The $p$-Adic Valuation Trees for Quadratic Polynomials for Odd Primes

TL;DR

This paper investigates the structure of $p$-adic valuation sequences of quadratic polynomials at odd primes using tree representations, linking polynomial coefficients to the trees' properties and sequence behavior.

Contribution

It introduces a method to determine whether the valuation trees are finite or infinite based on polynomial coefficients, and characterizes their structure and sequence properties.

Findings

Finite trees correspond to periodic valuation sequences.

Infinite trees indicate unbounded valuation sequences.

Polynomial coefficients determine the tree structure and sequence behavior.

Abstract

We examine the behavior of the sequences of $p$-adic valuations of quadratic polynomials with integer coefficients for an odd prime $p$ through tree representations. Under this representation, a finite tree corresponds to a periodic sequence, and an infinite tree corresponds to an unbounded sequence. We use the polynomial coefficients to determine whether the $p$-adic valuation trees are finite or infinite, the number of infinite branches, the number of levels, the valuations at terminating nodes, and their relationship to the corresponding sequences.