On The Smallest Houses Of Reciprocal Algebraic Integers

Let α be an algebraic integer of degrcc d. its conjugates are α1=α.2,…αd, and with b0= 1.bi ∈Z. its minimal polynomial. We denote, as usual, by the house of α. If the minimal polynomial is reciprocal.i.e. P(x)=P(1/x)xd. then the algebraic integer α is reciprocal.The smallest houses of algebraic integer was studied by many people. In 1985, Boyd [1] gavc a algorithm to search for the smallest house of algebraic integers of degree d (d< 12). and reciprocal algebraic integers of degrcc d (d≤16).In 2007. following the thought of Boyd. Rhin. Wu [5] computed the smallest houses of algebraic integers to the degree d=28 by using the theory of auxil-iary function, integer transfinite diameter. LLL algorithm and semi-infinite linear programming algorithm.In 2010, on the basis of Rhin,Wu [24], Fang. Li, Wu [14] computed the smallest houses of all reciprocal algebraic integers of degree d (d≤26). Moreover, they gave the smallest houses of d (28≤d≤40) with the condition of h=1.In this thesis, we improved the existing algorithm [14], especially for the con-struction of auxiliary function, which is used to seek for the boundary of Sk. Finally. we extended this computation of reciprocal algebraic integer of degree d (d≤42).