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There are Salem numbers with trace $-3$ and every degree at least $34$

Giacomo Cherubini, Pavlo Yatsyna

Abstract

We prove that there exist Salem numbers with trace $-3$ and every even degree $\geq 34$. Our proof combines a theoretical approach, which allows us to treat all sufficiently large degrees, with a numerical search for small degrees. Since it is known that there are no Salem numbers of trace $-3$ and degree $\leq 30$, our result is optimal up to possibly the single value $32$, for which it is expected there are no such numbers.

There are Salem numbers with trace $-3$ and every degree at least $34$

Abstract

We prove that there exist Salem numbers with trace and every even degree . Our proof combines a theoretical approach, which allows us to treat all sufficiently large degrees, with a numerical search for small degrees. Since it is known that there are no Salem numbers of trace and degree , our result is optimal up to possibly the single value , for which it is expected there are no such numbers.
Paper Structure (6 sections, 5 theorems, 12 equations, 2 tables)

This paper contains 6 sections, 5 theorems, 12 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Proof
  3. Interlacing polynomials and seven-tuples of primes
  4. Quadruples of primes
  5. Existing small-degree Salem polynomials
  6. Numerical search for missing degrees

Key Result

Theorem 1.1

For each even degree $d\ge 34$ there is Salem number of trace $-3$ and degree $d$.

Theorems & Definitions (7)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof