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 the resulting images, which we have called algebraic 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 natural $\text{PSL}(2;\mathbb{C})$ action. Hyperbolic geometry and the discriminant play an important role in low degree. In particular, we rediscover the quadratic and cubic root formulas as isometries of ${\mathbb{H}}^2$ and 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 of hyperbolic distance between roots in the complex plane and the discriminant as a measure of arithmetic height on a polynomial. In particular, we determine the supremum on the exponent k for which an algebraic target α has infinitely many approximations β whose hyperbolic distance from α does not exceed $\text{acosh}(1+1/|{\Delta }_{\beta }{|}^k)$. It turns out to fall into two cases, depending on whether α lies on the image of a plane of rational slope in coefficient space (a rational 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.

Keywords:

• algebraic numbers
• homogeneous geometry
• hyperbolic geometry
• Diophantine approximation

Mathematics Subject Classification:

• Primary: 11R04
• 11R11
• 11R16
• 11J04
• 11J68
• 11J87
• 53C30 Secondary: 11G50
• 11H99
• 54E99
• 57M99
• 11-02
• 51-02

Notes

1 For the polynomial $a{x}^2+bx+c=0$ the dot plotted will be at $\frac{-b+\sqrt{4ac-b^2}i}{2a}$ on the complex plane. Note this only considers polynomials with negative discriminant and thus complex roots. A similar image can be created for polynomials with only real roots, see Figure 9. The radius of the dot is proportional to $\frac{1}{\sqrt{4ac-b^2}}$ (one over the root discriminant) times the height above the real axis $\frac{\sqrt{4ac-b^2}}{2a}$ to adjust the radius to the hyperbolic metric. This gives dots with radius proportional to $\frac{1}{2a}$. If you are plotting the points yourself it can be useful to adjust the scale of the dots (keeping the same proportions) as more are added.

2 This space starts as ${\mathbb{R}}^n$, but multiplying all coefficients by a constant does not change the roots of the polynomials. It is natural, therefore to consider such polynomials as equivalent. This gives projective geometry one dimension lower, as discussed in Section 4.1.

3 The theorem as stated here is most commonly known as Dirichlet’s Approximation Theorem, but this theorem was already known to Legendre [39] as a result of the study of continued fractions. The proof we give in Section 5.1 is Dirichlet’s and actually gives a stronger asymptotic approximation result.

4 From this perspective the euclidean plane is ${\mathbb{R}}^2$ equipped with the group of rotations, translations and (glide) reflections. Projective geometry, hyperbolic geometry, and de Sitter space are other common examples, which we will encounter throughout our journey.

5 For example the linear polynomial $2x+3$ can be thought of as the homogeneous linear polynomial $2x+3y$, or the homogeneous quadratic $2xy+3y^2$, or the homogeneous cubic $2x{y}^2+3y^3$, etc.

6 Again taking the univariate polynomial $2x+3$ as an example, thought of as a linear homogeneous equation this has a single root. But thought of as a quadratic, such linear equations have an additional root “at infinity.”

7 In addition to managing to rigorously make sense of 1/0 of course!

8 However, it is certainly a visually natural thing to try given our representation of rationals as quotients - indeed probably the most common mistake when first learning arithmetic is to add rationals by taking their mediant!

9 As $\mathbb{C}$ is algebraically closed, the space of roots is easy to describe, defining the roots map does not require passing to a field extension, and the $\text{PSL}(2;\mathbb{C})$ symmetry can be exhibited at its most natural level of generality.

10 It is quick to check that as $r\to \infty$ the coefficients of the polynomial $(x-ry)(ax+by)$ converge projectively to $[0:a:b]$ and the roots to $\{[1:0],[-b:a]\}=\{\infty ,-b/a\}$. Thus, the quadratic formula extends continuously to linear equations interpreted as “quadratics with roots at infinity” as noted in Section 4.1.

11 This is an important argument in its own right, for it shows topologically the symmetric power ${\text{SP}}^2({\mathbb{S}}^2)$ is the complex projective plane in disguise. In particular, it is a closed manifold. Compare this with Remark 4.5 where ${\text{SP}}^2({\mathbb{S}}^1)$ is a manifold with boundary. This generalizes to arbitrary degree n, and the roots map provides a homeomorphism ${\mathcal{R}}_n:\mathbb{C}{\mathbb{P}}^n\stackrel{\sim }{\to }{\text{SP}}^n(\mathbb{C}{\mathbb{P}}^1)$.

12 This picture may already be familiar from representation theory: the action of $\text{PSL}(V)$ on a two-dimensional complex vector space V induces actions on symmetric powers of V, which are the irreducible representations of $\text{PSL}(V)$. Projectivizing this picture under an identification $V={\mathbb{C}}^2$ yields the result we exposit here.

13 Indeed there are many choices for A: if $a\in \mathbb{C}$ then $A=(\begin{array}{cc}1& a\\ 0& 1\end{array})$ also works.

14 On $\mathbb{R}{\mathbb{P}}_{\text{Coefs}}^2$, the orbit of quadratics with double root is exactly the discriminant locus (the set of polynomials whose discriminant is equal to zero); its complement is the union of the other two orbits.

15 In fact, from here, elementary topology completes the story as all separating circles divide $\mathbb{R}{\mathbb{P}}^2$ into a Möbius strip and a disk, giving the topological type of (2) and (3), respectively.

16 De Sitter geometry is a particular geometry not of space, but rather of spacetime. In this particular case, De Sitter 2-space describes a world with one space and one time dimension of positive curvature. Geometrically, this is just the hyperboloid of 1-sheet $x^2+y^2-z^2=1$ in ${\mathbb{R}}^3$ equipped with the action of $\text{SO}(2,1)$.

17 Hyperbolic geometry, commonly denoted ${\mathbb{H}}^2$, is the unique two dimensional geometry with constant negative curvature, and was the first non-Euclidean geometry discovered. Negative curvature implies that ${\mathbb{H}}^2$ violates Euclid’s fifth postulate with an infinitude of parallel lines to a given line through any point not on it. For an introductory treatment of the hyperbolic plane, see [3].

18 A Riemannian metric is a choice of inner product for each tangent space, which allows one to measure the length of vectors, and hence the arc length of curves.

19 The entire action of $\text{PSL}(2;\mathbb{R})$ on ${\mathbb{R}}^3$ via the representation ρ preserves the quadratic form Δ, and the inner product $〈(a_1,b_1,c_1),(a_2,b_2,c_2)〉=b_1{b}_2-2a_1{c}_2-2a_2{c}_1$ for which $\Delta (v)=〈v,v〉$. Thus the symmetries of ${\mathbb{H}}_{\text{Coefs}}^2$ are contained in the special orthogonal group of this form $\text{SO}(\Delta )$, the orthogonal group of a quadratic form q is the group of all matrices with determinant 1 whose action leaves q invariant: $\text{SO}(q)=\{A\in \text{SL}(n;\mathbb{R})|q(v,w)=q(Av,Aw)\}$. The indefinite orthogonal group $\text{SO}(\Delta )$ has two components, determined by whether or not a symmetry preserves or swaps the two sheets of the hyperboloid. The representation ρ is actually an isomorphism onto the connected component of the identity: $\text{PSL}(2;\mathbb{R})\cong {\text{SO}}_0(\Delta )$.

20 A similar description can be given for the two other geometries of quadratic polynomials. The $\mathbb{R}{\mathbb{P}}^1$ of polynomials with double roots is the zero set of Δ, or the projectivization of the light cone $b^2=4ac$. The space of polynomials with distinct real roots corresponds to points on which Δ is positive, which projectively forms a Möbius band. This space also has a natural notion of geometry, coming not from the hyperbolic plane but from relativity (it is called 1 + 1 dimensional de Sitter space, but is beyond the scope of this paper).

21 This is called the Hilbert metric. Such a metric may be defined for any convex subset $\Omega \subset \mathbb{R}{\mathbb{P}}^2$ not containing any entire projective line, and realizes a model of hyperbolic geometry precisely when Ω is bounded by a nonsingular conic section.

22 As particular examples, the 1-parameter families of quadratics with coefficients $[1:0:t]$, and $[1:t:1]$ project under the roots map to the vertical geodesic and unit circle through $i\in \mathbb{C}$, respectively.

23 In this way the orbits of rational geodesics under $\text{PSL}(2;\mathbb{Z})$ are identified with the narrow ideal classes of real quadratic fields K. See [21, section B.7].

24 Topologically, this describes a map which takes the projectivization of the exterior of the lightcone (a Möbius band) to the set of unordered pairs of distinct points on the circle (or ${\text{SP}}^2({\mathbb{S}}^1)$, which is also a Möbius band, as depicted in Figure 19).

25 Geometrically one may tell a beautiful story here quite analogous to Theorem 4.9, where the roots map is an isometry between a pair of Lorentzian metrics defined on each of these Möbius bands, though investigation of this would take us too far afield from the goals of this paper.

26 This action has finitely many orbits as $\text{PSL}(2;\mathbb{C})$ acts simply transitively on ordered triples of distinct elements of $\mathbb{C}{\mathbb{P}}^1$. Thus, some of these orbits are open. In higher degree, there are a continuum of orbits, which are parameterized by the moduli of n possibly indistinct points in $\mathbb{C}{\mathbb{P}}^1$ up to projective transformations (see also Remark 4.16).

27 Given a manifold X, the set of all tangent vectors to X at a point p is called the tangent space to X at p. You can think of this as a “linear approximation” to X near that point. If we restrict our attention to only unit vectors, we define the unit tangent space at p (for 2-dimensional geometries X, the unit tangent space at every point is just a circle). Collecting all the tangent spaces for every point of X gives the tangent bundle TX to X, and collecting only all unit vectors gives the unit tangent bundle $\text{UT}X$. For an introduction smooth manifolds and their tangent bundles, see [38].

28 For quadratics, the roots map is a homeomorphism on the entire space: there are no two-dimensional subfamilies, and all one-dimensional curves are geodesics.

29 Note this torus is not smoothly embedded in the space of coefficients, and is singular along the circle parameterizing cubics with a triple root.

30 Thus the space of cubics with three distinct roots is an infinite volume three dimensional orbifold with a geometric structure modeled on ${\widetilde{SL}}_2(\mathbb{R})$.

31 Computing expressions for $\Phi$ and its inverse we see they are compositions of elementary (continuous) functions: $\Phi (x\pm iy,r)={(v_1,v_2)}_{x+iy}={(\frac{w}{\sqrt{1+w^2}},\frac{1}{\sqrt{1+w^2}})}_{x+iy}$ for $w=\frac{2x(x-r)}{x^2+y^2-r^2}$ and ${\Phi }^{-1}{(v_1,v_2)}_{x+iy}=\{x\pm iy,r\}=\left\{x\pm iy,x+\left(\frac{v_2}{v_1}+\sqrt{1+{(\frac{v_2}{v_1})}^2}\right)y\right\}$

32 In fact, while any choice of inner product on the tangent space to our basepoint can be promoted to a Riemannian metric where $\text{PSL}(2;\mathbb{R})$ acts by isometries, this metric is the most symmetric possible choice: its isometry group is four-dimensional, whereas a generic inner product only leads to a three-dimensional isometry group.

33 More precisely, we take the Euclidean form of the metric on the affine patch where the following three curves of polynomials are orthogonal at the polynomial $x(x^2+1)$: (1)those with fixed imaginary root i, varying real root t, (2) those with fixed real root 0, varying real part of complex root, and (3) those with real root 0, varying imaginary part of complex root.

34 That is, choose v so that the geodesic γ_v with initial tangent v at z has $\underset{t\to \infty +}{\lim }\gamma (t)=r$.

35 Note however this homeomorphism is not an isometry of the metrics we have defined with the product metric on ${\mathbb{H}}^2\times \mathbb{R}$.

36 One shows the isometries of ${\mathbb{S}}^2$ act freely and transitively on the unit tangent bundle, which then provides a diffeomorphism from $\text{Isom}({\mathbb{S}}^2)=\text{SO}(3)$ and $\text{UT}{\mathbb{S}}^2$. Finally, we recall that $\text{SO}(3)$ is topologically real projective 3-space (for example, by noting that it is double covered by $\text{SU}(2)\cong {\mathbb{S}}^3$).

37 Using the coordinates $z=x+iy$ on the upper half plane, this vector is in the direction $(2y(r-x),{(r-x)}^2-y^2)$.

38 In light of Theorem 4.27 it cannot be anywhere else: as the only remaining portion to the cubic formula from this perspective is to solve the associated quadratic and linear equations: both of which have simple roots maps as we have seen before.

39 While the Riemannian metric here is easy to describe by translating the standard euclidean metric on some tangent space to a point $p\in \text{UT}{\mathbb{H}}_{\text{Coefs}}^2\subset \mathbb{R}{\mathbb{P}}^3$ around by action given by the representation ρ, its expression is complicated, making the computation of a distance function unwieldy.

40 One way to see this is to reduce to the unit circle; cubics cannot lie on the unit circle $\alpha \overline{\alpha }=1$ unless their real root r is rational, since the constant coefficient of the minimal polynomial $\alpha \overline{\alpha }r$ is rational.

41 That is, locally the hyperbolic metric and the euclidean metric are very nearly multiples of each other. This becomes exact at the level of tangent spaces for the Riemannian metric, where $d{s}_{\text{Hyp}}^2=\frac{1}{\text{Im}(z)}d{s}_{\text{Euc}}^2$.

42 Although the definition, and much of the discussion, works for $\mathbb{R}$ as well as $\mathbb{C},\text{PSL}(2;\mathbb{Z})$ doesn’t preserve any meaningful metric on the real line, and all of $\mathbb{Q}$ falls into a single orbit: this definition would lose too much information.

43 It generalizes the reduction theory of binary quadratic forms with respect to $\text{PSL}(2;\mathbb{Z})$ to general binary forms. It is not known how small a height one is guaranteed under their algorithm, nor where the minimum is attained.

44 We beg the reader’s forgiveness for the use of H for both naïve height of a polynomial and Weil height of a number; they do not satisfy $H(f_{\alpha })=H(\alpha )$, but the notation is standard in the literature.

45 This generalizes the rational approximation case, since there the angle θ between the projective lines $[\alpha :1]$ and $[p:q]$ satisfies

$\text{cos}\hspace{0.17em}\theta =\frac{〈(\alpha ,1),(p,q)〉}{\left|\right|(p,q)\left|\right|\left|\right|(\alpha ,1)\left|\right|}=\frac{|q\alpha -p|}{\left|\right|(p,q)\left|\right|\left|\right|(\alpha ,1)\left|\right|}.$

46 To see this, count the geodesics and assign a tubular neighborhood of width $1/n$ to the n-th such geodesic; require all terms β_k past the n-th to avoid the first n tubular neighborhoods; each term has finitely many restrictions placed upon it. (To make this work, the geodesics must be ordered as the terms are created, i.e., the n-th geodesic is always chosen so that the closure of its neighborhood does not include β_n .)

47 To see this for items (2) and (3) in Theorem 6.9 requires some consideration of the relationship between $H_{\text{PSL}}(f_{\alpha })$ and $H(f_{\alpha })$; we need to know the quotient is bounded in terms of the height of the relevant element of $\text{PSL}(2;\mathbb{Z})$, which is a constant in terms of α.

48 Indeed, by the action of $\text{PSL}(2;\mathbb{R})$, the pair of complex roots can be moved to any point in ${\mathbb{H}}^2$. Fixing this point, the remaining degree of freedom of the $\text{PSL}(2;\mathbb{R})$ action acts by rotation on the ideal boundary ${\partial }_{\infty }{\mathbb{H}}^2=\mathbb{R}{\mathbb{P}}^1$, and each $\text{PSL}(2;\mathbb{R})$ orbit is determined by the angle between these two points, measured between the tangent vectors at the complex root pointing to the real roots, as in equation (10).

49 The complex roots of Lehmer’s polynomial all lie on the unit circle, and its Mahler measure is a function of its real roots.

Additional information

Funding

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 and the Alfred P. Sloan Foundation award G-2019-11406 while the authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Illustrating Mathematics program. Katherine E. Stange is supported by NSF CAREER CNS-1652238.