On The Absolute Trace Of Totally Positive Reciprocal Algebraic Integer

Let a be a algebraic integer of degree d. with minimal polynomial and α1=α,α2,…,αd,its all conjugates. If the all conjugates of n are totally positive and the minimal polynomial of α verify P(x) = P(1/x)xd, we call α be totally positive reciprocal algebraic integer of degree d. The sum of all the conjugates called the trace of α, notes for the tr(α). tr(α)/d is called the absolute trace of α.On the absolute trace problem of algebraic integer, there is a long research history. and many research results are obtained. In this paper. according to the characteristics of totally positive reciprocal algebraic integer, we combined with the theory of the integer transfinite diameter, the auxiliary function. to discuss the lower bound of its absolute trace. We proved that all totally positive reciprocal algebraic integer of degree d have tr(α)/d > ρ,where ρ< 1.8945909…, unless the minimal polynomials of a are x2-3x+1,x4-7x3+13x2-7x+1, x8-15x7+83x6-220x5+ 303x4-220x3+83x2-15x+1.