On The Algorithm Of Totally Positive Algebraic Integer With Small Absolute Trace

Suppose α is a totally positive algebraic integer of degree d,that is to say,its conjugate roots α1=α,α,…αd are all positive real numbers,while its minimal polynomial is P(x)= xd + b1xd+…*+ bd-1x + bd.∑i=1d αi is the trace of α,noted by trace(α),and trace(α)/d is called the absolute trace of a.For trace(α)/d,there is a famous“Schur-Siegel-Smyth trace problem"(so called by P.Borwein[22]):Fix p<2.Then show that all but finitely many totally positive algebraic integers α have trace(α)/d>p.Finding all the totally positive algebraic integers with small absolute trace is a significant work for researching the above trace problem.In our research,we introduce a type of auxiliary functions combined with Chebyshev polynomials,in order to improve the bounds of Sk,where .With this method,we get better bounds of coefficients of P(x),and reduce computing time in finding all the totally positive algebraic integers with small absolute trace.According to the calculation results with the above method,we prove that there does not exist totally positive algebraic integer of degree d = 15 with absolute trace less than or equal to 1.8.And we improve the value of p to 1.792818 in the“Schur-Siegel-Smyth trace problem”.