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Replacement dynamics of binary quadratic forms

Raghav Bhutani, Frederick Saia

Abstract

For an $S$-valued function $f$ of $m \geq 1$ variables we consider the dynamical process in which the output $f(\overline{v})$ replaces exactly one entry of the input $\overline{v} \in S^m$ at each step. This can be viewed as a special case of multivariate polynomial semigroup dynamics, and our study focuses on periodic vectors with respect to this process. We define a stratification of periodic vectors according to their type, and characterize types for which the determination of periodic vectors comes down to dynamics of univariate polynomials. We then restrict to the case of a diagonal binary quadratic form $f$ over $\mathbb{Q}$, and classify rational periodic vectors for all types of period up to $5$. This includes two types, of periods $4$ and $5$, which do not arise from the univariate case, and we prove that there are no periodic vectors over the rationals of the single non-univariate type of period $4$.

Replacement dynamics of binary quadratic forms

Abstract

For an -valued function of variables we consider the dynamical process in which the output replaces exactly one entry of the input at each step. This can be viewed as a special case of multivariate polynomial semigroup dynamics, and our study focuses on periodic vectors with respect to this process. We define a stratification of periodic vectors according to their type, and characterize types for which the determination of periodic vectors comes down to dynamics of univariate polynomials. We then restrict to the case of a diagonal binary quadratic form over , and classify rational periodic vectors for all types of period up to . This includes two types, of periods and , which do not arise from the univariate case, and we prove that there are no periodic vectors over the rationals of the single non-univariate type of period .

Paper Structure

This paper contains 13 sections, 15 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Motivation: the univariate case
  3. Replacement dynamics
  4. Binary quadratic forms and main results
  5. Period type classes
  6. The binary case
  7. Replacement Dynamics for Binary quadratic forms
  8. Dynamic modular polynomials
  9. Period 1: self-loops
  10. Period 2
  11. Period 3
  12. Period 4
  13. Period 5

Key Result

Theorem 1.7

Let $f(x,y) = Cx^2+Dy^2$ be a fixed binary quadratic form over $\mathbb{Q}$ with $C, D \neq 0$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (40)

  • Conjecture 1.1: Poonen
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1
  • proof
  • ...and 30 more