The Smallest Pisot Number Of A Given Degree

A Pisot number is a real algebraic integer greater than 1 whose conjugates lie in the unit disk|z|<1.Based on Boyd’s algorithm,this paper improves the coefficient range of minimal polynomial of Pisot number by using its properties and auxiliary functions.Then we calculated all Pisot numbers of degree 3 to 10 in interval（1,2）and all Pisot numbers of degree 11 to 12 in interval（1.4142,1.6181）.We find that for 5≤d≤12,the minimal polynomial of the smallest Pisot number of degree d is x^d-x^d-1-x^d-2+x²-1.The smallest Pisot number increases with the degree,and tends to (（5）^1/2+1)/2=1.61803···which is the minimum limit point of Pisot number.If 5≤d≤12 is odd,there are three Pisot numbers of degree d less than (（5）^1/2+1)/2 which have the same minimal polynomial form.If 5≤d≤12 is even,there are one Pisot numbers of degree d less than (（5）^1/2+1)/2.With the above results,we give some conjectures on the smallest Pisot number of a given degree.