The Measure Of Totally Positive Algebraic Integer

Let a be a algebraic integer of degree d and P(x)be its minimal polynomial α1=α,α2,…,αd be its all conjugates.If all conjugates of a are totally pos-itive,we call α is a totally positive algebraic integer.If its minimal polynomial is reciprocal,i.e.P(x)= P(1/x)xd,then the algebraic integer a is reciprocal.The Mahler measure of algebraic integer a is the product of all conjugates of αthat have modulus at least 1,noted by M(α),M(α)1/d is called the absolute Mahler measure,noted by Ω(α).The length of algebraic integer a is the sum of the absolute value of all the cofficients of its minimal polynomial,noted by L(α).L(α)1/d is called the absolute length,noted by L(α).Our work is to discuss the lower bound of the Mahler measure of totally positive algebraic integer and the lower bound of the absolute length of totally positive reciprocal algebraic integer.We prove that all except finitely many positive algebraic integers α have Ω(α)≥ 1.722396 …,The result improve Flammang’s results of the lower bound of the Mahler measure of totally positive algebraic integer and we obtain a new exceptionnal element.We also prove that all except finitely many positive reciprocal algebraic integers a have L(α)>2.365855….Finally,we obtain two upper bounds of integer transfinite diameter on the real intervals.