Algebraic Number Starscapes
Algebraic Number Starscapes
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代数数星空
DOI：
10.1080/10586458.2022.2102094
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发表时间：
2022
影响因子：
0.5
通讯作者：
Trettel, Steve
中科院分区：
文献类型：
--
作者：
Harriss, Edmund;Stange, Katherine E.;Trettel, Steve
借助于广泛的计算机可视化，我们研究了复平面上的代数数的几何以及它们的丢番图逼近。在我们称之为代数星景的结果图像的激励下，我们描述了从多项式的系数空间到根空间的映射的几何，重点讨论了二次和三次情况。几何学描述和解释了插图的显著特征，并激发了对复杂平面的丢番图近似中的基本结果进行几何思维的重塑。同时，这些图像为插图和研究的共生提供了一个案例研究，也为更广泛的受众提供了一个几何和数论的入口点。具体地说，这篇论文的目的是为齐次几何和丢番图逼近的研究提供一个容易理解的介绍。研究了在自然作用下根空间和系数空间的齐次几何。双曲几何和判别式在低阶次中起着重要作用。特别地，我们重新发现了作为单位切丛及其单位切丛的等距的二次和三次根公式。利用这个几何学，我们确定发送某些多项式族到其复根的地图(我们的星景图像)何时是嵌入的。从发展的几何观点出发，重新考虑了有界次代数数对复数的丢番图逼近的基本问题。在二次情形(用二次无理函数逼近)中，我们考虑用复平面上根之间的双曲距离来逼近，而这个判别式是多项式上算术高度的一种度量。特别地，我们确定了代数目标α具有与β的双曲距离不超过α的无穷多个逼近的指数的上确界。事实证明，根据α是否位于系数空间中有理斜率平面(自测地线)的像上，它可以分为两种情况。这一结果是施密特子空间定理的应用。我们的结果恢复了Bugeaud和Evertse的结果的二次情形，并对他们发现的二分性给出了一些几何解释。在区分目标或近似是否位于有理测地线上方面，我们的陈述更进一步。这篇论文随附了软件，并以各种各样的公开问题结束。
We study the geometry of algebraic numbers in the complex plane, and their Diophantine approximation, aided by extensive computer visualization. Motivated by the resulting images, which we have calledalgebraic starscapes, we describe the geometry of the map from the coefficient space of polynomials to the root space, focusing on the quadratic and cubic cases. The geometry describes and explains the notable features of the illustrations, and motivates a geometric-minded recasting of fundamental results in the Diophantine approximation of the complex plane. Meanwhile, the images provide a case-study in the symbiosis of illustration and research, and an entry-point to geometry and number theory for a wider audience. In particular, the paper is written to provide an accessible introduction to the study of homogeneous geometry and Diophantine approximation. We investigate the homogeneous geometry of root and coefficient spaces under the naturalaction. Hyperbolic geometry and the discriminant play an important role in low degree. In particular, we rediscover the quadratic and cubic root formulas as isometries ofand its unit tangent bundle, respectively. Utilizing this geometry, we determine when the map sending certain families of polynomials to their complex roots (our starscape images) are embeddings. We reconsider the fundamental questions of the Diophantine approximation of complex numbers by algebraic numbers of bounded degree, from the geometric perspective developed. In the quadratic case (approximation by quadratic irrationals), we consider approximation in terms ofhyperbolicdistance between roots in the complex plane and thediscriminantas a measure of arithmetic height on a polynomial. In particular, we determine the supremum on the exponentkfor which an algebraic targetαhas infinitely many approximationsβwhose hyperbolic distance fromαdoes not exceed. It turns out to fall into two cases, depending on whetherαlies on the image of a plane of rational slope in coefficient space (arational geodesic). The result comes as an application of Schmidt’s subspace theorem. Our results recover the quadratic case of results of Bugeaud and Evertse, and give some geometric explanation for the dichotomy they discovered. Our statements go a little further in distinguishing approximability in terms of whether the target or approximations lie on rational geodesics. The paper comes with accompanying software, and finishes with a wide variety of open problems.
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影响因子：
0.8
作者：
Matthew Bates
通讯作者：
Matthew Bates
线性递推序列和多维连分数的周期性
DOI：
10.1007/s11139-016-9820-2
发表时间：
2017
期刊：
The Ramanujan Journal
影响因子：
--
作者：
Nadir Murru
通讯作者：
Nadir Murru
影响因子：
0.7
作者：
J. Wolfskill
通讯作者：
J. Wolfskill
影响因子：
0.7
作者：
H. Davenport;W. Schmidt
通讯作者：
W. Schmidt
数论中的可见结构
DOI：
10.1080/00029890.2001.11919824
发表时间：
2001
期刊：
The American Mathematical Monthly
影响因子：
--
作者：
P. Borwein;Loki Jörgenson
通讯作者：
Loki Jörgenson