# Optimal bounds for B\"uchi's problem in modular arithmetic II

## TL;DR

This paper establishes upper bounds on the length of sequences of squares generated by quadratic polynomials over modular rings, extending B"uchi's problem to higher powers of primes.

## Contribution

It provides the first bounds for B"uchi's problem in modular arithmetic for prime powers, generalizing previous results for prime moduli.

## Key findings

- Existence of a uniform bound M for sequences of squares from quadratic polynomials over Z/p^sZ.
- Reduction to polynomials with invertible dominant coefficients simplifies the analysis.
- The bounds depend only on p, s, and the polynomial's properties.

## Abstract

Given a prime $p\ge5$ and an integer $s\ge1$, 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.