On The Totally Positive Algebraic Integer

Let a be an algebraic integer, if its conjugates are all real numbers, then a is a totally real algebraic integer; if its conjugates are all positive numbers, then a is a totally positive algebraic integer. The research of totally real algebraic integer is an important topic in computational number theory.In this work, we find all totally positive algebraic integers of degree d=9,10, with trace2d-1. Our computation use a method adapted which concerned the Newton formula, integer transfinite diameter, auxiliary function, Chebyshev Poly-nomial and semi-linear programming. These tools are used to give better bounds of the coefficients of the minimal polynomial of totally positive algebraic integers concerned. With the transformation x=z+1/z+2, we get also the salem numbers of degree18and20.