Totally positive algebraic integers with small trace

The “Schur-Siegel-Smyth trace problem” is a famous open problem that has existed for nearly 100 years. To study this problem with the known methods, we need to find all totally positive algebraic integers with small trace. In this work, on the basis of the classical algorithm, we construct a new type of explicit auxiliary functions related to Chebyshev polynomials to give better bounds for S k S_k , and reduce sharply the computing time. We are then able to push the computation to degree 15 15 and prove that there is no such totally positive algebraic integer with absolute trace 1.8 1.8 . As an application, we improve the lower bound for the absolute trace of totally positive algebraic integers to 1.793145 ⋯ 1.793145\cdots .