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Real and finite field versions of Chebotarev's theorem

Tarek Emmrich, Stefan Kunis

TL;DR

The paper addresses extending Chebotarev-type nonvanishing-minors results for the Fourier matrix of prime size to real and finite-field settings. The main approach uses Schur polynomials and a cyclotomic/Galois framework to relate minors to evaluations at algebraic numbers and to divisibility by the cyclotomic polynomial $\Phi_p(X)$. For the real version, it proves that the Vandermonde-like matrix with entries $2\cos(2\pi j/p)$ has all minors nonzero, by exploiting the minimal polynomial $P(X)$ of $2\cos(2\pi/p)$ with $P(2)=p$. For finite fields, it proves nonvanishing of minors of the $p\times p$ Fourier matrix over $\mathbb{F}_{q^{p-1}}$ under a sharp order-condition-based bound on $q$, and discusses subsequent work that removes this order condition in some cases. They also outline a partial extension to nonprimitive-order cases and discuss applications to real phase retrieval and Riesz bases of exponentials, highlighting the practical impact for computations over real arithmetic and signal reconstruction.

Abstract

Chebotarev's theorem on roots of unity states that all minors of the Fourier matrix of prime size are non-vanishing. This result has been rediscovered several times and proved via different techniques. We follow the proof of Evans and Isaacs and generalize the original result to a real version and a version over finite fields. For the latter, we are able to remove an order condition between the characteristic of the field and the size of the matrix as well as decrease a sufficient lower bound on the characteristic by Zhang considerably. Direct applications include a specific real phase retrieval problem as well as a recent result for Riesz bases of exponentials.

Real and finite field versions of Chebotarev's theorem

TL;DR

The paper addresses extending Chebotarev-type nonvanishing-minors results for the Fourier matrix of prime size to real and finite-field settings. The main approach uses Schur polynomials and a cyclotomic/Galois framework to relate minors to evaluations at algebraic numbers and to divisibility by the cyclotomic polynomial . For the real version, it proves that the Vandermonde-like matrix with entries has all minors nonzero, by exploiting the minimal polynomial of with . For finite fields, it proves nonvanishing of minors of the Fourier matrix over under a sharp order-condition-based bound on , and discusses subsequent work that removes this order condition in some cases. They also outline a partial extension to nonprimitive-order cases and discuss applications to real phase retrieval and Riesz bases of exponentials, highlighting the practical impact for computations over real arithmetic and signal reconstruction.

Abstract

Chebotarev's theorem on roots of unity states that all minors of the Fourier matrix of prime size are non-vanishing. This result has been rediscovered several times and proved via different techniques. We follow the proof of Evans and Isaacs and generalize the original result to a real version and a version over finite fields. For the latter, we are able to remove an order condition between the characteristic of the field and the size of the matrix as well as decrease a sufficient lower bound on the characteristic by Zhang considerably. Direct applications include a specific real phase retrieval problem as well as a recent result for Riesz bases of exponentials.

Paper Structure

This paper contains 5 sections, 13 theorems, 71 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Real version of Chebotarëv's theorem
  4. Chebotarëv's theorem over finite fields
  5. Characteristic $q$ not primitive modulo $p$

Key Result

Theorem 1

Let $p\in\mathbb{N}$ be prime and $\omega_p=\exp(2\pi i / p)\in\mathbb{C}$ be a $p$-th root of unity, then all minors of the Fourier matrix are nonzero.

Figures (3)

  • Figure 2.1: The Substitutions of the Schur Polynomial in the proof of \ref{['Chebotarev']}.
  • Figure 4.1: The $p$-th cyclotomic polynomial over $\mathbb{Z}$ and $\mathbb{F}_q$, with $\operatorname{ord}_p(q)=p-1$.
  • Figure 4.2: The $p$-th cyclotomic polynomial over $\mathbb{Z}$ and $\mathbb{F}_q$, with $\operatorname{ord}_p(q)=r<p-1$.

Theorems & Definitions (32)

  • Theorem 1: See e.g. EvansStevenhagenfrenkelTao; see \ref{['sec:pre']}
  • Theorem 2: See \ref{['sect:real']}
  • Theorem 3: Informal version; see \ref{['sec:finitefields']}
  • Lemma 4: See MitchellEvans
  • proof : Proof of \ref{['Chebotarev']}
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Theorem 7
  • ...and 22 more