Algebraic Number Starscapes
Abstract
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 geometricminded recasting of fundamental results in the Diophantine approximation of the complex plane. The images provide a casestudy in the symbiosis of illustration and research, and an entrypoint 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, 333368). 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2020
 DOI:
 10.48550/arXiv.2008.07655
 arXiv:
 arXiv:2008.07655
 Bibcode:
 2020arXiv200807655H
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Differential Geometry;
 Primary: 11R04;
 11R11;
 11R16;
 11J04;
 11J68;
 11J87;
 53C30 Secondary: 11G50;
 11H99;
 54E99;
 57M99
 EPrint:
 67 pages, 36 figures