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Angle duality and a gap principle for convex combinations of incomplete polynomials on the unit circle

Teng Zhang

TL;DR

This work extends two key qualitative results of Ge and Gonek to convex combinations of incomplete polynomials tied to a unit-circle polynomial. By introducing a geometric angle gain mechanism and a robust $oldsymbol{\mathcal{D}}$-region framework, the authors prove an angle duality identity for zeros of $oldsymbol{\mathcal{L}}_oldsymbol{\lambda}$ and establish a quantitative gap principle that bounds zeros away from the unit circle in terms of the largest zero-gap of the original polynomial. The positivity of the convex weights is essential, and the analysis relies on sharp trigonometric inequalities and local boundary geometry rather than coarse convex-hull constraints. The results deepen understanding of how local geometry governs the distribution of zeros and critical-like points for convex combinations of incomplete polynomials on the unit circle, with potential relevance to Dirichlet L-functions, unitary-characteristic polynomials, and statistical-mechanics partition functions.

Abstract

In this paper, we establish an angle duality and a gap principle for convex combinations of incomplete polynomials, extending two results of Ge and Gonek in [IMRN, 2024].Our approach is geometric: we introduce an ``angle gain'' mechanism for points inside the lune region and quantify how moving away from the unit circle forces a definite increase in the relevant angle functional.This yields a robust lower bound that is uniform under convex mixing and leads to the desired separation phenomenon.The main difficulty is that incomplete polynomials and their convex combinations may have highly nonuniform root distributions on the unit circle, so classical convex-hull type constraints are too coarse; one must instead control the local geometry of chords and boundary arcs and relate it to critical-point behavior through sharp trigonometric inequalities.

Angle duality and a gap principle for convex combinations of incomplete polynomials on the unit circle

TL;DR

This work extends two key qualitative results of Ge and Gonek to convex combinations of incomplete polynomials tied to a unit-circle polynomial. By introducing a geometric angle gain mechanism and a robust -region framework, the authors prove an angle duality identity for zeros of and establish a quantitative gap principle that bounds zeros away from the unit circle in terms of the largest zero-gap of the original polynomial. The positivity of the convex weights is essential, and the analysis relies on sharp trigonometric inequalities and local boundary geometry rather than coarse convex-hull constraints. The results deepen understanding of how local geometry governs the distribution of zeros and critical-like points for convex combinations of incomplete polynomials on the unit circle, with potential relevance to Dirichlet L-functions, unitary-characteristic polynomials, and statistical-mechanics partition functions.

Abstract

In this paper, we establish an angle duality and a gap principle for convex combinations of incomplete polynomials, extending two results of Ge and Gonek in [IMRN, 2024].Our approach is geometric: we introduce an ``angle gain'' mechanism for points inside the lune region and quantify how moving away from the unit circle forces a definite increase in the relevant angle functional.This yields a robust lower bound that is uniform under convex mixing and leads to the desired separation phenomenon.The main difficulty is that incomplete polynomials and their convex combinations may have highly nonuniform root distributions on the unit circle, so classical convex-hull type constraints are too coarse; one must instead control the local geometry of chords and boundary arcs and relate it to critical-point behavior through sharp trigonometric inequalities.
Paper Structure (5 sections, 11 theorems, 86 equations, 4 figures)

This paper contains 5 sections, 11 theorems, 86 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:incomplete_dual']}
  4. Two geometric lemmas for the gap principle
  5. Proof of Theorem \ref{['thm:incomplete_gap']}

Key Result

Theorem 1.2

Let $z=e^{i\theta}$ and $z^{+}=e^{i\theta^{+}}$ be two consecutive distinct zeros of $\mathcal{L}(u)$. Then where the critical points $z_k'$ are counted with multiplicity and angles at endpoints are interpreted using Definition def:endpoint when needed.

Figures (4)

  • Figure 1: An illustration of Theorem \ref{['thm:Ge_Gonek']} for $\mathcal{L}(u)=(u-1)(u-i)(u+i)$.
  • Figure 2: Illustration of Theorem \ref{['thm:Ge_Gonek_gap']} for $\mathcal{L}(u)=(u-1)(u-i)(u+i)$.
  • Figure 3: A schematic illustration of the $\mathcal{D}$-region $\mathcal{D}(z,z^{+})=\overline{\mathbb{D}}\cap H(z,z^{+})$.
  • Figure 4: Auxiliary configuration in proof of Lemma \ref{['lem:angle-gain']}.

Theorems & Definitions (23)

  • Definition 1.1: Endpoint convention
  • Theorem 1.2: Ge--Gonek
  • Theorem 1.3: Ge--Gonek
  • Theorem 1.4
  • Remark 1.5: A zero-weight counterexample
  • Theorem 1.6
  • Remark 1.7
  • Example 1.8
  • Definition 2.1: $\mathcal{D}$-region
  • Lemma 2.2: DE08Zha24
  • ...and 13 more