The Property Of Totally Positive Algebraic Integers And The Integer Transfinite Diameter

Let α be an algebraic integer of degree d with minimal polynomial P（x）,α₁,…,α_d are its conjugates.If all the conjugates of α are real,then α is called totally real.If all its conjugates are positive,then α is called totally positive.The absolute trace of α,is defined by the mean value of all its conjugates.Let T denotes the set of absolute traces of totally positive algebraic integers.The famous Schur-Siegel-Smyth trace problem is about finding the smallest limit point L of T.It has existed for more than 100 years and is still unsolved.To study this trace problem,in Chapter 3,we try to use an exhaustive research method to find all totally positive algebraic integers with small absolute trace and given degree.Combined with Chebyshev polynomials,we construct a new type of explicit auxiliary functions to give better bounds for the coefficients of P（x）,and sharply reduce the computing time.Then,we push the computation to degree 15,and prove that there does not exist such totally positive algebraic integers.As an application,we improve the lower bound for the Schur-SiegelSmyth trace problem to 1.793145….The absolute S_k-measure of α is defined by the mean value of the kth power ofα_i（i=1,…,d）.Let T_k denotes the set of absolute S_k-measures of totally positive algebraic integers.For each k>0,the smallest limit point L_k of T_k is unknown.This problem is a generalization of the Schur-Siegel-Smyth trace problem.In Chapter 4,we compute the lower bounds v_k of L_k for each integer in the range 2≤k≤15 with the method of explicit auxiliary functions.Then we give a function to approximatively express the values of v_k for integers 2≤k≤15,and to predict the behaviour of v_k as k tends to infinity.Moreover,we give results for lower bounds of L_k for all real numbers k>2.The house（foundation）of α is defined by the maximal（minimal）modulus of α_i（i=1,…,d）.The problem of investigating the distribution of conjugates of algebraic integers involves important measures such as houses,foundations and absolute traces.For instance,Schinzel-Zassenhaus Conjecture is about studying the lower bound of the houses of algebraic integers.In Chapter 5,for the first time we give lower bounds for the houses of totally positive algebraic integers on the condition of bounded absolute traces.We also improve the upper bounds for the foundations of totally positive algebraic integers with bounded absolute traces.As an application,for some certain values of σ（0<σ<1）,we give the lower and upper bounds for tr（α）,where α is a totally positive algebraic integer with conjugates lying in（σ,4+σ-ε）.Explicit auxiliary functions,introduced into number theory by Smyth,play an important role in the study of the above problems.Wu connected it to integer transfinite diameters.The integer Chebyshev problem related to integer transfinite diameter is also an old problem in number theory,it is to find a polynomial q_n（x）∈Z[x],with degree less than or equal to n,of minimal supremum norm on unit interval[0,1],and to analyze the behaviour of its supremum norm as n tends to infinity.The integer Chebyshev problem on[0,1]originates from an elementary proof of the Prime Number Theorem.In fact,it is unsolved for any intervals with length less than 4.In Chapter 6,we focus on the Farey intervals and[0,（（√n-√m））²],on which the integer transfinite diameters are related to Schur-Siegel-Smyth trace problem and irrationality approximation of logarithms of rational numbers.For these intervals,we give new results on the upper bounds for integer transfinite diameters,and analyse the characteristics of the integer Chebyshev polynomials.In particular,we improve the upper bound to 0.422678 for the integer transfinite diameter on[0,1].As for the research methods,we introduce a new approach in Chapter 3 for the exhaustive research for algebraic integers with certain characteristics.Additionally,the new results for the problems in Chapter 4,5 and 6 are contributed from the improvement in the Wu’s algorithm and analysis of the related algebraic integers.Furthermore,the new results in Chapter 6 also benefit from our consideration of the relationship between algebraic integers with small trace and the integer transfinite diameters on Farey intervals and on[0,（（√n-√m））²].