【作者】 张卓；

【导师】 吴强；

【作者基本信息】 西南大学 ， 计算数学， 2015， 硕士

【摘要】 设α为d次代数整数,其极小多项式为其中b0=1.bi∈Z,α1=α.α2,…αd为α的所有共轭根.我们将α的所有共轭根模的最大值记作同,并将其形象的称为代数整数α的房子.即若P(x)是互反的,即满足P(x)=P(1/x)xd则称α是互反代数整数.关于代数整数的最小房子问题,很多人对其进行了研究.1985年Boyd [5]结合牛顿公式计算出了次数为d(d≤12)的代数整数的最小房子以及次数为d(d≤16)的互反代数整数的最小房子.2007年Rhin,Wu[24]沿用Boyd的思路.并结合辅助函数.整超限直径.LLL算法以及半无限线性规划算法等理论和算法将代数整数的最小房子计算到了28次.2010年Fang.Li,Wu[14]在Rhin,Wu算法的基础上.对互反代数整数的最小房子进行了讨论.得到了次数为d(d≤26)的最小房子.同时,计算出了次数为d(28≤d≤40)且高度为1的互反代数整数的最小房子本文通过构造新的辅助函数.进一步改善Sk的界,并结合改进后的算法,得到了次数为d((d≤42)的互反代数整数的最小房子.更多还原

【Abstract】 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).更多还原