Abstract
In the late nineteenth century, Felix Klein revived the problem of solving the quintic equation from the moribund state into which Galois had placed it. Klein’s approach was a mix of algebra and geometry built on the structure of the regular icosahedron. His method’s key feature is the connection between the quintic’s Galois group and the rotational symmetries of the icosahedron. Roughly a century after Klein’s work, Doyle and McMullen developed an algorithm for solving the quintic that also exploited icosahedral symmetry. Their innovation was to employ a symmetrical dynamical system in one complex variable. In effect, the dynamical behavior provides for a partial breaking of the polynomial’s symmetry and the extraction of two roots following one iterative run of the map. The recent discovery of a map whose dynamics breaks all of the quintic’s symmetry allows for five roots to emerge from a single pass. After sketching some algebraic and geometric background, the discussion works out an explicit procedure that deploys the special map in order to solve the quintic in a complete sense.
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1 Overview
Solving a polynomial equation calls for a means to overcome the polynomial’s symmetry. In the case of the fifthdegree 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 facecenters 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 twocycles 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 quinticsolving 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 basinsofattraction 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.
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:
where (x, y) 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 facecenters \(f_k\), and 30 edgemidpoints \(e_k\) respectively. For ease of reference, call the members of these sets “12points, 20points, and 30points.” 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:
Accordingly, F and H are algebraically independent, while an algebraic combination of the two generators in degree 60 vanishes (with multiplicity two) at the 30points:
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 20points 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 nontrivial multiplicative character \(\lambda _T\) appears under a group action \(\mathcal {G}\). That is, for all \(T\in \mathcal {G},\)
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 degreefour 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 facecenters (antipodes of the vertices). The results are complex conjugate forms
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 Klein4 subgroup \(\mathcal {V}\). As for the action of \(\mathcal {T}_5\) on \(q_5\):
and similarly for \(\hat{q}_5\).
An absolute \(\mathcal {T}_5\) invariant results when we take the form that vanishes at the sixpoint tetrahedral orbit associated with edges:
In this case, \(t_5(A(x,y))=t_5(x,y)\) for all \(A\in \mathcal {T}_5\). Furthermore, the product of the degreefour forms yields a degreeeight \(\mathcal {T}_5\) invariant:
Since the eight zeroes of \(u_5\) have orderthree symmetry, they are vertices of an inscribed cube as well as eight of the icosahedron’s facecenters. Note that the vertices of such a cube coincide with the vertices and facecenters 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 \(12=208\):
The tetrahedral invariants satisfy relations in degrees 12 and 24:
Applying powers of an orderfive element \(P\in \mathcal {I}\) manufactures the remaining \(\mathcal {T}_k\) invariants:
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
Using standard invarianttheoretic 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
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
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
yields a \(\mathcal {G}\)equivariant whose degree is \(r1\).
Proof
Express the cross operator as
Take \(\gamma \in \mathcal {G}\) with \(x=\gamma w\). A calculation remains.
\(\square \)
Applying the operator to the generating invariants of \(\mathcal {I}\) produces maps that generate the module of \(\mathcal {I}\)equivariants over \(\mathcal {I}\)invariants:
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 fullmeasure subsets of \(\mathbf {CP}^1\). The DoyleMcMullen iteration uses \(\phi \) whose attracting set is a special orbit—namely, the facecenters. 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 60point \(\mathcal {I}\)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 “31maps” 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 periodfive critical points as well as of other criticallyfinite maps in degree 31. Now, take for g an \(\mathcal {I}\)map whose critical points belong to twelve fivecycles each of which resides at consecutive pentagonal vertices on a \(B_{62}\). Figure 2 depicts a \(B_{62}\) configuration realized by the critical set.
Its analytic form appears once a root
of a homogeneous degree24 equation has been approximated:
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 fivefold, threefold, and twofold rotational symmetry respectively. Figure 3 shows the output of an algorithm worked out in [3] that constructs an approximation to the edgesystem overlaid on a coloring scheme determined by the map’s topological behavior.
Appearing in Fig. 4 are basinofattraction plots that reveal g’s symmetry and global dynamics.
By criticalfiniteness, 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 fivecycle:
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 quinticsolving procedure. The presence of a periodfive attracting set makes for an elegant algorithm.
4 Solving the Quintic
Say that you want to solve the quintic
L. Dickson reduced the general fiveparameter 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 oneparameter 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 invarianttheoretic structure due to the icosahedral action \(\mathcal {I}\) on \(\mathbf {CP}^1\) as described in Sect. 2. Let
where, for clarity’s sake, \(x=(x_1,x_2)\) are homogeneous coordinates replacing the former (x, y). By construction, the degreeeight tetrahedral forms \(u_k\), hence, \(\rho _k\), experience \(\mathcal {A}_5\) permutation under the action of \(\mathcal {I}\).
Next, take the degreezero rational functions \(\rho _k\) as five roots of a polynomial:
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
Since \(\sum _{k=1}^5 u_k\) is degreeeight and there are no such \(\mathcal {I}\)invariants, it turns out that \(b_4=0\). Similarly,
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,
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 \(Z=\frac{F^5}{H^3}\):
Next, we have
where the sum produces an \(\mathcal {I}\)invariant of degree 32, a term expressible as \(b\,F H\). Hence,
Finally, the zerothorder term is
The product of the \(u_i\) has degree 40, giving an \(\mathcal {I}\)invariant \(c H^2\) so that
Evaluating the factors a, b, and c yields a oneparameter family of quintic resolvents
equivalent to Dickson’s fifthdegree resolvent.
Turning to the construction of a quinticsolving 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 selfparametrizing 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
that’s linear in \(w=(w_1,w_2)\) and degree31 in \(y=(y_1,y_2)\). Using the substitution
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:
Figure 5 shows each \(S_y\) as a coordinate change from the yparametrized wspace and icosahedral action \(\mathcal {I}_y^w\) to the fixed xspace with action \(\mathcal {I}^x\).
With coordinate transformation \(S_y\) in hand, we can construct the generating invariants and equivariants under \(\mathcal {I}_y^w\). Taking the degree12 invariant
the result is a polynomial whose wdegree is 12 while each \(a_k(y)\) has a ydegree 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
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 degreezero 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 Zparametrized function:
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
whose wdegree is 20. Taking the cross of the parametrized forms produces the generating maps on \(\mathbf {CP}^1_w\):
Next, we develop a Zparametrized 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 \(x=Aw\):
Proof
\(\square \)
We can regard the constant \(A^{1}\) as projectively meaningless so that the transformation rule establishes a semiconjugacy:
Applying the formula derived in Theorem 4.1 to the basic icosahedral maps yields
and
These transformation properties uncover a map on the wspace that is dynamically equivalent to g(x).
Theorem 4.2
With \(\alpha \) and \(\beta \) as determined in Sect. 3, the map
satisfies a projective conjugacy with g(x):
Proof
Using identities previously derived for the parametrization of basic invariants and equivariants,
\(\square \)
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 fivecycle
determined by \(\mathcal {I}_y^w\). Naturally, \((r_1,\dots ,r_5)\) is a fivecycle 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 RootSelection
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 degree12 family of \(\mathcal {T}_k\) invariants
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 twelveelement subset of \(\mathcal {C}_{g}\) that \(\mathcal {T}_k\) preserves and tune one of the parameters \(\gamma _k\) or \(\theta _k\) so that
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” degree48 form
Spending the parameters that remain, we obtain a normalized degreezero function
with specific behavior on the critical set of g(x):
In practice, forcing \(\tilde{h}_k\) to have the desired properties is more convenient when working with undetermined coefficients in the degree48 family of tetrahedral invariants in five homogeneous parameters:
Pairing the yparametrized \(B_k(S_y w)\) with the roots \(\rho _k\) of the resolvent \(R_Z\) leads to a rootextraction function in the Z parameter:
Here, we take a bare summation to run from 1 to 5.
Proposition 4.3
The factor
is \(\mathcal {I}_y\)invariant.
Proof
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 \(u_k(y)\),
where \(\sigma \) is a permutation on \(\{1,\dots ,5\}\). \(\square \)
By \(\mathcal {I}_y\)invariance, we obtain a Zparametrized function
that’s degreezero in y. Finally, we take
as a rootextractor associated with the resolvent \(R_Z\). Note that this rational function is degreezero 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
Evaluating the extraction function at q gives
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.
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
$$\begin{aligned} R_{Z_0}=v^540 Z_0 v^25 Z_0 vZ_0 \end{aligned}$$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
$$\begin{aligned} Z^{1}(Z_0)=\{x\in \mathbf {CP}^1 \ \vert \ F(x)^5Z_0 H(x)^3=0\}. \end{aligned}$$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 60fold \(\mathcal {A}_5\) 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.
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 criticallyfinite 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.
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 fivecycle \((\omega _1,\dots ,\omega _5).\) 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 5cycle is on the order of squaring from one member of a cycle to the next.

4.
The final mechanism is the rootextractor \(\varGamma _{Z_0}(w)\) 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 quinticsolving procedure, the interested reader can obtain a Mathematica notebook with supporting data at [8].
Change history
13 September 2022
A Correction to this paper has been published: https://doi.org/10.1007/s4400702200032z
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Acknowledgements
The work described in this paper benefited significantly from discussions with Peter Doyle.
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Crass, S. Completely Solving the Quintic by Iteration. La Matematica 1, 829–847 (2022). https://doi.org/10.1007/s4400702200027w
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DOI: https://doi.org/10.1007/s4400702200027w