### Algebraic Number Starscapes

Joint with Edmund Harris and Kate Stange

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
PSL(2;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.