论文标题
代数数星形
Algebraic Number Starscapes
论文作者
论文摘要
我们通过广泛的计算机可视化研究了复杂平面中代数数的几何形状及其二磷酸近似值。由这些图像的动机，称为代数Starscapes，我们描述了地图的几何形状，从多项式的系数到根空间，集中在二次和立方体情况上。几何形状描述并解释了插图的显着特征，并激发了对复杂平面的二磷酸近似值的基本结果的几何重塑。这些图像在插图和研究的共生中提供了一个案例研究，并为更广泛的受众提供了几何学和数字理论的入口。该论文的编写是为了对均质几何学和二芬太汀近似的研究提供可访问的介绍。 We investigate the homogeneous geometry of root and coefficient spaces under the natural $\operatorname{PSL}(2;\mathbb{C})$ action, especially in degrees 2 and 3. We rediscover the quadratic and cubic root formulas as isometries, and determine when the map sending certain families of polynomials to their complex roots (our starscape images) are embeddings. 我们将二次非理性的复杂二磷酸近似值视为双曲线距离和判别因子，是算术高度的量度。我们恢复了Bugeaud和Evertse结果的二次案例，并为它们发现的二分法给出了一些几何解释（Bugeaud，Y。和Evertse，J.-H。，由代数数量通过有限程度的代数数量的复杂代数数近似，Ann。Sc.norm。Sc.norm.Sund。Subssuper。Super。Pisacl。Sci。Sci。（5）8（2009年）。我们的陈述在区分目标还是近似值的近似值方面进一步差异。 该论文带有随附的软件，并带有各种各样的开放问题。
We study the geometry of algebraic numbers in the complex plane, and their Diophantine approximation, aided by extensive computer visualization. Motivated by these images, called algebraic starscapes, we describe the geometry of the map from the coefficient space of polynomials to the root space, focussing on the quadratic and cubic cases. The geometry describes and explains notable features of the illustrations, and motivates a geometric-minded recasting of fundamental results in the Diophantine approximation of the complex plane. 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. 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 natural $\operatorname{PSL}(2;\mathbb{C})$ action, especially in degrees 2 and 3. We rediscover the quadratic and cubic root formulas as isometries, and determine when the map sending certain families of polynomials to their complex roots (our starscape images) are embeddings. We consider complex Diophantine approximation by quadratic irrationals, in terms of hyperbolic distance and the discriminant as a measure of arithmetic height. We recover the quadratic case of results of Bugeaud and Evertse, and give some geometric explanation for the dichotomy they discovered (Bugeaud, Y. and Evertse, J.-H., Approximation of complex algebraic numbers by algebraic numbers of bounded degree, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8 (2009), no. 2, 333-368). 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.
