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Classifying minimal vanishing sums of roots of unity

Louis Christie, Kenneth J. Dykema, Igor Klep

TL;DR

This work advances the classification of minimal vanishing sums of roots of unity by completing a hand-derived catalog for weights up to 16, revealing new phenomena beyond prior classifications. It formalizes a type-theoretic framework, analyzes parities and heights, and proves a comprehensive list of 76 types with height 1 for weight ≤16. To push beyond, the authors develop a TypeGen-based algorithm that explores weights up to 21, producing conjectural classifications and parity data, while highlighting height-2 examples at weight 21 and the broader landscape of vanishing sums. The results, supported by computational methods and accompanied by an open-source codebase, provide a roadmap for extending the classification and understanding the structure of vanishing sums in cyclotomic settings.

Abstract

A vanishing sum of roots of unity is called minimal if no proper, nonempty sub-sum of it vanishes. This paper classifies all minimal vanishing sums of roots of unity of weight at most 16 by hand, thereby uncovering new phenomena beyond the earlier 1998 classification of Poonen and Rubinstein (SIAM J. Discrete Math.) that went up to weight 12. The paper also develops an algorithm to explore higher weights up to 21, yielding a conjectural extension of the classification.

Classifying minimal vanishing sums of roots of unity

TL;DR

This work advances the classification of minimal vanishing sums of roots of unity by completing a hand-derived catalog for weights up to 16, revealing new phenomena beyond prior classifications. It formalizes a type-theoretic framework, analyzes parities and heights, and proves a comprehensive list of 76 types with height 1 for weight ≤16. To push beyond, the authors develop a TypeGen-based algorithm that explores weights up to 21, producing conjectural classifications and parity data, while highlighting height-2 examples at weight 21 and the broader landscape of vanishing sums. The results, supported by computational methods and accompanied by an open-source codebase, provide a roadmap for extending the classification and understanding the structure of vanishing sums in cyclotomic settings.

Abstract

A vanishing sum of roots of unity is called minimal if no proper, nonempty sub-sum of it vanishes. This paper classifies all minimal vanishing sums of roots of unity of weight at most 16 by hand, thereby uncovering new phenomena beyond the earlier 1998 classification of Poonen and Rubinstein (SIAM J. Discrete Math.) that went up to weight 12. The paper also develops an algorithm to explore higher weights up to 21, yielding a conjectural extension of the classification.

Paper Structure

This paper contains 10 sections, 5 theorems, 37 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions and reader's guide
  3. Notation and Preliminaries
  4. Types and parities
  5. A few examples
  6. Classification of sorou of weight $\leq16$
  7. Towards a classification of vanishing sorou of weight $>16$
  8. Heights and equisigned sum of roots of unity
  9. Computational results and conjectures.
  10. Conjectured list of all minimal vanishing types of length $\leq 21$

Key Result

Lemma 2.1

The relative order of a minimal vanishing sorou $h$ is a product $p_1p_2\cdots p_s$ of distinct primes $p_1<p_2<\cdots<p_s$.

Theorems & Definitions (15)

  • Lemma 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Remark 2.5
  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • ...and 5 more