Salem numbers less than 49/37
Jean-Marc Sac-Épée
TL;DR
The paper addresses extending the catalog of Salem numbers below $\eta=49/37$ by transforming the search into a half-degree problem and solving a series of integer linear feasibility problems via random root-separator sampling. It reduces the Salem polynomial search to finding monic polynomials $q(x)$ with all real roots, where $d-1$ lie in $(-2,2)$ and one root lies in $(2,\eta+1/\eta)$, and enforces linear sign-alternation constraints on evaluations at chosen separators. The authors implement this approach in Julia, solve ILPs with Gurobi, and reconstruct Salem polynomials from the feasible $q$—recovering all known numbers below $1.3$ and discovering 10 new numbers in $(1.3,\eta)$ with degrees $26$–$44$, while discussing evidence for list completeness beyond certifiable degrees. The work demonstrates the viability of ILP-based strategies for Salem-number discovery and provides explicit new examples, contributing to the understanding of Salem-number distribution near the plastic constant and informing completeness questions for existing catalogs.
Abstract
A certain number of lists of small Salem numbers, some of which are certified as complete, are available online. Notably, the website of M. J. Mossinghoff features a list of 47 Salem numbers smaller than 1.3, as well as complete lists of Salem numbers of fixed degrees that are smaller than various bounds. The objective of this work is to advance the understanding of Salem numbers by providing a list of Salem numbers smaller than a threshold of 49/37 (≈ 1.324324) through the implementation of a method based on integer linear programming and uniform sampling of root separators. Beyond the intrinsic interest of the newly detected Salem numbers, the rediscovery of already known Salem numbers through this alternative method could offer valuable insights into the potential completeness of already existing lists for degrees where it has not yet been proven.