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Smyth's conjecture and a non-deterministic Hasse principle

Jordan S. Ellenberg, Will Hardt

Abstract

In a 1986 paper, Smyth proposed a conjecture about which integer-linear relations were possible among Galois-conjugate algebraic numbers. We prove this conjecture. The main tools (as Smyth already anticipated) are combinatorial rather than number-theoretic in nature. For instance, the question can be reinterpreted as a question about the possible eigenvalues of a specified linear combination of permutation matrices. What's more, we reinterpret Smyth's conjecture as a local-to-global principle for a "non-deterministic system of equations" where variables are interpreted as compactly supported K-valued random variables (for K a local or global field) rather than as elements of K.

Smyth's conjecture and a non-deterministic Hasse principle

Abstract

In a 1986 paper, Smyth proposed a conjecture about which integer-linear relations were possible among Galois-conjugate algebraic numbers. We prove this conjecture. The main tools (as Smyth already anticipated) are combinatorial rather than number-theoretic in nature. For instance, the question can be reinterpreted as a question about the possible eigenvalues of a specified linear combination of permutation matrices. What's more, we reinterpret Smyth's conjecture as a local-to-global principle for a "non-deterministic system of equations" where variables are interpreted as compactly supported K-valued random variables (for K a local or global field) rather than as elements of K.

Paper Structure

This paper contains 11 sections, 15 theorems, 100 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. Non-deterministic local-to-global principles
  4. Acknowledgments
  5. Non-deterministic solutions over local fields
  6. The case of global function fields
  7. Non-deterministic solutions over non-archimedean local fields
  8. Non-deterministic solutions over global function fields
  9. Approximate local-to-global for non-deterministic solutions over $\mathbb{Q}$
  10. Balanced and approximately balanced weightings on hypergraphs
  11. Proof of Theorem \ref{['th:main']}

Key Result

Theorem 1

Let $a_1, \ldots, a_r$ be a set of elements of $\mathbb{Q}$. Then there exists a probability distribution $(X_1, \ldots, X_r)$ on $\mathbb{Q}^r$, supported on a finite subset of the hyperplane $\sum a_i x_i = 0$, and whose marginals $X_i$ are equal in distribution, if and only if the following local

Figures (1)

  • Figure 1: An approximate solution for $5L_1+6L_2+7L_3=0$

Theorems & Definitions (38)

  • Theorem 1
  • Proposition 2
  • proof
  • Theorem 3
  • proof
  • Remark 4
  • Remark 5
  • Remark 6
  • Example 7
  • Remark 8
  • ...and 28 more