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On the Rigidity of the Roots of Power Series with Constrained Coefficients

Jacob Kewarth

TL;DR

This work investigates the rigidity of root sets Σ_S of power series whose coefficients lie in a finite, symmetric, normalized set S. It develops recursive 1-step and 2-step criteria to certify whether a given λ ∈ D is a root, and analyzes how depth into the unit disk and angular position influence membership, including Bandt’s algorithm for visualization. A key contribution is a sharp rigidity theorem: if Σ_S = Σ_T for S,T with identical maxima, then all small positive integers up to { $2\sqrt{\max(S)}+1$ } must coincide between S and T, with a quasi-rigidity principle extending to nearby coefficients; the paper also classifies connectedness for Σ_S under gap and maximum conditions and provides geometric interpretation via spike structures. These results advance understanding of how coefficient choices constrain root geometry and have implications for the structure of Σ_S across coefficient sets.

Abstract

Here we consider the set $Σ_S$ of roots of power series whose coefficients lie in a given set $S$ and how such sets of roots vary as the set $S$ varies. We give an estimate of the depth that complex roots can reach into the disc, offer some criterion for the set of roots to be connected or disconnected, and show that for two finite symmetric sets $S$ and $T$ of integers containing $1$, if $Σ_S = Σ_T$ then all of their elements between $1$ and $2\sqrt{\max(S)}+1$ must agree.

On the Rigidity of the Roots of Power Series with Constrained Coefficients

TL;DR

This work investigates the rigidity of root sets Σ_S of power series whose coefficients lie in a finite, symmetric, normalized set S. It develops recursive 1-step and 2-step criteria to certify whether a given λ ∈ D is a root, and analyzes how depth into the unit disk and angular position influence membership, including Bandt’s algorithm for visualization. A key contribution is a sharp rigidity theorem: if Σ_S = Σ_T for S,T with identical maxima, then all small positive integers up to { } must coincide between S and T, with a quasi-rigidity principle extending to nearby coefficients; the paper also classifies connectedness for Σ_S under gap and maximum conditions and provides geometric interpretation via spike structures. These results advance understanding of how coefficient choices constrain root geometry and have implications for the structure of Σ_S across coefficient sets.

Abstract

Here we consider the set of roots of power series whose coefficients lie in a given set and how such sets of roots vary as the set varies. We give an estimate of the depth that complex roots can reach into the disc, offer some criterion for the set of roots to be connected or disconnected, and show that for two finite symmetric sets and of integers containing , if then all of their elements between and must agree.

Paper Structure

This paper contains 12 sections, 32 theorems, 46 equations, 8 figures.

Table of Contents

  1. Introduction
  2. History and Outline
  3. Definitions and Basic Properties
  4. Recursive Criterions
  5. Complex Roots
  6. Polynomials
  7. How Deep Can The Roots Get?
  8. The Depth of Complex Roots
  9. Depth At Each Angle
  10. Connectedness
  11. Rigidity
  12. Geometric Interpretation for the early rigidity

Key Result

Theorem \ref{weakrigidity}

Let $S$ and $T$ be finite non-empty symmetric normalized sets of integers. If $\Sigma_S = \Sigma_T$ and $k$ is an integer in $(1, 2\sqrt{\max(S)}+1)$ then $k\in S$ if and only if $k \in T$.

Figures (8)

  • Figure 1: $\Sigma_{\{0, \pm 1 \} }$ in the first quadrant
  • Figure 2: $\Sigma_{\{0, \pm 1 , \pm 3\} }$ in the first quadrant. Even though the coefficients skip 2 we have $\Sigma_{\{0, \pm 1 , \pm 3\} } \cap [0, \infty)$ is connected.
  • Figure 3: $\Sigma_{\{0, \pm 1, \pm 4 \} }$ in the first quadrant.
  • Figure 4: The set of roots of power series with coefficients any integer from $-10$ to $10$.
  • Figure 5: In black we see the set of roots of power series with coefficients any integer from $-10$ to $10$ except $\pm 2$. In grey are the roots we get if we also allow $\pm 2$.
  • ...and 3 more figures

Theorems & Definitions (62)

  • Theorem \ref{weakrigidity}
  • Definition 1.1
  • Theorem \ref{annulus}
  • Definition 1.2
  • Remark
  • Theorem \ref{prod ineq}
  • Theorem \ref{rho-out}
  • Theorem \ref{conn-crit}
  • Theorem \ref{quasirigidity}
  • Definition 2.1
  • ...and 52 more