2-adic Valuations of Quadratic Sequences

TL;DR

This paper characterizes the 2-adic valuation sequences of quadratic polynomials, detailing their boundedness, periodicity, and structure using tree and sequence frameworks, providing a comprehensive understanding of their properties.

Contribution

It offers a complete description of 2-adic valuation sequences for quadratic polynomials, including conditions for boundedness and periodicity, based on polynomial coefficients.

Findings

Characterization of bounded and unbounded valuation sequences

Complete description of periodic sequences in the bounded case

Framework using trees and sequences to analyze valuations

Abstract

Properties of 2-adic valuation sequences for general quadratic polynomials with integer coefficients are determined directly from the coefficients. These properties include boundedness or unboundedness, periodicity, and valuations at terminating nodes. We completely describe the periodic sequences in the bounded case. Throughout, we frame results in terms of trees and sequences.