The Sum Of The K-power Of Conjugates Of Totally Positive Algebraic Integers

Let ? be a algebraic integer of degree d,with minimal polynomial P(x)=xd+ad-1xd-1+…+a1x+a0,and ?1=?,?2,…,?d be its all conjugates.If the all conjugates of ? are totally positive,we call ? be a totally positive algebraic integer of degree d.We denote(?).When k=1 the S1 is the trace of ?,and S1/d is the absolute trace of?.For the absolute trace of a totally positive algebraic integer,there is a famous“Schur-Siegel-S myth trace problem":Fix p<2.Then show that all but finitely many totally positive algebraic integers ? have S1/d>?.In this work,we study the lower bound of Sk/d with the integer transfinite diameter and the auxiliary functions.We improve the lower bound of S2/d and S3/d,and get the lower bound of Sk/d for 4 ?k?12.Then we obtain an estimation formula of the lower bound of Sk/d for 2 ?k?9.We conjecture that this estimation formula is also valued for all the lower bound of Sk/d for k?2.