Salem numbers with minimal trace

In this paper, a new method to compute lower and upper bounds for Salem numbers with a given trace and a given degree is given. With this method, it is proven that the smallest trace of Salem numbers of degree 22 22 is − 1 -1 . Further, new lower bounds for degree of Salem numbers with minimal trace − 5 -5 and − 6 -6 are given. All Salem numbers of trace − 2 -2 and degree 24 24 , 26 26 are given. This includes 7 7 additional Salem numbers of degree 26 26 beyond what was previously known. The auxiliary functions related to Chebyshev polynomials, which are adapted to Salem number, are used in this work.