1 Overview

Solving a polynomial equation calls for a means to overcome the polynomial’s symmetry. In the case of the fifth-degree equation, the general symmetry group is the symmetric group \(\mathcal {S}_5\). In terms of Galois theory, we can reduce the symmetry to that of the alternating group \(\mathcal {A}_5\) by adjoining the square root of the polynomial’s discriminant to the coefficient field. Our reward for this reduction is that we can realize \(\mathcal {A}_5\) as the rotational symmetries of the regular icosahedral configuration on the complex projective line \(\mathbf {CP}^1\)—that is, the Riemann sphere.

Exploiting icosahedral structure, Doyle and McMullen constructed a quintic algorithm at the core of which is a map \(\phi \) on \(\mathbf {CP}^1\) that respects the \(\mathcal {A}_5\) symmetry [1]. The map is strongly critically finite, meaning that its critical set \(\mathcal {C}_\phi \)—the twenty face-centers of the icosahedron—is \(\phi \)-invariant; that is, \(\phi (\mathcal {C}_\phi )=\mathcal {C}_\phi \). In particular, each superattracting critical point has period two. It follows that \(\phi \)’s global dynamics is reliable—meaning that a full measure’s worth of points in \(\mathbf {CP}^1\) belongs to the basins of attraction associated with the two-cycles in \(\mathcal {C}_\phi \). Their procedure employs \(\phi \)’s dynamics in a way that partially breaks the \(\mathcal {A}_5\) symmetry and, with one iterative run, computes two roots.

Recent computational work [2, 3] determined all maps with icosahedral symmetry whose critical sets have size 60 and are internally periodic—meaning that the map acts on its critical set as a permutation. The article [2] gave a cursory discussion of how an iterative procedure can rely on such a map in order to approximate all of a quintic’s roots. Here, we provide a detailed account of how to design a quintic-solving device around the dynamics of one such map g whose critical points have period five. Since the superattracting set has generic size, the dynamics of g effectively breaks all of an equation’s \(\mathcal {A}_5\) symmetry. Accordingly, the algorithm produces all five roots with a single iterative run. Unlike the method developed in [4], the current approach explicitly draws upon icosahedral symmetry in algebraic, geometric, and dynamical settings.

The discussion first lays out the necessary ingredients derived from the icosahedral action on \(\mathbf {CP}^1\). Then from an understanding of the \(\mathcal {A}_5\)-invariant forms and \(\mathcal {A}_5\)-equivariant maps, the special map g emerges. In conclusion, the study assembles the moving parts from icosahedral algebra and the reliable dynamics of g into a procedure that solves equations in a specific family of quintics.

Computational results, whether exact or approximate, are products of Mathematica, as was some graphical content. The exposition makes clear which computations are exact or approximate. The program Dynamics 2 generated basins-of-attraction plots [5].

2 Icosahedral Algebra: Invariants and Equivariants

An account of the algebraic objects that arise from the icosahedral action on \(\mathbf {CP}^1\) appears in other places. [1, 2, 6] Here, results relevant to the task at hand appear without discussion.

Fig. 1
figure 1

Configuration of five sets of tetrahedral vertices: each color corresponds to four vertices of a tetrahedron when projected onto the sphere. Taking the tetrahedrons’ face-centers instead would produce the same set of points, but a different color scheme

Denote by \(\mathcal {I}\) the \(\mathcal {A}_5\)-isomorphic group of 60 rotational symmetries of the regular icosahedron as a graph structure on the sphere. We can express such a rotation as either a Mobius transformation or a member of \(\mathbf {PSL}_2(\mathbf {C})\), that is, a \(2\times 2\) matrix whose determinant equals 1 [6] (Fig. 1 renders this graph on \(\mathbf {C}\).). Three polynomials generate the ring of \(\mathcal {I}\)-invariants:

$$\begin{aligned} \mathbf {C}[x,y]^\mathcal {I}=\left<F(x,y), H(x,y), T(x,y)\right> \end{aligned}$$

where (xy) are homogeneous coordinates on \(\mathbf {CP}^1\). The forms F, H, and T vanish at the special \(\mathcal {I}\)-orbits: 12 vertices \(v_k\), 20 face-centers \(f_k\), and 30 edge-midpoints \(e_k\) respectively. For ease of reference, call the members of these sets “12-points, 20-points, and 30-points.” In coordinates where a pair of antipodal vertices reside at the north and south poles (0 and \(\infty \) in \(\mathbf {C}\cup \{\infty \}\)), we can express the generating invariants as products over special orbits:

$$\begin{aligned} F&=\prod _{k=1}^{12} (x-v_k y)= x y \left( x^{10}-11x^5 y^5-y^{10}\right) \\ H&=\prod _{k=1}^{20} (x-f_k y)= x^{20}+228x^{15} y^5+494x^{10} y^{10}-228x^5 y^{15}+y^{20} \\ T&=\prod _{k=1}^{30} (x-e_k y)= x^{30}-522x^{25} y^5-10005x^{20} y^{10}-10005x^{10} y^{20}\\&\quad +522x^5 y^{25}+y^{30}. \end{aligned}$$

Accordingly, F and H are algebraically independent, while an algebraic combination of the two generators in degree 60 vanishes (with multiplicity two) at the 30-points:

$$\begin{aligned} T^2=H^3-1728\,F^5. \end{aligned}$$

We also need the system of invariants for each of five tetrahedral subgroups denoted \(\mathcal {T}_1,\dots ,\mathcal {T}_5\) in \(\mathcal {I}\). Each \(\mathcal {T}_k\) is a rotational symmetry group of a regular tetrahedron and acts as an alternating group \(\mathcal {A}_4\) on two sets of four 20-points whose union forms eight vertices of a \(\mathcal {T}_k\)-invariant cube. Overall, these disjoint sets of four points occupy the vertices of five regular tetrahedra that the icosahedron circumscribes in two chirally distinct ways. Figure 1 shows an icosahedral net in \(\mathbf {C}\) and one way to decompose the vertices into tetrahedral sets distinguished by color. Indices for the tetrahedral groups correspond to the colors. Under \(\mathcal {I}\) the five tetrahedra (and cubes) realize \(\mathcal {A}_5\) behavior.

Definition 1

A relative invariant is a form \(\varPhi \) for which a non-trivial multiplicative character \(\lambda _T\) appears under a group action \(\mathcal {G}\). That is, for all \(T\in \mathcal {G},\)

$$\begin{aligned} \varPhi \circ T = \lambda _T\,\varPhi \quad \text {such that}\ \lambda _T \not = 1\ \text {for some}\ T\in \mathcal {G}. \end{aligned}$$

An invariant is absolute when \(\lambda _T= 1\ \text {for all}\ T \in \mathcal {G}\).

Using \(\mathcal {T}_5\) as a reference group, there are two degree-four relative \(\mathcal {T}_5\) invariants: one, \(q_5\), given by the product that involves the tetrahedral vertices and the other, \(\hat{q}_5\), given by the product that uses the tetrahedral face-centers (antipodes of the vertices). The results are complex conjugate forms

$$\begin{aligned} q_5=&\ \frac{1}{2} \bigl (2 x^4+(1+i \sqrt{15}) x^3 y+(3-i \sqrt{15}) x^2 y^2-(1+i \sqrt{15}) x y^3+2 y^4\bigr )\\ \hat{q}_5=&\ \frac{1}{2} \bigl (2 x^4+ (1-i \sqrt{15})x^3 y)+(3+i \sqrt{15}) x^2 y^2- (1-i\sqrt{15}) x y^3+2 y^4\bigr ). \end{aligned}$$

The rationale for using the index 5 will be evident presently. To illustrate relative invariance, recall that \(\mathcal {T}_5\simeq \mathcal {A}_4\) and that \(\mathcal {A}_4\) contains a Klein-4 subgroup \(\mathcal {V}\). As for the action of \(\mathcal {T}_5\) on \(q_5\):

$$\begin{aligned} q_5(A(x,y))={\left\{ \begin{array}{ll} q_5(x,y)&{}A\in \mathcal {V}\subset \mathcal {T}_5\\ -q_5(x,y)&{}A\in \mathcal {T}_5-\mathcal {V} \end{array}\right. } \end{aligned}$$

and similarly for \(\hat{q}_5\).

An absolute \(\mathcal {T}_5\) invariant results when we take the form that vanishes at the six-point tetrahedral orbit associated with edges:

$$\begin{aligned} t_5= x^6-2 x^5 y-5 x^4 y^2-5 x^2 y^4+2 x y^5+y^6. \end{aligned}$$

In this case, \(t_5(A(x,y))=t_5(x,y)\) for all \(A\in \mathcal {T}_5\). Furthermore, the product of the degree-four forms yields a degree-eight \(\mathcal {T}_5\) invariant:

$$\begin{aligned} u_5=q_5 \hat{q}_5=x^8+x^7 y+7 x^6 y^2-7 x^5 y^3+7 x^3 y^5+7 x^2 y^6-x y^7+y^8. \end{aligned}$$

Since the eight zeroes of \(u_5\) have order-three symmetry, they are vertices of an inscribed cube as well as eight of the icosahedron’s face-centers. Note that the vertices of such a cube coincide with the vertices and face-centers of a tetrahedron. Alternatively, these points are vertices of two tetrahedra in opposite chiral systems. Hence, the icosahedral invariant H is divisible by \(u_5\) and the quotient is a \(\mathcal {T}_5\) invariant of degree \(12=20-8\):

$$\begin{aligned} m_5=\frac{H}{u_5}=&\ x^{12}-x^{11} y-6 x^{10} y^2+20 x^9 y^3+15 x^8 y^4+24 x^7 y^5+11 x^6 y^6\\&\ -24 x^5 y^7+15 x^4 y^8 -20 x^3 y^9-6 x^2 y^{10}+x y^{11}+y^{12}. \end{aligned}$$

The tetrahedral invariants satisfy relations in degrees 12 and 24:

$$\begin{aligned} q_5^3=&\ 80 t_5^2 - (95 - 9 \sqrt{15} i)m_5\\ 64 m_5^2=&\ 95 t_5^2 m_5-40 t_5^4+9 u_5^3. \end{aligned}$$

Applying powers of an order-five element \(P\in \mathcal {I}\) manufactures the remaining \(\mathcal {T}_k\) invariants:

$$\begin{aligned} t_k=t_5\circ P^k \qquad u_k=u_5\circ P^k \qquad m_k=m_5\circ P^k \qquad k=1,\dots ,4. \end{aligned}$$

In the chosen coordinates, we can take \(P(x,y)=(\epsilon ^3 x,\epsilon ^2 y)\) where \(\epsilon =e^{2 \pi i/5}\). Significantly, the action of \(\mathcal {I}\) permutes each of these three sets of five tetrahedral invariants as \(\mathcal {A}_5\) objects.

Given a group action \(\mathcal {G}\) on a space X, a natural symmetry for a map \(f:X\rightarrow X\) is known as \(\mathcal {G}\)-equivariance and satisfies

$$\begin{aligned} f\circ \gamma = \gamma \circ f\qquad \text {for all}\ \gamma \in \mathcal {G}. \end{aligned}$$

Using standard invariant-theoretic techniques, a generating \(\mathcal {G}\)-invariant gives rise to a \(\mathcal {G}\)-equivariant (or \(\mathcal {G}\)-map) of one less degree.

For present purposes, set

$$\begin{aligned} x=\begin{pmatrix}x_1\\ x_2\end{pmatrix}\quad w=\begin{pmatrix}w_1\\ w_2\end{pmatrix}\quad \text {and} \quad J=\begin{pmatrix}0&{}-1\\ 1&{}0\end{pmatrix}. \end{aligned}$$

A bit later, we call on these coordinates again. There is a simple commutation rule for J and a linear change of coordinate \(x=Aw\) as follows.

Lemma 2.1

Let A be the matrix of a linear transformation on \(\mathbf {C}^2\) with \(A^T\) its transpose and \(\det {A}=1\). Then

$$\begin{aligned} J (A^T)^{-1} = A J. \end{aligned}$$

Now, for the equivariance claim.

Theorem 2.2

Let P(x) be an invariant of degree r under the action on \(\mathbf {CP}^1\) of a group \(\mathcal {G}\subset \mathbf {PSL}_2(\mathbf {C})\). The “cross” operator

$$\begin{aligned} \times P(x)=(-\partial _{x_2} P,\partial _{x_1} P) \end{aligned}$$

yields a \(\mathcal {G}\)-equivariant whose degree is \(r-1\).


Express the cross operator as

$$\begin{aligned} f(x)=\times _x P(x) = J \nabla _x P(x)\qquad \text{ where }\ \nabla _x=(\partial _{x_1},\partial _{x_2}). \end{aligned}$$

Take \(\gamma \in \mathcal {G}\) with \(x=\gamma w\). A calculation remains.

$$\begin{aligned} P(\gamma w)=&\ P(w)\qquad [\gamma \in \mathcal {G}\text { and }P\text { is }\mathcal {G}-\text {invariant}]\\ r\,P(x)=&\ r\,P(w)\\ \nabla _x P(x)^T x=&\ \nabla _w P(w)^Tw\qquad [\text {homogeneity of }P(w)]\\ \nabla _x P(x)^T x=&\ \nabla _w P(w)^T \gamma ^{-1} \gamma w\\ \nabla _x P(x)^T=&\ \nabla _w P(w)^T \gamma ^{-1}\qquad [x=\gamma w]\\ \nabla _x P(x)=&\ (\gamma ^{-1})^T\nabla _w P(w)\\ J \nabla _x P(x)=&\ J(\gamma ^T)^{-1}\nabla _w P(w)\\ J \nabla _x P(x)=&\ \gamma J\,\nabla _w P(w)\qquad [\text {Lemma}~2.1]\\ \times _x P(x)=&\ \gamma (\times _w P(w))\\ f(x)=&\ \gamma \,f(w)\\ f(\gamma w)=&\ \gamma f(w). \end{aligned}$$

\(\square \)

Applying the operator to the generating invariants of \(\mathcal {I}\) produces maps that generate the module of \(\mathcal {I}\)-equivariants over \(\mathcal {I}\)-invariants:

$$\begin{aligned} \phi&=\times F= \bigl (-x^{11}+66 x^6 y^5+11 x y^{10},11 x^{10} y-66 x^5 y^6-y^{11}\bigr )\\ \eta&=\times H= 20 \bigl (-57 x^{15} y^4-247 x^{10} y^9+171 x^5 y^{14}-y^{19}, x^{19}+171 x^{14} y^5\\&\quad +247 x^9 y^{10}-57 x^4 y^{15}\bigr ). \end{aligned}$$

These exceptional maps exhibit elegant behavior: \(\phi \) twists and wraps a dodecahedral face \(\mathcal {F}\) onto the 11 faces in the complement of the face antipodal to \(\mathcal {F}\) while \(\eta \) does the analogous twisting and wrapping for a face of the icosahedron. For edges of the respective polyhedra we can take great circle arcs between vertices to obtain sets that are forward invariant under the respective map. Call this structure a dynamical polyhedron. Moreover, each map expands the internal angle of a face in its dynamical polyhedron onto an external angle of the antipodal face. The vertices are thereby periodic critical points and their superattracting basins are full-measure subsets of \(\mathbf {CP}^1\). The Doyle-McMullen iteration uses \(\phi \) whose attracting set is a special orbit—namely, the face-centers. Hence, \(\mathcal {A}_5\) symmetry is partially broken allowing for the extraction of two roots.

To break \(\mathcal {A}_5\) symmetry fully, we look for a map g whose critical set \(\mathcal {C}_g\) is a generic 60-point \(\mathcal {I}\)-orbit that is permuted under the action of g. All maps of this sort have degree 31 and are classified in [3]. Excepting two cases, the dynamical polyhedra associated with these special “31-maps” derive from the icosahedral structure in the following way: they consist of regions that correspond to twelve pentagons, twenty triangles, and thirty quadrilaterals. The resulting configuration is called a \(B_{62}\). (It also goes by the awkward name rhombicosidodecahedron.).

3 A Special Map

The paper [3] discusses the discovery and geometric behavior of 24 \(\mathcal {I}\)-maps with period-five critical points as well as of other critically-finite maps in degree 31. Now, take for g an \(\mathcal {I}\)-map whose critical points belong to twelve five-cycles each of which resides at consecutive pentagonal vertices on a \(B_{62}\). Figure 2 depicts a \(B_{62}\) configuration realized by the critical set.

Fig. 2
figure 2

A \(B_{62}\) arrangement of g’s critical set on both \(\mathbf {CP}^1\) and \(\mathbf {C}\) is chiral, meaning the cluster of points has only rotational symmetry. Under iteration, a critical point superattracts as it cycles through consecutive vertices of the pentagon on which it resides

Its analytic form appears once a root

$$\begin{aligned} (\alpha ,\beta )\approx (19,-10.82535-1.09144 i) \end{aligned}$$

of a homogeneous degree-24 equation has been approximated:

$$\begin{aligned} {g}&= \alpha H\cdot \phi \ +\ \beta F\cdot \eta \\&\approx \Bigl (-19 x \bigl ( x^{30}-(487.52150+65.48659 i) x^{25} y^5-(10234.85663-436.57731 i)x^{20} y^{10}\\&\quad -(1781.38888-3383.47417 i)x^{15} y^{15}-(9016.01160+1878.43133 i)x^{10} y^{20}\\&\quad +(618.781738-183.822026 i)x^5y^{25} + (0.39511+1.14888 i)y^{30} \bigr ),\\&\quad -19y\bigl ( (0.39511+1.14888 i)x^{30}-(618.78173-183.82202 i) x^{25}y^5\\&\quad -(9016.01160+1878.43133 i) x^{20}y^{10} +(1781.38888-3383.47417 i) x^{15}y^{15}\\&\quad -(10234.85662-436.57731 i) x^{10}y^{20}+(487.52150+65.48659 i) x^5 y^{25}+y^{30} \bigr )\Bigr ). \end{aligned}$$

Between the map’s two components, a kind of duality appears in the coefficients as a manifestation of the \(\times \) operation when deriving the generating maps \(\phi \) and \(\eta \).

As discussed in [2] and [3], the combinatorial geometry of g’s behavior gives rise to a polyhedral system of forward invariant “edges” \(\mathcal {E}_{g}\) with vertices at the critical points. This collection of edges fills in the \(B_{62}\) structure whose faces consist of twelve pentagons, twenty triangles, and thirty quadrilaterals that realize five-fold, three-fold, and two-fold rotational symmetry respectively. Figure 3 shows the output of an algorithm worked out in [3] that constructs an approximation to the edge-system overlaid on a coloring scheme determined by the map’s topological behavior.

Fig. 3
figure 3

On the right we see an approximation of a \(B_{62}\) in \(\mathbf {C}\). The plot exhibits a color-luminosity (CL) field in which each point z receives a (C(z), L(z)) coordinate determined by \((\arg (z),|z |)\). The plot on the left reveals the combinatorial behavior of g in which a point z is colored as (C(g(z)), L(g(z))). So, the map’s behavior is evident by matching color-luminosity values between left and right fields. The overlaid curves (gray) in the left plot outline the regions—call them “pre-faces”—that map to the respective types of face (outlined in white) on the right. The algorithm that generates the edges relies on backward iteration; hence, the appearance of gaps around the critical points where the pullback is highly expansive. Specifically, by noting which pre-faces reside in a face, we can discern that the image of a pentagon covers a pentagon, a triangle covers three pentagons, one triangle, and three quadrilaterals, while a quadrilateral covers ten pentagons, 20 triangles, and 29 quadrilaterals

Appearing in Fig. 4 are basin-of-attraction plots that reveal g’s symmetry and global dynamics.

Fig. 4
figure 4

Basins of attraction for twelve critical five-cycles. A point is colored (or shaded) according to which of the critical 5-cycles its orbit approaches asymptotically. The top right plot is a magnified view of the region in the square box shown in the top left plot while the bottom left image displays the region bounded by the square in the plot at top right and also displays the edge-system \(\mathcal {E}_{g}\). Associated with a critical cycle are five large basin components in black that accumulate at infinity—a fixed 12-point. For a global view, at bottom right we see the basins under a projection of the plane onto a disk. Note that the coloring is inconsistent between the plots. The map’s chirality is evident

By critical-finiteness, g is expanding relative to the hyperbolic metric on \(\mathbf {CP}^1-\mathcal {C}_{g} \). Hence, the orbit of almost every \(p\in \mathbf {CP}^1\) tends to a critical five-cycle:

$$\begin{aligned} {g}^k(p)\overset{k\rightarrow \infty }{\longrightarrow } (r_1,r_2,r_3,r_4,r_5)\subset \mathcal {C}_{g}. \end{aligned}$$

Because g cycles the adjacent pentagonal vertices \((r_1,\ldots ,r_5)\), we can take each \(r_k\) to be a vertex of the tetrahedron invariant under \(\mathcal {T}_k\)—evident in Fig. 1. This dynamical outcome lies at the core of a quintic-solving procedure. The presence of a period-five attracting set makes for an elegant algorithm.

4 Solving the Quintic

Say that you want to solve the quintic

$$\begin{aligned} z^5+a_4\,z^4+a_3\,z^3+a_2\,z^2+a_1\,z+a_0=0. \end{aligned}$$

L. Dickson reduced the general five-parameter equation to a resolvent using one parameter in such a way that, from a solution to the resolvent, you can recover a solution to the original equation. [7, Ch. XIII] The procedure about to be developed solves equations in Dickson’s special one-parameter family. Accordingly, it performs an essential role in the general solution. We now work out the steps that culminate in a quintic algorithm.

4.1 Resolvent

First, we create a parametrized family of quintic equations that our dynamical algorithm will solve. The parametrization makes use of the invariant-theoretic structure due to the icosahedral action \(\mathcal {I}\) on \(\mathbf {CP}^1\) as described in Sect. 2. Let

$$\begin{aligned} \rho _k(x)=\frac{F(x)u_k(x) }{H(x)}\qquad k=1,\dots ,5 \end{aligned}$$

where, for clarity’s sake, \(x=(x_1,x_2)\) are homogeneous coordinates replacing the former (xy). By construction, the degree-eight tetrahedral forms \(u_k\), hence, \(\rho _k\), experience \(\mathcal {A}_5\) permutation under the action of \(\mathcal {I}\).

Next, take the degree-zero rational functions \(\rho _k\) as five roots of a polynomial:

$$\begin{aligned} R_x(v)=\prod _{k=1}^5 (v-\rho _k(x))=\sum _{j=0}^5 b_j(x) v^j. \end{aligned}$$

Being symmetric functions in the \(\rho _k\), the coefficients \(b_j\) are \(\mathcal {I}\)-invariant and thereby expressible in terms of F and H.

Note that some of the coefficients vanish due to their degree. For instance, the coefficient of \(v^4\) is

$$\begin{aligned} b_4=\frac{F}{H} \sum _{k=1}^5 u_k. \end{aligned}$$

Since \(\sum _{k=1}^5 u_k\) is degree-eight and there are no such \(\mathcal {I}\)-invariants, it turns out that \(b_4=0\). Similarly,

$$\begin{aligned} b_3=\frac{F^2}{H^2} \sum _{1\le k< \ell \le 5} u_k u_\ell , \end{aligned}$$

but the degree of \(u_k u_\ell \) is 16 for which no \(\mathcal {I}\)-invariant exists so that \(b_3=0\). Continuing in this fashion,

$$\begin{aligned} b_2=\frac{F^3}{H^3} \sum _{1\le k< \ell < m \le 5} u_k u_\ell u_m. \end{aligned}$$

In this case, the forms \(u_k u_\ell u_m\) have degree 24 and so, the sum gives an \(\mathcal {I}\)-invariant \(a\,F^2\). Accordingly, the coefficient admits expression in the icosahedral parameter \(Z=\frac{F^5}{H^3}\):

$$\begin{aligned} b_2=a\,\frac{F^5}{H^3}=a\,Z. \end{aligned}$$

Next, we have

$$\begin{aligned} b_1=\frac{F^4}{H^4} \sum _{1\le k< \ell< m<n \le 5} u_k u_\ell u_m u_n \end{aligned}$$

where the sum produces an \(\mathcal {I}\)-invariant of degree 32, a term expressible as \(b\,F H\). Hence,

$$\begin{aligned} b_1=\frac{F^4}{H^4} b\,F H=b\,\frac{F^5}{H^3}=b\,Z. \end{aligned}$$

Finally, the zeroth-order term is

$$\begin{aligned} b_0=\frac{F^5}{H^5} u_1 u_2 u_3 u_4 u_5. \end{aligned}$$

The product of the \(u_i\) has degree 40, giving an \(\mathcal {I}\)-invariant \(c H^2\) so that

$$\begin{aligned} b_0=\frac{F^5}{H^5}c\,H^2=c\,\frac{F^5}{H^3}=c\,Z. \end{aligned}$$

Evaluating the factors a, b, and c yields a one-parameter family of quintic resolvents

$$\begin{aligned} R_Z(v)=v^5-40 Z v^2-5 Z v-Z \end{aligned}$$

equivalent to Dickson’s fifth-degree resolvent.

Turning to the construction of a quintic-solving algorithm, the key step occurs when we connect a specific resolvent \(R_Z\) with a map \({g}_Z\) that’s conjugate to the special \(\mathcal {I}\)-map g. That step is taken by self-parametrizing the icosahedral group. Finally, we build a function—also parametrized by Z—that converts the outcome of \({g}_Z\)’s superattracting behavior into the roots of a chosen \(R_Z\).

4.2 Parametrization

To begin the parametrization process, consider the family of transformations

$$\begin{aligned} x=S_y{w}=H(y) \phi (y) w_1 + F(y) \eta (y) w_2 \end{aligned}$$

that’s linear in \(w=(w_1,w_2)\) and degree-31 in \(y=(y_1,y_2)\). Using the substitution

$$\begin{aligned} (x_1,x_2)\rightarrow (y_1,y_2), \end{aligned}$$

the coordinate y replaces x verbatim, giving an associated icosahedral group \(\mathcal {I}_y\) with y serving as a parameter. Accordingly, the transformation enjoys an equivariance property:

$$\begin{aligned} S_{Ay}=AS_y \qquad \text {for all }A\in \mathcal {I}_y. \end{aligned}$$

Figure 5 shows each \(S_y\) as a coordinate change from the y-parametrized w-space and icosahedral action \(\mathcal {I}_y^w\) to the fixed x-space with action \(\mathcal {I}^x\).

Fig. 5
figure 5

Parametrizing the icosahedral action. From \(\mathbf {CP}^1_y\), the map S picks up a variable y as a parameter. That is, each y determines a change of variable transformation \(S_y\) from w-space to x-space. For a given y, the group \(\mathcal {I}_y^w\) is the icosahedral action on w-space

With coordinate transformation \(S_y\) in hand, we can construct the generating invariants and equivariants under \(\mathcal {I}_y^w\). Taking the degree-12 invariant

$$\begin{aligned} F(x)=F(S_y{w})=\sum _{k=0}^{12} a_k(y) w_1^{12-k}w_2^k, \end{aligned}$$

the result is a polynomial whose w-degree is 12 while each \(a_k(y)\) has a y-degree of \(12\cdot 31\). Moreover, each \(a_k(y)\) is invariant under \(\mathcal {I}_y\) and thereby expressible as a polynomial \(\hat{a}_k(F(y),H(y))\). Hence, we get

$$\begin{aligned} F(S_y{w})=\sum _{k=0}^{12} \hat{a}_k(F(y),H(y)) w_1^{12-k}w_2^k. \end{aligned}$$

Note that, by degree considerations, the form T(y), having degree 30, cannot appear to an odd power in the invariant expression for \(a_k(y)\) whereas T(y) raised to an even power converts to a polynomial in F(y) and H(y), courtesy of the relation mentioned in Sect. 2. Dividing by \(F(y)^{31}\) “normalizes” F(x) to a degree-zero rational function in y. Recalling that the coordinates x and y behave identically, we can take \(Z=\frac{H(y)^3}{F(y)^5}\) to obtain a Z-parametrized function:

$$\begin{aligned} F_Z(w)=&\frac{F(x)}{F(y)^{31}}= \frac{F(S_y{w})}{F(y)^{ 31}}\biggr |_{H(y)^3\rightarrow F(y)^5 Z}\\ =&\ Z^{-6}\bigl ( 4096000000000000 w_1^{12} Z^3 (16 Z (432 Z (432 Z-95)-437)+57)\\&-204800000000000 w_1^{11} w_2 Z^2 (132 Z (864 Z (216 Z+5)-47)-1)\\&-112640000000000 w_1^{10} w_2^2 Z^2 (8 Z (864 Z (4104 Z+245)-3443)-11)\\&-28160000000000 w_1^9 w_2^3 Z^2 (32 Z (216 Z (3456 Z+833)-4961)-121)\\&-4224000000000 w_1^8 w_2^4 Z^2 (864 Z (20952 Z-1147)-1331)\\&-337920000000 w_1^7 w_2^5 Z^2 (432 Z (131328 Z-18053)-18287)\\&-704000000 w_1^6 w_2^6 Z (48 Z (432 Z (138240 Z-76183)-140479)+1)\\&+211200000 w_1^5 w_2^7 Z (432 Z (3314304 Z+28501)-11)\\&+26400000 w_1^4 w_2^8 Z (432 Z (4202496 Z+89177)-121)\\&+1760000 w_1^3 w_2^9 Z (13824 Z (138240 Z+11477)-1331)\\&+8553600 w_1^2 w_2^{10} Z (6027264 Z-113)\\&+w_1 w_2^{11} (69120 Z (84049920 Z-3077)-20)\\&+w_2^{12} (1769472 Z (172800 Z-11)-11) \bigr ). \end{aligned}$$

To convey a sense of the result, the exact expression is quoted here. The lengthy formulas for subsequent computations are suppressed and can be found at [8]. Applying the same technique generates a function

$$\begin{aligned} H_Z(w)= \frac{H(x)}{H(y)^{31}}= \frac{H(S_y{w})}{H(y)^{31}}\biggr |_{H(y)^3\rightarrow F(y)^5 Z} \end{aligned}$$

whose w-degree is 20. Taking the cross of the parametrized forms produces the generating maps on \(\mathbf {CP}^1_w\):

$$\begin{aligned} \phi _Z(w)= \times _w F_Z(w)\qquad \eta _Z(w)= \times _w H_Z(w). \end{aligned}$$

Next, we develop a Z-parametrized version of g defined on \(\mathbf {CP}^1_w\) the first step of which is to capture how the cross operator transforms under a linear change of coordinates on \(\mathbf {C}^2\). Let \(|A |\) denote the determinant and operator subscripts specify differentiation variables.

Theorem 4.1

The derivative operator \(\times \) satisfies a transformation rule under the coordinate change \(x=Aw\):

$$\begin{aligned} \times _x P(x)= |A|^{-1} A (\times _w P(Aw)). \end{aligned}$$


$$\begin{aligned} \times _x P(x)=&\ J \nabla _x P(Aw)\\ =&\ J \vert A |^{-1} (A^T)^{-1} \nabla _w P(Aw)\qquad [\text {transformation rule for }\nabla ]\\ =&\ |A|^{-1} J (A^T)^{-1} \nabla _w P(Aw)\\ =&\ |A|^{-1} A\, J\, \nabla _w P(Aw)\qquad [\text {Lemma}~2.1]\\ =&\ |A|^{-1} A (\times _w P(Aw)). \end{aligned}$$

\(\square \)

We can regard the constant \(|A|^{-1}\) as projectively meaningless so that the transformation rule establishes a semi-conjugacy:

$$\begin{aligned} f(Aw)=|A|^{-1} A \hat{f}(w)\quad \text {with}\ f(x)= \times _x P(x)\ \text {and}\ \hat{f}(w)=\times _w P(Aw). \end{aligned}$$

Applying the formula derived in Theorem 4.1 to the basic icosahedral maps yields

$$\begin{aligned} \phi (x)=&\ \times _x F(x)\\ =&\ \times _x F(S_y w)\\ =&\ |S_y |^{-1} S_y (\times _w (F(y)^{31} F_Z(w)))\\ =&\ F(y)^{31} |S_y |^{-1} S_y (\times _w F_Z(w))\\ =&\ F(y)^{31} |S_y |^{-1} S_y (\phi _Z(w)) \end{aligned}$$


$$\begin{aligned} \eta (x)=&\ \times _x H(x)\\ =&\ \times _x H(S_y w)\\ =&\ |S_y |^{-1} S_y (\times _w (H(y)^{31} H_Z(w)))\\ =&\ H(y)^{31} |S_y |^{-1} S_y (\times _w H_Z(w))\\ =&\ H(y)^{31} |S_y |^{-1} S_y (\eta _Z(w)). \end{aligned}$$

These transformation properties uncover a map on the w-space that is dynamically equivalent to g(x).

Theorem 4.2

With \(\alpha \) and \(\beta \) as determined in Sect. 3, the map

$$\begin{aligned} {g}_Z(w):=\alpha \,F_Z(w)\eta _Z(w) + \beta \,H_Z(w) \phi _Z(w) \end{aligned}$$

satisfies a projective conjugacy with g(x):

$$\begin{aligned} {g}(S_y w) = \lambda (y)\, S_y g_Z(w)\qquad \lambda (y)\in \mathbf {C}. \end{aligned}$$


Using identities previously derived for the parametrization of basic invariants and equivariants,

$$\begin{aligned} {g}(x)=&\ \alpha \,F(x) \eta (x) + \beta \,H(x) \phi (x)\\ =&\ \alpha \,F(y)^{31} F_Z(w) H(y)^{31} |S_y |^{-1} S_y (\eta _Z(w))\\&+\ \beta \, H(y)^{31} H_Z(w) F(y)^{31}|S_y |^{-1} S_y (\phi _Z(w))\\ {g}(S_y w)=&\ F(y)^{31} H(y)^{31} |S_y |^{-1} S_y \bigl [\alpha \,F_Z(w)\eta _Z(w) + \beta \,H_Z(w) \phi _Z(w))\bigr ]\\ {g}(S_y w)=&\ \lambda (y)\,S_y {g}_Z(w). \end{aligned}$$

\(\square \)

In this formula, y is effectively fixed by the selection of Z. The conjugacy implies that the \({g}_Z\)-orbit of a random initial condition \(w_0\) in \(\mathbf {CP}^1_w\) is asymptotic to a superattracting five-cycle

$$\begin{aligned} (\omega _1, \dots , \omega _5)=(S^{-1}r_1,\dots ,S^{-1}r_5) \end{aligned}$$

determined by \(\mathcal {I}_y^w\). Naturally, \((r_1,\dots ,r_5)\) is a five-cycle of adjacent pentagonal vertices in \(\mathcal {C}_{g}\) under both g(x) and the action of \(\mathcal {I}^x\) on \(\mathbf {CP}^1_x\).

4.3 Root-Selection

The final step is the assembly of an algorithm that uses the global attracting dynamics of \({g}_Z\) and the random nature of the initial condition \(w_0\) to effectively break \(\mathcal {A}_5\) symmetry entirely—precisely the state required in order to obtain all of \(R_Z\)’s roots. To that end, we now fabricate a tool that’s configured to output the roots of a chosen resolvent following a single iterative run of \({g}_Z\).

For each tetrahedral subgroup \(\mathcal {T}_k\), consider the degree-12 family of \(\mathcal {T}_k\) invariants

$$\begin{aligned} h_k(x)=\gamma _k\,t_k(x)^2+\theta _k\,m_k(x). \end{aligned}$$

While \(q_k(x)^3\) is also \(\mathcal {T}_k\)-invariant, we need not include it here in light of the relation that exists between the three tetrahedral forms in degree twelve. Let \(\mathcal {C}_k\) be the twelve-element subset of \(\mathcal {C}_{g}\) that \(\mathcal {T}_k\) preserves and tune one of the parameters \(\gamma _k\) or \(\theta _k\) so that

$$\begin{aligned} h_k(p)={\left\{ \begin{array}{ll}0&{}p\in \mathcal {C}_k\\ c(p)\ne 0&{}p\in \mathcal {C}_{g}-\mathcal {C}_k \end{array}\right. }. \end{aligned}$$

Note that c(p) depends on the remaining parameter as well as which of the 48 points in \(\mathcal {C}_{g}-\mathcal {C}_k\) is selected.

Next, define the “complementary” degree-48 form

$$\begin{aligned} \tilde{h}_k(x)= \frac{\prod _{\ell =1}^5 h_\ell (x)}{h_k(x)} =h_1(x)\dots h_{k-1}(x) h_{k+1}(x)\dots h_5(x). \end{aligned}$$

Spending the parameters that remain, we obtain a normalized degree-zero function

$$\begin{aligned} B_k(x)= \frac{\tilde{h}_k(x)}{F(x)^4} \end{aligned}$$

with specific behavior on the critical set of g(x):

$$\begin{aligned} B_k(p)={\left\{ \begin{array}{ll}1&{}p\in \mathcal {C}_k\\ 0&{}p\in \mathcal {C}_{g}-\mathcal {C}_k \end{array}\right. }. \end{aligned}$$

In practice, forcing \(\tilde{h}_k\) to have the desired properties is more convenient when working with undetermined coefficients in the degree-48 family of tetrahedral invariants in five homogeneous parameters:

$$\begin{aligned} \alpha _1 t_k(x)^8 + \alpha _2 t_k(x)^4 u_k(x)^3 + \alpha _3 u_k(x)^6 + \alpha _4 t_k(x)^6 m_k(x) +\alpha _5 t_k(x)^2 u_k(x) m_k(x). \end{aligned}$$

Pairing the y-parametrized \(B_k(S_y w)\) with the roots \(\rho _k\) of the resolvent \(R_Z\) leads to a root-extraction function in the Z parameter:

$$\begin{aligned} \varGamma _y(w)=&\ \sum B_k(x) \rho _k(y)=\sum B_k(S_y w) \frac{u_k(y) F(y)}{H(y)}\\ =&\ \sum \frac{\tilde{h}_k(S_y w)}{F(S_y w)^4} \frac{u_k(y) F(y)}{H(y)}\\ =&\ \sum \frac{\tilde{h}_k(S_y w)}{(F(y)^{31} F_Z(w))^4} \frac{u_k(y) F(y)}{H(y)}\qquad [F(S_y w)=F(y)^{31} F_Z(w)]\\ =&\ \frac{F(y)}{F(y)^{124} F_Z(w)^4 H(y)} \sum \tilde{h}_k(S_y w) u_k(y)\\ =&\ \frac{\sum \tilde{h}_k(S_y w) u_k(y)}{F(y)^{123} H(y)} \frac{1}{F_Z(w)^4}. \end{aligned}$$

Here, we take a bare summation to run from 1 to 5.

Proposition 4.3

The factor

$$\begin{aligned} \varLambda (y,w)=\sum \tilde{h}_k(S_y w) u_k(y) \end{aligned}$$

is \(\mathcal {I}_y\)-invariant.


Let \(A\in \mathcal {I}_y\). By the \(\mathcal {I}_y\) equivariance of \(S_y w\) as well as the congruent permutation action of \(\mathcal {I}_y\) on \(\tilde{h}_k(S_y w)\) and \(u_k(y)\),

$$\begin{aligned} \varLambda (Ay,w)=&\ \sum \tilde{h}_k(S_{A y} w) u_k(A y)\\ =&\ \sum \tilde{h}_k(A S_y w) u_{\sigma (k)}(y)\\ =&\ \sum \tilde{h}_{\sigma (k)}(S_y w) u_{\sigma (k)}(y)\\ =&\ \varLambda (y,w) \end{aligned}$$

where \(\sigma \) is a permutation on \(\{1,\dots ,5\}\). \(\square \)

By \(\mathcal {I}_y\)-invariance, we obtain a Z-parametrized function

$$\begin{aligned} L_Z(w)=\frac{\varLambda (y,w)}{F(y)^{123} H(y)} \end{aligned}$$

that’s degree-zero in y. Finally, we take

$$\begin{aligned} \varGamma _Z(w)=\frac{L_Z(w)}{F_Z(w)^4} \end{aligned}$$

as a root-extractor associated with the resolvent \(R_Z\). Note that this rational function is degree-zero in w.

To see how the extraction process works, fix a value for y and let \(q\in \mathcal {C}_{{g}_Z}\subset \mathbf {CP}^1_w\) so that

$$\begin{aligned} q=S_y^{-1} p\quad \text {for some}\ p\in \mathcal {C}_j\subset \mathbf {CP}^1_x. \end{aligned}$$

Evaluating the extraction function at q gives

$$\begin{aligned} \varGamma _Z(q)=&\ \frac{L_Z(q)}{F_Z(q)^4}= \frac{\sum \tilde{h}_k(S_y S_y^{-1} p) u_k(y)}{F(y)^{123} H(y) F_Z(q)^4}\\ =&\ \sum \frac{\tilde{h}_k(p)}{F(y)^{124} F_Z(q)^4} \frac{u_k(y) F(y)}{H(y)}\\ =&\ \sum \frac{\tilde{h}_k(p)}{F(y)^{31\cdot 4} F_Z(q)^4} \rho _k(y)\\ =&\ \sum \frac{\tilde{h}_k(p)}{(F(y)^{31} F_Z(q))^4} \rho _k(y)\\ =&\ \sum \frac{\tilde{h}_k(p)}{F(p)^4} \rho _k(y)\qquad [F(x)=F^{31}(y)F_Z(w)]\\ =&\ \sum B_k(p) \rho _k(y)\\ =&\ \rho _j \qquad [p\in \mathcal {C}_j]. \end{aligned}$$

4.4 Algorithm

Machinery for the production of roots to the quintic resolvent \(R_Z\) is now in place. All that remains is to turn the crank.

  1. 1.

    Every mechanism relies on the icosahedral parameter \(Z=\frac{F^5}{H^3}\). First, select a random or arbitrary value \(Z_0\) for this parameter and so, from the family \(R_Z\) obtain a specific quintic

    $$\begin{aligned} R_{Z_0}=v^5-40 Z_0 v^2-5 Z_0 v-Z_0 \end{aligned}$$

    whose roots the procedure estimates. Remark. Solving \(R_{Z_0}=0\) amounts to inverting the quotient map given by \(Z(x)=Z_0\) in as much as the solutions form a single icosahedral orbit

    $$\begin{aligned} Z^{-1}(Z_0)=\{x\in \mathbf {CP}^1 \ \vert \ F(x)^5-Z_0 H(x)^3=0\}. \end{aligned}$$

    That is, with the elements \(x\in Z^{-1}(Z_0)\), we can produce the roots of \(R_{Z_0}\) by evaluating \(\rho _k(x)\) for \(k=1,\dots ,5\). Such an inversion requires that 60-fold \(\mathcal {A}_5\) symmetry fully breaks—an outcome that’s achieved dynamically. Ultimately, this result amounts to the inversion of the elementary symmetric functions that make up the coefficients of the general quintic. [9]

  2. 2.

    Compute the invariants \(F_{Z_0}(w)\) and \(H_{Z_0}(w)\) from which the equivariants \(\phi _{Z_0}(w)\) and \(\eta _{Z_0}(w)\) follow. Then determine the critically-finite map

    $$\begin{aligned} {g}_{Z_0}(w)= \alpha \,F_{Z_0}(w) \eta _{Z_0}(w) + \beta \,H_{Z_0}(w) \phi _{Z_0}(w) \end{aligned}$$

    on \(\mathbf {CP}^1_w\). Remark. The parameter Z is the harness that attaches \(R_Z\) to \({g}_Z\).

  3. 3.

    Randomly select an initial condition \(w_0\in \mathbf {CP}^1_w\) and compute the orbit \(\bigl (g_{Z_0}^k(w_0)\bigr )\) until it produces values \(({\tilde{\omega }}_1,\dots ,{\tilde{\omega }}_5)\) that approximate to high precision a superattracting five-cycle \((\omega _1,\dots ,\omega _5).\) Remark. Random selection of \(w_0\) is the device that breaks \(\mathcal {A}_5\) symmetry in a way that’s analogous to coloring g’s basins. Since the critical points are simple, local convergence to a superattracting 5-cycle is on the order of squaring from one member of a cycle to the next.

  4. 4.

    The final mechanism is the root-extractor \(\varGamma _{Z_0}(w)\) which returns an estimate to arbitrary precision for the roots \(r_k\) of \(R_{Z_0}\):

    $$\begin{aligned} r_k=\varGamma _{Z_0}({\tilde{\omega }}_k),\quad k=1,\dots ,5. \end{aligned}$$

To implement the quintic-solving procedure, the interested reader can obtain a Mathematica notebook with supporting data at [8].