Slicing the stars: counting algebraic numbers, integers, and units by degree and height
Slicing the stars: counting algebraic numbers, integers, and units by degree and height
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DOI：
10.2140/ant.2017.11.1385
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发表时间：
2017-01-01
影响因子：
1.3
通讯作者：
Gunther, Joseph
中科院分区：
文献类型：
--
作者：
Grizzard, Robert;Gunther, Joseph
Masser和Vaaler给出了一个渐近公式的数量代数数的程度d和增加的高度。这个问题通过计算Rd+1中均匀膨胀的星星体中的格点（对应于Z上的最小多项式）来解决。体积的这星星机构计算陈省身和Vaaler，谁也计算了体积的余维一“片”对应的monic多项式，这导致结果Barroero计数代数整数。我们展示了如何估计更高余维切片的体积，这使我们能够计算单位，给定范数的代数整数，迹，范数和迹，等等。我们还改进了Chern-Vaaler的格点计数参数，以获得具有更好功耗节省的显式误差项，这导致Masser-Vaaler和Barroero的一些结果的显式版本。
Masser and Vaaler have given an asymptotic formula for the number of algebraic numbers of given degree d and increasing height. This problem was solved by counting lattice points (which correspond to minimal polynomials over Z) in a homogeneously expanding star body in Rd+1. The volume of this star body was computed by Chern and Vaaler, who also computed the volume of the codimension-one "slice" corresponding to monic polynomials; this led to results of Barroero on counting algebraic integers. We show how to estimate the volume of higher-codimension slices, which allows us to count units, algebraic integers of given norm, trace, norm and trace, and more. We also refine the lattice point-counting arguments of Chern-Vaaler to obtain explicit error terms with better power savings, which lead to explicit versions of some results of Masser-Vaaler and Barroero.