Algebraic Number Starscapes
Edmund Harriss, Katherine E. Stange, Steve Trettel

TL;DR
This paper explores the geometry of algebraic numbers in the complex plane through visualizations called algebraic starscapes, linking geometric structures with Diophantine approximation and providing new insights into algebraic root behavior.
Contribution
It introduces a geometric framework for understanding algebraic numbers and their approximations, rediscovering classical formulas as isometries and analyzing embedding properties of root maps.
Findings
Identifies when root maps are embeddings in quadratic and cubic cases.
Provides geometric explanations for approximation dichotomies.
Recovers and extends results on complex Diophantine approximation.
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 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…
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Taxonomy
TopicsAdvanced Numerical Analysis Techniques · Mathematical Dynamics and Fractals · History and Theory of Mathematics
