A Salem number τ of degree 2d is a real algebraic integer greater than 1 whose other conjugates all lie in the closed disc |z|≤1, with at least one on the unit circle. The transformation α= τ+1/τ+2 produces a totally positive algebraic integer α of degree d whose all zeros but one are in the interval (0,4). In this talk, a new method is given to optimize the lower and upper bounds for such totally positive algebraic integer α with given trace and given degree, and then the bounds for sk. Therefore all Salem numbers of degree 32, 40 and 62 with minimal trace −3, −4 and −6 respectively are found. Consequently, the new lower bounds for degrees of Salem numbers with these minimal traces are given. This is a joint work with Qiong Chen.
