Algebraic Integers With Smallest House And The Smallest Perron Numbers

Let ? be an algebraic integer of degree d.The house of ? is the largest modules of its conjugates,i.e.(?).For the smallest house of algebraic integers of degree d,saying m(d),there is a famous conjecture of Schinzel-Zassenhaus:m(d)?1+c1/d,where c1 is a positive constant.Let ?p be an algebraic integer of degree d.A Perron number is a real algebraic integer ?p of degree d?2,whose conjugates are ?Pi,such that(?).For the smallest Perron numbers,there is a famous conjecture of Lind-Boyd:The smallest Perron number of degree d?2 has minimal polynomial#12These two conjectures are classical problems in number theory,and the studies are related with the search of the smallest house of algebraic integer and the smallest Perron number.So far,the smallest house of algebraic integers of degree d?28 and the smallest Perron numbers of degree d ?27 has been founded.Moreover,the constant c1 in Schinzel-Zassenhaus conjecture has been improved and the Lind-Boyd conjecture is satisfied for d?27.In our work,we establish a new auxiliary function using quasi Chebyshev poly-nomial for computing the upper and lower bounds of Sk.As consequence,we compute the smallest house of algebraic integers of degree 29?d?32,and the smallest Perron numbers of degree 28?d?31.We study the behavior of smallest house of algebraic integer based on the numerical results.