TL;DR
This paper classifies Salem polynomials of length 5, describes the structure of length 6 Salem polynomials, and provides a comprehensive list of known Salem numbers below the smallest Pisot number.
Contribution
It offers a complete classification for length 5 and nearly complete for length 6 Salem polynomials, including a list of Salem numbers below the smallest Pisot number.
Findings
Complete classification of Salem polynomials of length 5.
Most length 6 Salem polynomials belong to 12 infinite families.
Table of all known Salem numbers below the smallest Pisot number.
Abstract
We give a complete classification of all Salem polynomials of length 5. For length 6 we show that all but finitely many Salem polynomials lie in one of 12 infinite families, and subject to Lehmer's Conjecture we give a complete list of the 126 exceptions. We provide a table of short polynomials for all known Salem numbers below the smallest Pisot number.
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