that for any quadratic polynomial $f$ with coefficients in the ring of integers modulo $p^s$, such that $f$ is not a square, if a sequence $(f(1),\dots ,f(N))$ is a sequence of squares, then $N$ is at most $M$. We obtain this result by reducing to the case where $f$ has an invertible dominant coefficient.

accepted in Canadian Mathematical Bulletin