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Next: Appendix. Up: Solving the quintic by Previous: Towers of Algorithms.

Rational Maps with Symmetry.

To compute tex2html_wrap_inline2493 extensions, one must use rational maps with icosahedral symmetry. In this section will construct all rational maps with given symmetries, using invariant polynomials. We then give a conceptual proof of the existence of the map claimed in Lemma 4.5, and also obtain concrete formulas for use in the solution of the quintic.

Let tex2html_wrap_inline2495 be a finite group of Möbius transformations. How can we construct rational maps such that tex2html_wrap_inline2497 ?

Here are three ways to construct such f.

I. Projectively Natural Newton's Method. Ordinary Newton's method applied to a rational function p(z) can be thought of as the map which sends z to tex2html_wrap_inline2505 , where A(z) is the unique automorphism of tex2html_wrap_inline2509 whose 1-jet matches that of p at z. If one replaces A(z) by the unique Möbius\ transformation of tex2html_wrap_inline2517 whose 2-jet agrees with that of p, then the resulting iteration,


is `projectively natural', in the sense that tex2html_wrap_inline2521 for any Möbius transformation tex2html_wrap_inline2523 . Thus tex2html_wrap_inline2525 contains tex2html_wrap_inline2527 whenever p(z) is tex2html_wrap_inline2531 -invariant (and such p are easily constructed).

II. Geometric Constructions. Consider, for example, the case of the icosahedral group. Tile the Riemann sphere by congruent spherical pentagons, in the configuration of a regular dodecahedron (the dual to the icosahedron). Construct a conformal map from each face of the dodecahedron to the complement of its opposite face, taking vertices to opposite vertices. (See Figure 4.) The maps piece together across the boundaries of the faces, yielding a degree 11 rational map f(z) with fixed points at the face centers and critical points at each vertex. Since the notions of `opposite face' and `opposite vertex' are intrinsic, the map commutes with the icosahedral group.

Figure 4: Geometric construction of a rational map.

This construction has many variants. For example, it can be applied to the 20 faces of the icosahedral triangulation, giving a rational map of degree 19, or to the tiling by 30 rhombuses, giving a map of degree 29. (This last tiling, which may be unfamiliar, is by Dirichlet fundamental domains for the 30 edge-midpoints of the dodecahedron. Each rhombus marks the territory which is closer (in the spherical metric) to one of the 30 points than to any other.)

III. Algebraic Constructions. Our final method suffices to produce all rational maps with given symmetries. It will make clear, for example, that the three maps just constructed, together with the identity, are the only maps of degree < 31 with icosahedral symmetry.

Let E be a 2-dimensional complex vector space.

A point p on tex2html_wrap_inline2543 corresponds to a line in E hence to a linear functional with this line as its kernel. A collection of n points corresponds to a homogeneous polynomial of degree n, vanishing along the lines corresponding to the n points. Like the linear map corresponding to a single point, this polynomial is only well-defined up to multiplication by an element of tex2html_wrap_inline2553 .

A rational map tex2html_wrap_inline2555 corresponds to a homogeneous polynomial map tex2html_wrap_inline2557 . X can be obtained by homogenizing the numerator and denominator of f.

Since the tangent space to any point of E is canonically isomorphic to E, X can also be considered as a homogeneous vector field on E.

Now let tex2html_wrap_inline2571 be a finite group, tex2html_wrap_inline2573 its pre-image in the group of linear maps of determinant 1. A vector field X on E is invariant if there exists a character tex2html_wrap_inline2579 such that tex2html_wrap_inline2581 for all tex2html_wrap_inline2583 in tex2html_wrap_inline2585 . X is absolutely invariant if the character is trivial.

The action of tex2html_wrap_inline2589 on vector fields goes over to the action of tex2html_wrap_inline2591 by conjugation on rational maps, establishing:

P 5.1. tex2html_wrap_inline2593 contains tex2html_wrap_inline2595 if and only if the corresponding vector field X(v) is tex2html_wrap_inline2599 -invariant.



1. The possibility of a character arises because f(z) determines X(v) only up to scale.

2. For a 2-dimensional vector space, tex2html_wrap_inline2611 and tex2html_wrap_inline2613 are canonically isomorphic; thus a rational map tex2html_wrap_inline2615 also determines a homogeneous 1-form tex2html_wrap_inline2617 , unique up to scale.

3. A rational map of degree n determines a 1-form tex2html_wrap_inline2621 which is homogeneous of degree n+1; the converse is true unless tex2html_wrap_inline2625 for some homogeneous polynomial g and 1-form tex2html_wrap_inline2629 with tex2html_wrap_inline2631 . In this case the numerator and denominator of the corresponding rational function are not relatively prime.

4. A homogeneous polynomial h(v) determines an exact 1-form dh(v); thus a configuration of n+1 points on tex2html_wrap_inline2639 naturally determines a rational map of degree n.

Let x and y be a basis for tex2html_wrap_inline2647 . The 1-form


is an absolute SL(E) invariant, as well as a primitive for the invariant volume form tex2html_wrap_inline2649 . The rational map corresponding to tex2html_wrap_inline2651 is the identity ( tex2html_wrap_inline2653 annihilates the line through v).

T 5.2. A homogeneous 1-form tex2html_wrap_inline2657 is invariant if and only if


where f and g are invariant homogeneous polynomials with the same character and tex2html_wrap_inline2663 .

Proof. Suppose tex2html_wrap_inline2665 is invariant. The exterior derivative tex2html_wrap_inline2667 , where h(v) is a homogeneous polynomial. Since tex2html_wrap_inline2671 is an absolute invariant of tex2html_wrap_inline2673 , h(v) is invariant with the same character as tex2html_wrap_inline2677 . Setting tex2html_wrap_inline2679 , it is easy to check that tex2html_wrap_inline2681 and hence tex2html_wrap_inline2683 is closed. Integrating this closed form along lines from the origin yields its unique homogeneous primitive g (v); by uniqueness, g(v) is invariant with the same character as tex2html_wrap_inline2689 .

The converse is clear; the condition on degrees assures that the sum is homogeneous.

The construction of invariant rational maps is thus reduced to the problem of invariant homogeneous polynomials. The latter correspond simply to finite sets of points on tex2html_wrap_inline2691 , invariant under tex2html_wrap_inline2693 , and are easily described.

Example: The Icosahedral Group.


Identify the Riemann sphere with a round sphere in tex2html_wrap_inline2709 so that 0 and tex2html_wrap_inline2713 are poles and |z| = 1 is the equator. Inscribe a regular icosahedron in the sphere normalized so one vertex is at 0 and an adjacent vertex lies on the positive real axis (in tex2html_wrap_inline2719 ). Then the isometries of the icosahedron act on tex2html_wrap_inline2721 by a group tex2html_wrap_inline2723 isomorphic to tex2html_wrap_inline2725 . This particular normalization agrees with the conventions of [8] and [2].

Since the abelianization of the binary icosahedral group tex2html_wrap_inline2727 is zero, every invariant is an absolute invariant.

We identify tex2html_wrap_inline2729 with tex2html_wrap_inline2731 , and choose a basis tex2html_wrap_inline2733 for tex2html_wrap_inline2735 such that the coordinate z on tex2html_wrap_inline2739 is equal to x/y.

There are three special orbits for the action of tex2html_wrap_inline2743 : the 12 vertices, 20 face-centers and 30 edge-midpoints of the icosahedron. The corresponding invariant polynomials, derived in [8], are:




Every other orbit has cardinality 60, and corresponds to a linear combination of the degree 60 invariants tex2html_wrap_inline2745 , tex2html_wrap_inline2747 and tex2html_wrap_inline2749 (which satisfy the relation tex2html_wrap_inline2751 ). Thus every homogeneous polynomial invariant under the binary icosahedral group is a polynomial in f, H and T.

P 5.3. There are exactly four rational maps of degree < 31 which commute with the icosahedral group. Then these four maps, of degree 1, 11, 19, and 29 respectively are:





Proof. An invariant rational map of degree < 31 corresponds to an invariant 1-form of degree < 32. The only invariant homogeneous polynomials of degree < 32 are f, H and T. Since no two of their degrees differ by 2, we conclude from Theorem 5.2 that the invariant 1-forms of degree < 32 are proportional to either tex2html_wrap_inline2775 or dg (v), where g is equal to f, H or T. The rational maps corresponding tex2html_wrap_inline2787 are the identity, while those corresponding to df, dH and dT are the other three maps computed above.

Remark. One may glean from the footnote on page 345 of [9] that these maps were known as well to Klein.

Proof of lemma 4.5 (Existence of an Icosahedral Algorithm). Consider the map tex2html_wrap_inline2795 . We claim the critical points of tex2html_wrap_inline2797 reside at the 20 vertices of a spherical regular dodecahedron, and are each mapped to their antipodal vertices under one iteration. This is clear from the geometric construction of tex2html_wrap_inline2799 (method II above).

It can also be verified by counting. tex2html_wrap_inline2801 has 20 critical points, which must be a union of orbits of tex2html_wrap_inline2803 ; the only such orbit corresponds to the vertices of a dodecahedron. Each vertex has an tex2html_wrap_inline2805 stabilizer in tex2html_wrap_inline2807 ; since tex2html_wrap_inline2809 commutes with the group action, the image vertex is fixed by the same subgroup. A simple critical point which is fixed cannot commute with the tex2html_wrap_inline2811 action; hence the corresponding critical value must be the antipodal vertex.

Thus tex2html_wrap_inline2813 is critically finite, and almost every point is attracted to periodic cycles of order two lying at pairs of antipodal vertices. The map tex2html_wrap_inline2815 satisfies the hypotheses of the Lemma.



1. There is a one-parameter family of invariant rational maps of degree 31, which will be used to construct tex2html_wrap_inline2819 in tex2html_wrap_inline2821 in our explicit solution of the quintic.

2. Let p(z) be a polynomial of degree d. Consider radically modified Newton's method:


tex2html_wrap_inline2827 is the unique rational map of degree (d-1) with fixed points at the roots of p and derivative (1-d) at each fixed point. When d = 2, tex2html_wrap_inline2837 is a Möbius transformation of order two fixing the roots of p; for d > 2 the roots are repelling. (Thus tex2html_wrap_inline2843 is not suggested as a method to find roots of p.)

tex2html_wrap_inline2847 coincides with the rational map naturally associated to the roots of p by exterior derivative of the corresponding homogeneous polynomial, as discussed above. This observation will simplify the description of our explicit iterative scheme for the quintic: we need only specify p.

next up previous
Next: Appendix. Up: Solving the quintic by Previous: Towers of Algorithms.

Peter Doyle