# Solving the octic by iteration in six dimensions

Scott Crass  Mathematics Department
California State University, Long Beach
Long Beach, CA 90840
<EMAIL_ADDRESS>
###### Abstract.

The requirement for solving a polynomial is a means of breaking its symmetry,
which in the case of the octic, is that of the symmetric group
$\mathcal{S}_{8}$. Its eight-dimensional linear permutation representation
restricts to a six-dimensional projective action. A mapping of complex
projective $6$-space with this $\mathcal{S}_{8}$ symmetry can provide the
requisite symmetry-breaking tool.

The article describes some of the $\mathcal{S}_{8}$ geometry in
$\mathbf{CP}^{6}$ as well as a special $\mathcal{S}_{8}$-symmetric rational
map in degree four. Several basins-of-attraction plots illustrate the map’s
geometric and dynamical properties. The work culminates with an explicit
algorithm that uses this map to solve a general octic. A concluding discussion
treats the generality of this approach to equations in higher degree.

###### Key words and phrases:

complex dynamics, equivariant maps, polynomial equations, symmetric group

##  1\. Overview

In [Crass 2001], I develop a solution to the quintic that relies on a single
iteration in three dimensions. Given almost any quintic $p$, there is a map on
complex projective $3$-space $\mathbf{CP}^{3}$ whose dynamics provides for a
root of $p$. This approach is geometric: the map has the $\mathcal{S}_{5}$
symmetry of the general quintic.

The present paper extends this work to the eighth-degree equation. At its core
is an $\mathcal{S}_{8}$-symmetric map on $\mathbf{CP}^{6}$ whose geometric
behavior is connected to a special configuration of lines. Motivating this
general project is a desire to develop solutions to equations that utilize
symmetrical and geometrically elegant dynamical systems. I do not consider the
numerical aspects of the algorithm. However, since the map converges very
rapidly, numerical considerations might well be of interest. Indeed, the
geometry associated with the map accounts for the rapid convergence.

In addition, the work establishes the existence of a method for all degrees
greater than four that is analogous to the eighth-degree case. This involves
showing that there is an infinite family of maps—one for each dimension
greater than two—with special geometric properties.

Finally, these maps add to the examples of complex dynamics in several
dimensions. This recently-active and difficult field seems to be in need of
examples that are not concocted for purposes of illustration.

The work unfolds in four stages: 1) some background geometry, 2) a special map
with $\mathcal{S}_{8}$ symmetry, 3) a solution to the octic based on the
preceding stages, 4) a consideration of whether the octic algorithm
generalizes to higher degree equations.

Section 2: $\mathcal{S}_{8}$ geometry. The setting here is $\mathbf{CP}^{6}$
upon which the symmetric group $\mathcal{S}_{8}$ acts. Finding a map with
special $\mathcal{S}_{8}$ geometry requires some familiarity with this action.
We will consider some features associated with the map that emerges in the
second stage. Indeed, the discovery of this map derives from an awareness of
the algebraic and geometric surroundings:

  * •

coordinate systems on $\mathbf{CP}^{6}$

  * •

the structure of certain special orbits of points, lines, planes, and
hyperplanes

  * •

the system of $\mathcal{S}_{8}$-invariant polynomials—the building-blocks for
maps that are $\mathcal{S}_{8}$-symmetric.

Section 3: Maps with $\mathcal{S}_{8}$ symmetry. At this stage, we exploit our
geometric understanding to discover empirically a map associated with the
complete graph on eight vertices—an $8$-point $\mathcal{S}_{8}$ orbit. The
discussion turns to its geometric and dynamical behavior—empirical testing
suggests that the $8$-point orbit is the only attractor. However, whether it
possesses this or another desired global dynamical property is not known. In
light of substantial experimental and graphical evidence, I attribute these
properties to the map in conjectures.

Section 4: Dynamical solution to the octic. A special family of octics
corresponds to a _rigid_ family $\mathcal{E}$ of $\mathcal{S}_{8}$-symmetric
maps on $\mathbf{CP}^{6}$. ‘Rigidity’ means that each member of $\mathcal{E}$
is conjugate to a single reference map $f$. Thus, associated with an octic $p$
is a map $g_{p}=\phi_{p}\,f\,\phi_{p}^{-1}$ that we iterate. Using
$\mathcal{S}_{8}$ tools, the dynamical output—conjecturally, a single
$\mathcal{S}_{8}$ orbit—provides for an approximate solution to $p=0$. Since
almost any octic $p$ transforms into the special family, the solution is
general.

Note: Up to this point, the exposition follows that of [Crass 2001] which the
reader can consult for details.

Section 5: Generalization: Solving the $n$th-degree equation by iteration in
$n-2$ dimensions. The geometric and dynamical description of the octic-solving
map has an analogue for each permutation-based $\mathcal{S}_{n}$ action on
$\mathbf{CP}^{n-2}$ with $n\geq 5$. Here, we can show that there is always a
map with the desired special properties. Given such a map for which the
$n$-point orbit is the attractor, the solution algorithm for the octic
generalizes to one for the $n$th degree equation.

##  2\. $\mathcal{S}_{8}$ acts on $\mathbf{CP}^{6}$

The search for a special $\mathcal{S}_{8}$-symmetric map begins with a
faithful action of $\mathcal{S}_{8}$. Klein’s approach to the $n$th degree
equation was to look for the lowest dimensional faithful action of
$\mathcal{S}_{n}$ or the alternating group $\mathcal{A}_{n}$. For $n<8$, there
are special actions of either $\mathcal{S}_{n}$ or $\mathcal{A}_{n}$; that is,
there are faithful representations that do not derive directly from
permutations on $\mathbf{C}^{n}$. However, special geometry—at least for
linear actions—ends at $n=7$. When $n>7$, the space of least dimension on
which $\mathcal{S}_{n}$ or $\mathcal{A}_{n}$ acts faithfully is
$\mathbf{CP}^{n-2}$. [Wiman 1899]

The permutation action of the symmetric group $\mathcal{S}_{8}$ on
$\mathbf{C}^{8}$ preserves the hyperplane

$\mathcal{H}_{x}=\Biggl{\\{}\sum_{k=1}^{8}x_{k}=0\Biggr{\\}}\simeq\mathbf{C}^{7}$

and, thereby, restricts to a faithful seven-dimensional irreducible
representation. This $\mathbf{C}^{7}$ action projects one-to-one to a group
$\mathcal{G}_{8!}$ on $\mathbf{CP}^{6}$.

###  2.1. Coordinates

For many purposes, the most perspicuous geometric description of
$\mathcal{G}_{8!}$ employs eight coordinates that sum to zero. One advantage
is the simple expression of the $\mathcal{S}_{8}$-duality between points and
hyperplanes. In general, for a finite action $\mathcal{G}$ whose matrix
representatives are unitary, a point $a$ is $\mathcal{G}$-_dual_ to the
hyperplane

$\mathcal{L}=\\{\overline{a}\cdot x=0\\}.$

Consequently, $a$ and $\mathcal{L}$ have the same stabilizer in $\mathcal{G}$.
Since the action of $\mathcal{S}_{8}$ on $\mathbf{C}^{7}$ is orthogonal, a
point

$a=[a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},a_{7},a_{8}]_{\sum
a_{k}=0}\in\mathbf{CP}^{6}$

corresponds to the hyperplane

$\\{a\cdot x=0\\}=\Biggl{\\{}\sum_{k=1}^{8}a_{k}\,x_{k}=0\Biggr{\\}}.$

(Square brackets indicate homogeneous coordinates.)

A system of seven _hyperplane coordinates_ describes the hyperplane
$\mathcal{H}_{u}$. It arises from the “hermitian” change of variable

$u=H\,x\qquad x=\overline{H^{T}}u\qquad
H=\frac{1}{2\,\sqrt{2}}\begin{pmatrix}1&\mathbf{\omega}&i&\mathbf{\omega}^{3}&-1&\mathbf{\omega}^{5}&-i&\mathbf{\omega}^{7}\\\
1&i&-1&-i&1&i&-1&-i\\\
1&\mathbf{\omega}^{3}&-i&\mathbf{\omega}&-1&\mathbf{\omega}^{7}&i&\mathbf{\omega}^{5}\\\
1&-1&1&-1&1&-1&1&-1\\\
1&\mathbf{\omega}^{5}&i&\mathbf{\omega}^{7}&-1&\mathbf{\omega}&-i&\mathbf{\omega}^{3}\\\
1&-i&-1&i&1&-i&-1&i\\\
1&\mathbf{\omega}^{7}&-i&\mathbf{\omega}^{5}&-1&\mathbf{\omega}^{3}&i&\mathbf{\omega}\end{pmatrix}$

where $\mathbf{\omega}=e^{\pi\,i/4}$ and the choice of scalar factor gives

(2.1) $H\,\overline{H^{T}}=I_{7}\qquad\overline{H^{T}}H=(a_{ij})\qquad
a_{ij}=\begin{cases}-\frac{7}{8}&i=j\\\ \frac{1}{8}&i\neq j\end{cases}\qquad
i,j=1,\dots 8.$

###  2.2. Invariant polynomials

According to the fundamental result on symmetric functions the $n$ elementary
symmetric functions of degrees one through $n$ generate the ring of
$\mathcal{S}_{n}$-invariant polynomials. Since the $\mathcal{S}_{8}$ action on
$\mathbf{CP}^{6}$ occurs where the degree-$1$ symmetric polynomial vanishes,
there are seven generating $\mathcal{G}_{8!}$-invariants. By Newton’s
identities, the power sums

$F_{k}(x)=\sum_{\ell=1}^{8}x_{\ell}^{k}\qquad k=2,\dots,8$

also generate the $\mathcal{G}_{8!}$ invariants. In hyperplane coordinates,
the _forms_ in degrees two and three are

$\displaystyle\Phi_{2}(u)=$ $\displaystyle\ F_{2}(\overline{H^{T}}u)=\
u_{4}^{2}+2\,u_{3}\,u_{5}+2\,u_{2}\,u_{6}+2\,u_{1}\,u_{7}$
$\displaystyle\Phi_{3}(u)=$ $\displaystyle\
\frac{3}{2\,\sqrt{2}}\,\bigl{(}u_{2}\,u_{3}^{2}+u_{2}^{2}\,u_{4}+2\,u_{1}\,u_{3}\,u_{4}+2\,u_{1}\,u_{2}\,u_{5}+u_{1}^{2}\,u_{6}+u_{5}^{2}\,u_{6}+u_{4}\,u_{6}^{2}$
$\displaystyle\
+2\,u_{4}\,u_{5}\,u_{7}+2\,u_{3}\,u_{6}\,u_{7}+u_{2}\,u_{7}^{2}\bigr{)}$

The remaining five generating invariants $\Phi_{k}(u)$ arise algebraically
from these two. Classical techniques show that a _relative invariant_
—invariant up to a multiplicative character—results from taking the
determinant of the _bordered hessian_

$BH(F,G,J)=\begin{pmatrix}&&&\frac{\partial{G}}{\partial{x_{1}}}\\\
&H(F)&&\vdots\\\ &&&\frac{\partial{G}}{\partial{x_{n}}}\\\
\frac{\partial{J}}{\partial{x_{1}}}&\dots&\frac{\partial{J}}{\partial{x_{n}}}&0\end{pmatrix}$

of three $\mathrm{GL}_{n}(\mathbf{C})$ invariants $F$, $G$, and $J$. The
$n\times n$ matrix $H(F)$ is the hessian of $F$.

######  Proposition 2.1.

Given $T\in\mathrm{GL}_{n}(\mathbf{C})$ and $B(x)=\det{(BH(F(x),G(x),J(x)))}$
for invariants $F,G,J$,

$\displaystyle B(Tx)$ $\displaystyle=(\det{T})^{-2}B(x).$

For the permutation action of $\mathcal{S}_{8}$, this results in an _absolute_
invariant that is expressible in terms of the generators $\Phi_{k}$. The
following result will serve a subsequent computational purpose. (Many of this
work’s results arise from _Mathematica_ computations.)

######  Proposition 2.2.

With

$\displaystyle G_{4}$ $\displaystyle=\det{(BH(\Phi_{2},\Phi_{3},\Phi_{3}))}$
$\displaystyle G_{5}$ $\displaystyle=\det{(BH(\Phi_{2},\Phi_{3},\Phi_{4}))}$
$\displaystyle G_{6}$ $\displaystyle=\det{(BH(\Phi_{2},\Phi_{4},\Phi_{4}))}$
$\displaystyle G_{7}$ $\displaystyle=\det{(BH(\Phi_{2},\Phi_{4},\Phi_{5}))}$
$\displaystyle G_{8}$ $\displaystyle=\det{(BH(\Phi_{2},\Phi_{5},\Phi_{5}))},$

the “power-sum” invariants are given by

$\displaystyle\Phi_{4}$
$\displaystyle=\frac{1}{576}\bigl{(}72\,\Phi_{2}^{2}+G_{4}\bigr{)}$
$\displaystyle\Phi_{5}$
$\displaystyle=\frac{1}{768}\bigl{(}96\,\Phi_{2}\,\Phi_{3}+G_{5}\bigr{)}$
$\displaystyle\Phi_{6}$
$\displaystyle=\frac{1}{960}\bigl{(}120\,\Phi_{2}\,\Phi_{4}+G_{6}\bigr{)}$
$\displaystyle\Phi_{7}$
$\displaystyle=\frac{1}{1280}\bigl{(}160\,\Phi_{3}\,\Phi_{4}+G_{7}\bigr{)}$
$\displaystyle\Phi_{8}$
$\displaystyle=\frac{1}{1600}\bigl{(}200\,\Phi_{4}^{2}+G_{8}\bigr{)}.$

###  2.3. Special orbits

The $6$-dimensional $\mathcal{S}_{8}$ action comes in both real and complex
versions. This means that, in the standard $x$ coordinates, $\mathcal{G}_{8!}$
acts on $\mathcal{R}$—the $\mathbf{RP}^{6}$ of points with real components.
Table 2.1 enumerates some special orbits contained in $\mathcal{R}$. For ease
of expression, I will refer to special points (or lines, planes, etc.) in
terms of the orbit size: “$8$-points” ($28$-lines, $56$-planes,
$28$-hyperplanes). Also, these points receive a symbolic description in
reference to orbit size (superscript) and coordinate expression (subscript).

Corresponding to each special point $a$ is the hyperplane $\\{a\cdot x=0\\}$.
In the case of the $28$-points

$[1,-1,0,0,0,0,0,0],\ \dots\ ,[0,0,0,0,0,0,1,-1],$

there are the $28$-hyperplanes

$\mathcal{L}^{5}_{28_{12}}=\\{x_{1}=x_{2}\\},\ \dots\
,\mathcal{L}^{5}_{28_{78}}=\\{x_{7}=x_{8}\\}.$

The involutions

$x_{1}\leftrightarrow x_{2},\ \dots\ ,x_{7}\leftrightarrow x_{8}$

pointwise fix the respective hyperplanes. These $28$ transpositions generate
$\mathcal{G}_{8!}$ so that it acts as both a _real_ and _complex reflection
group_. (See [Shephard and Todd 1954].)

Various special planes and lines appear as intersections of the
$28$-hyperplanes. Table 2.2 and Table 2.3 summarize the situation. Of
particular dynamical significance is the collection of $28$-lines
$\mathcal{L}^{1}_{28_{ij}}$. This configuration forms the complete graph on
the $8$-points. (See Figure 1 for two views.)

Figure 1. Configuration of $28$-lines and $8$-points Table 2.1. Three special
orbits
$\begin{array}[]{c|c|c|c}\text{Size}&\text{Representative}&\text{Descriptor}&\text{Stabilizer}\\\
\hline\cr&&&\\\ 8&[-7,1,1,1,1,1,1,1]&p^{8}_{1}&\mathcal{S}_{7}\\\\[8.0pt]
28&[0,0,0,0,0,0,1,-1]&p^{28}_{78}&\mathcal{S}_{6}\times\mathbf{Z}_{2}\\\\[8.0pt]
28&[1,1,1,1,1,1,-3,-3]&q^{28}_{78}&\mathcal{S}_{6}\times\mathbf{Z}_{2}\\\\[8.0pt]
\end{array}$ Table 2.2. Some fundamental $\mathbf{CP}^{2}$ orbits
$\begin{array}[]{c|c|c|c|c}&&\text{Set-wise}&\text{Point-
wise}&\text{Restricted}\\\ \text{Geometric
definition}&\text{Descriptor}&\text{stabilizer}&\text{stabilizer}&\text{action}\\\
\hline\cr&&&&\\\
\mathcal{L}^{5}_{28_{ij}}\cap\mathcal{L}^{5}_{28_{ik}}\cap\mathcal{L}^{5}_{28_{i\ell}}\cap\mathcal{L}^{5}_{28_{im}}&\mathcal{L}^{2}_{56}&\mathcal{S}_{5}\times\mathcal{S}_{3}&\mathcal{S}_{5}&\mathcal{S}_{3}\\\\[8.0pt]
\mathcal{L}^{5}_{28_{ij}}\cap\mathcal{L}^{5}_{28_{k\ell}}\cap\mathcal{L}^{5}_{28_{mn}}\cap\mathcal{L}^{5}_{28_{pq}}&\mathcal{L}^{2}_{105}&\mathcal{S}_{4}\times\mathbf{Z}_{2}^{4}&\mathbf{Z}_{2}^{4}&\mathcal{S}_{4}\\\\[8.0pt]
\mathcal{L}^{5}_{28_{ij}}\cap\mathcal{L}^{5}_{28_{ik}}\cap\mathcal{L}^{5}_{28_{\ell
m}}\cap\mathcal{L}^{5}_{28_{\ell
n}}&\mathcal{L}^{2}_{280}&\mathcal{S}_{3}^{2}\times\mathbf{Z}_{2}^{2}&\mathcal{S}_{3}^{2}&\mathbf{Z}_{2}^{2}\\\\[8.0pt]
\mathcal{L}^{5}_{28_{ij}}\cap\mathcal{L}^{5}_{28_{k\ell}}\cap\mathcal{L}^{5}_{28_{km}}\cap\mathcal{L}^{5}_{28_{kn}}&\mathcal{L}^{2}_{420}&\mathcal{S}_{4}\times\mathbf{Z}_{2}^{2}&\mathcal{S}_{4}\times\mathbf{Z}_{2}&\mathbf{Z}_{2}\\\\[8.0pt]
\mathcal{L}^{5}_{28_{ij}}\cap\mathcal{L}^{5}_{28_{k\ell}}\cap\mathcal{L}^{5}_{28_{mn}}\cap\mathcal{L}^{5}_{28_{mp}}&\mathcal{L}^{2}_{840}&\mathcal{S}_{3}\times\mathbf{Z}_{2}^{3}&\mathcal{S}_{3}\times\mathbf{Z}_{2}^{2}&\mathbf{Z}_{2}\par\end{array}$
Table 2.3. Special $\mathbf{CP}^{1}$ orbits
$\begin{array}[]{c|c|c|c|c}&&\text{Set-wise}&\text{Point-
wise}&\text{Restricted}\\\ \text{Geometric
definition}&\text{Descriptor}&\text{stabilizer}&\text{stabilizer}&\text{action}\\\
\hline\cr&&&&\\\
\mathcal{L}^{5}_{28_{ij}}\cap\mathcal{L}^{5}_{28_{ik}}\cap\mathcal{L}^{5}_{28_{i\ell}}\cap\mathcal{L}^{5}_{28_{im}}\cap\mathcal{L}^{5}_{28_{in}}&\mathcal{L}^{1}_{28_{pq}}&\mathcal{S}_{6}\times\mathbf{Z}_{2}&\mathcal{S}_{6}&\mathbf{Z}_{2}\\\
&\scriptstyle{p,q\neq i,\dots,n}&&&\\\\[8.0pt]
\mathcal{L}^{5}_{28_{ij}}\cap\mathcal{L}^{5}_{28_{k\ell}}\cap\mathcal{L}^{5}_{28_{km}}\cap\mathcal{L}^{5}_{28_{kn}}\cap\mathcal{L}^{5}_{28_{kp}}&\mathcal{L}^{1}_{168}&\mathcal{S}_{5}\times\mathbf{Z}_{2}&\mathcal{S}_{5}\times\mathbf{Z}_{2}&\mathbf{Z}_{1}\\\\[8.0pt]
\mathcal{L}^{5}_{28_{ij}}\cap\mathcal{L}^{5}_{28_{k\ell}}\cap\mathcal{L}^{5}_{28_{mn}}\cap\mathcal{L}^{5}_{28_{mp}}\cap\mathcal{L}^{5}_{28_{mq}}&\mathcal{L}^{1}_{210}&\mathcal{S}_{4}\times\mathbf{Z}_{2}^{3}&\mathcal{S}_{4}\times\mathbf{Z}_{2}^{2}&\mathbf{Z}_{2}\\\\[8.0pt]
\mathcal{L}^{5}_{28_{ij}}\cap\mathcal{L}^{5}_{28_{k\ell}}\cap\mathcal{L}^{5}_{28_{km}}\cap\mathcal{L}^{5}_{28_{np}}\cap\mathcal{L}^{5}_{28_{nq}}&\mathcal{L}^{1}_{280}&\mathcal{S}_{3}^{2}\times\mathbf{Z}_{2}^{2}&\mathcal{S}_{3}\times\mathbf{Z}_{2}&\mathbf{Z}_{2}\\\\[8.0pt]
\mathcal{L}^{5}_{28_{ij}}\cap\mathcal{L}^{5}_{28_{ik}}\cap\mathcal{L}^{5}_{28_{\ell
m}}\cap\mathcal{L}^{5}_{28_{\ell n}}\cap\mathcal{L}^{5}_{28_{\ell
p}}&\mathcal{M}^{1}_{280}&\mathcal{S}_{4}\times\mathcal{S}_{3}&\mathcal{S}_{4}\times\mathcal{S}_{3}&\mathbf{Z}_{1}\\\\[8.0pt]
\end{array}$

##  3\. Equivariant maps

The primary tool to be used in solving the general octic is a rational map $f$
with $\mathcal{S}_{8}$ symmetry. In algebraic terms, this means that

$f\circ T=T\,f\quad\text{for all}\ T\in\mathcal{G}_{8!}$

while the geometric upshot is that $f$ sends $\mathcal{G}_{8!}$-orbits to
$\mathcal{G}_{8!}$-orbits. Furthermore, the map should have _reliable
dynamics_ : its attractor

  1. 1)

is a _single_ $\mathcal{G}_{8!}$ orbit

  2. 2)

has a corresponding basin with full measure in $\mathbf{CP}^{6}$ (_strongly_
reliable)

  3. $2^{\prime}$)

alternatively, has a corresponding basin that is dense in $\mathbf{CP}^{6}$
(_weakly_ reliable).

###  3.1. Basic maps

A finite group action $\mathcal{G}$ on $\mathbf{C}^{n}$ induces an action on
the associated exterior algebra. Moreover, $\mathcal{G}$-invariant
$(n-1)$-forms correspond to $\mathcal{G}$-equivariant maps. (See [Crass
1999].)

For a reflection group, the number of generating $0$-forms (that is,
polynomials) is the dimension of the action. [Shephard and Todd 1954, p. 282]
From a result in complex reflection groups, this is also the number of
generating $1$-forms and $(n-1)$-forms. [Orlik and Terao 1992, p. 232] Indeed,
the generating $1$-forms are exterior derivatives of the $0$-forms while the
generating $(n-1)$-forms are wedge products of $1$-forms.

######  Proposition 3.1.

With $X^{k}_{i}=-7\,x_{i}^{k}+\sum_{j\neq i}x_{j}^{k}$, the seven maps

$f_{k}(x)=\bigl{[}X^{k}_{1},X^{k}_{2},X^{k}_{3},X^{k}_{4},X^{k}_{5},X^{k}_{6},X^{k}_{7},X^{k}_{8}\bigr{]}\qquad
k=1,\dots,7$

generate the module of $\mathcal{G}_{8!}$ equivariants over the ring of
$\mathcal{G}_{8!}$ invariants.

These maps are projections onto the hyperplane $\mathcal{H}_{x}$ along
$[1,1,1,1,1,1,1,1]$ of the power maps

$\bigl{[}x_{1}^{k},x_{2}^{k},x_{3}^{k},x_{4}^{k},x_{5}^{k},x_{6}^{k},x_{7}^{k},x_{8}^{k}\bigr{]}.$

######  Proposition 3.2.

Under an orthogonal action an invariant $F(x)$ gives rise to an equivariant
$f(x)$ by means of a formal gradient

$f(x)=\nabla_{x}F(x)=\biggl{[}\frac{\partial F}{\partial
x_{1}}(x),\dots,\frac{\partial F}{\partial x_{n}}(x)\biggr{]}.$

###### Proof.

See [Crass 2001]. ∎

Note that the $\mathcal{G}_{8!}$-equivariant $f_{k}(x)$ is _not_ equal to
$\nabla_{x}F_{k+1}(x)$, but is a multiple of

$\nabla_{x}F_{k+1}(x)|_{x_{i}^{k}\rightarrow X_{i}^{k}}.$

While this may be a source of confusion, it does not cause problems since we
are working on the hyperplane $\mathcal{H}_{x}$. When using hyperplane
coordinates on $\mathcal{H}_{u}$, the discrepancy manifests itself in the
appearance of a constant $-\tfrac{8}{k+1}$ for each map $\phi_{k}(u)$. (See
below.)

Recalling the change of coordinates from $\mathcal{H}_{x}$ to
$\mathcal{H}_{u}$, a map on $\mathcal{H}_{x}$ expresses itself as a map

$\phi(u)=Hf(\overline{H^{T}}u)$

on $\mathcal{H}_{u}$. Having these maps in terms of the basic $u$-invariants
$\Phi_{k}(u)$ will be useful.

######  Definition 3.1.

Let

$R=(r_{ij})=\begin{cases}1&i+j=8\\\
0&\text{otherwise}\end{cases}\quad\text{and}\quad\nabla_{u}^{r}F(u)=R\,\nabla_{u}F(u)$

represent the _reversed identity_ and _reversed gradient_.

######  Proposition 3.3.

In $\mathcal{H}_{u}$ coordinates, the map $\phi(u)=H\,f(\overline{H^{T}}u)$ is
given by

$\phi(u)=\nabla_{u}^{r}\Phi(u)$

where $\Phi(u)=F(\overline{H^{T}}u)=F(x)$ and $f(x)=\nabla_{x}F(x)$.

###### Proof.

[Crass 2001]. ∎

Thus, the generating equivariants $\phi_{k}(u)=Hf_{k}(\overline{H^{T}}u)$ are

(3.2) $\phi_{k}(u)=-\frac{8}{k+1}\,\nabla_{u}^{r}\Phi_{k+1}(u).$

Although the factors $-\tfrac{8}{k+1}$ have no projective effect on the maps
individually, they do play a role when forming combinations of maps from
parametrized families that do not have a common degree. (See Section 4.6.)
Explicit expressions for the maps of degrees one and two are

$\displaystyle\phi_{1}(u)=$ $\displaystyle\
-8\,[u_{1},u_{2},u_{3},u_{4},u_{5},u_{6},u_{7}]$ $\displaystyle\phi_{2}(u)=$
$\displaystyle\
-2\,\sqrt{2}\,\bigl{[}2\,(u_{4}\,u_{5}+u_{3}\,u_{6}+u_{2}\,u_{7}),u_{1}^{2}+u_{5}^{2}+2\,u_{4}\,u_{6}+2\,u_{3}\,u_{7},$
$\displaystyle\
2\,(u_{1}\,u_{2}+u_{5}\,u_{6}+u_{4}\,u_{7}),u_{2}^{2}+2\,u_{1}\,u_{3}+u_{6}^{2}+2\,u_{5}\,u_{7},2\,(u_{2}\,u_{3}+u_{1}\,u_{4}+u_{6}\,u_{7}),$
$\displaystyle\
u_{3}^{2}+2\,u_{2}\,u_{4}+2\,u_{1}\,u_{5}+u_{7}^{2},2\,(u_{3}\,u_{4}+u_{2}\,u_{5}+u_{1}\,u_{6})\bigr{]}.$

The lengthy results for the remaining maps are available at [Crass 2000].

###  3.2. A fixed point property

For a $\mathcal{G}_{8!}$-equivariant $f$ and a point $a$ that an element
$T\in\mathcal{G}_{8!}$ fixes,

$T\,f(a)=f(Ta)=f(a).$

Hence, equivariants preserve fixed points of a group element.

Being pointwise fixed by the involution

$x_{i}\longleftrightarrow x_{j},$

a $28$-hyperplane

$\mathcal{L}^{5}_{28_{ij}}=\\{x_{i}-x_{j}=0\\}$

either maps to itself or collapses to its companion $28$-point

$p^{28}_{ij}=[\dots 0\dots,\overbrace{1}^{i},\dots
0\dots,\overbrace{-1}^{j},\dots 0\dots]\notin\mathcal{L}^{5}_{28_{ij}}.$

In the former generic case, the map preserves the planes and lines that are
intersections of $28$-hyperplanes. (See Tables 2.2 and 2.3.)

###  3.3. Families of equivariants

The $\mathcal{G}_{8!}$ equivariants form a degree-graded module over the
$\mathcal{G}_{8!}$ invariants. This means that for an invariant $F_{\ell}$ and
equivariant $g_{m}$ of degrees $\ell$ and $m$, the product

$F_{\ell}\cdot g_{m}$

is an equivariant of degree $\ell+m$. When looking for a map in a certain
degree $k$ with special geometric or dynamical properties, my approach is to
express the entire family of “$k$-maps” and by manipulation of parameters,
locate a subfamily with the desired behavior.

###  3.4. A special map in degree four

In the configuration of $28$-lines $\mathcal{L}^{1}_{28_{ij}}$ each $8$-point
lies at the intersection of seven lines while each $\mathcal{L}^{1}_{28_{ij}}$
contains the points $p^{8}_{i}$ and $p^{8}_{j}$. (See Section 2.3.) Moreover,
these are the only intersections of $28$-lines. We can attempt to exploit this
structure by looking for a map with superattracting “pipes” along the
$28$-lines: this means that, at each point on the line, the map is critical in
all five “off-line” directions. Under such a map, the $8$-points would be
superattracting in all directions. In addition, we want the map’s degree to be
as small as possible. Degrees two and three avail us of too few parameters.

The family of $4$-maps has (homogeneous) dimension three:

$\alpha_{1}\,\Phi_{3}\,\phi_{1}+\alpha_{2}\,\Phi_{2}\,\phi_{2}+\alpha_{3}\,\phi_{4}.$

Of course, choosing two parameter values determines a map on projective space.
Obtaining a map $g_{4}$ for which the $28$-lines are critical in the off-line
directions requires two parameters. With the third parameter, we choose a lift
of the map to $\mathbf{C}^{7}$ that fixes the $8$-points. The result is

(3.3) $g_{4}=-2\,\Phi_{3}\,\phi_{1}-9\,\Phi_{2}\,\phi_{2}+84\,\phi_{4}.$

The central dynamical role played by the $8$-points suggests that a good
choice of coordinates for this map places these points at

$[1,0,0,0,0,0,0],\ \dots\ ,[0,0,0,0,0,0,1],[1,1,1,1,1,1,1].$

Using $[v_{1},\dots,v_{7}]$ for this system, the map takes the form

$g_{4}(v)=[v_{1}\,T_{1}(v),\dots,v_{7}\,T_{7}(v)]$

where

$T_{k}(v)=7\,v_{k}^{3}-4\,v_{k}^{2}\,S_{1}(\widehat{v_{k}})+2\,v_{k}\,S_{2}(\widehat{v_{k}})-S_{3}(\widehat{v_{k}}),$

$S_{k}$ is the degree-$k$ elementary symmetric function in six variables, and

$\widehat{v_{k}}=(\dots,v_{k-1},v_{k+1},\dots).$

In the $v$-coordinates, the equations

$v_{i}=x,\ v_{j}=y,\ v_{k}=0\qquad\text{for}\ k\neq i,j$

determine $21$ of the $28$-lines $\mathcal{L}^{1}_{28_{ij}}$. For all $k$,
each term in $S_{2}(\widehat{v_{k}})$ and $S_{3}(\widehat{v_{k}})$ contains at
least one $v_{\ell}\ (\ell\neq i,j)$. Thus,

$S_{2}(\widehat{v_{k}})|_{\mathcal{L}^{1}_{28_{ij}}}=0\quad\text{and}\quad
S_{3}(\widehat{v_{k}})|_{\mathcal{L}^{1}_{28_{ij}}}=0$

so that in the inhomogeneous coordinate $z=\tfrac{x}{y}$ on the line the map
restricts to

$z\longrightarrow-z^{3}\,\frac{7\,z-4}{4\,z-7}$

while the pair of $8$-points appear at $0$ and $\infty$.

######  Proposition 3.4.

The $8$-points $p^{8}_{i}$, $p^{8}_{j}$ are the attractor for the restriction
$g=g_{4}|_{\mathcal{L}^{1}_{28_{ij}}}$. Furthermore, in the coordinates used
above, the Julia set $J_{g}$ is the unit circle.

###### Proof.

Since the mobius transformation

$\frac{7\,z-4}{4\,z-7}$

preserves the unit disk,

$|g(z)|<|z|^{3}$

for $|z|<1$. If $|z|<1$, iteration yields

$|g^{n}(z)|<|g^{n-1}(z)|^{3}<\dots<|g(z)|^{3^{n-1}}<|z|^{3^{n}}.$

Thus, every point in the disk belongs to the basin of the superattracting
fixed point $0$. Since $g$ is symmetric under

$z\longrightarrow\frac{1}{z},$

the basin of $\infty$ is $\\{|z|>1\\}$.

The complete invariance of $\\{|z|=1\\}$ implies that it contains and, indeed,
is the map’s Julia set.

∎

Thus, the basins of the $8$-points contain the $28$-lines excepting the
$\mathbf{RP}^{1}$ equator—the unit circle in the coordinates above. Along this
circle $C$, $g_{4}$ has periodic, preperiodic, and what we might call chaotic
saddle points. Attached to each point $z$ on $C$ is a $5$-dimensional stable
manifold $W_{z}$ consisting of points attracted to $C$, that is, to the
trajectory of $z$. Locally, the stable manifolds are mutually disjoint and,
collectively over the circle, give a stable manifold $W_{C}$ of $C$ whose
real-dimension is eleven and that belongs to the basin boundaries of the pair
of $8$-points. We can see (real) $2$-dimensional slices of this stable
manifold by plotting basins of attraction on spaces that are $g_{4}$-invariant
and intersect $W_{C}$. (See Appendix 6.)

By construction, $g_{4}$ self-maps each $\mathcal{S}_{6}$-symmetric
$28$-hyperplane $\mathcal{L}^{5}_{28_{ij}}$ and hence, preserves each
$\mathbf{CP}^{1}$ and $\mathbf{CP}^{2}$ intersection of hyperplanes. Denote
these $1$\- and $2$-dimensional spaces by $\mathcal{L}^{1}_{m}$ and
$\mathcal{L}^{2}_{n}$. Furthermore, $g_{4}$ is equivariant under the
antiholomorphic transformation

$x\longrightarrow\bar{x}$

and, thereby, preserves$\mathcal{R}$—the $\mathcal{S}_{8}$-symmetric
$\mathbf{RP}^{6}$. We can get a picture of the map’s _restricted dynamics_ by
plotting basins of attraction on $\mathcal{L}^{1}_{m}$—a $\mathbf{CP}^{1}$—and
on the $\mathbf{RP}^{2}$ intersections

$\mathcal{L}^{2}_{n}\cap\mathcal{R}.$

The basin portraits appear in Appendix 6. Graphical and experimental evidence
support a claim of reliability for $g_{4}$.

######  Conjecture 3.1.

The attractor for $g_{4}$ is the $8$-point orbit.

######  Conjecture 3.2.

Under $g_{4}$, the basins of the $8$-points fill up $\mathbf{CP}^{6}$ in
measure.

Finally, since $g_{4}$ has real coefficients (see (3.3)), it preserves the
$\mathbf{RP}^{6}$ whose points have real $u$ coordinates. This is _not_ the
$\mathcal{S}_{8}$-symmetric $\mathcal{R}$. Rather it has the $\mathcal{S}_{7}$
symmetry of $p^{8}_{1}$ which is $[1,1,1,1,1,1,1]$ in the $u$ space.
Accordingly, there are eight spaces of this type.

##  4\. Solving the octic

To compute a root of a polynomial, one must overcome its symmetry. For a
general equation of degree $n$ the obstacle is $\mathcal{S}_{n}$. Klein
described a means to this end: given values for an independent set of
$\mathcal{S}_{n}$-invariant homogeneous polynomials

$a_{1}=G_{1}(x)\ \dots\ a_{m}=G_{m}(x),$

find the $\mathcal{S}_{n}$ orbits of solutions $x$ to these equations. [Klein
1913, pp. 69ff] This task of inverting the $G_{k}$ is the _form problem on
$\mathcal{S}_{n}$_. It also has an inhomogeneous manifestation: for $m-1$
given values, invert $m-1$ invariant rational functions of degree zero.

By iterating a reliable $\mathcal{S}_{n}$ equivariant we can break the
obstructing symmetry. In effect, the dynamics provides a mechanism for solving
the form problem and, hence, the $n$th degree equation.

###  4.1. Parameters

The $\mathcal{G}_{8!}$ rational form problem is to solve

$\displaystyle K_{1}$
$\displaystyle=\frac{\Phi_{3}(u)^{2}}{\Phi_{2}(u)^{3}}\quad$ $\displaystyle
K_{2}$ $\displaystyle=\frac{\Phi_{4}(u)}{\Phi_{2}(u)^{2}}\quad$ $\displaystyle
K_{3}$ $\displaystyle=\frac{\Phi_{5}(u)}{\Phi_{2}(u)\,\Phi_{3}(u)}$
$\displaystyle K_{4}$ $\displaystyle=\frac{\Phi_{6}(u)}{\Phi_{2}(u)^{3}}\quad$
$\displaystyle K_{5}$
$\displaystyle=\frac{\Phi_{7}(u)}{\Phi_{2}(u)\,\Phi_{5}(u)}\quad$
$\displaystyle K_{6}$ $\displaystyle=\frac{\Phi_{8}(u)}{\Phi_{2}(u)^{4}}.$

As functions, the $K_{i}$ define the $\mathcal{G}_{8!}$ quotient map

$\displaystyle[K_{1},K_{2},K_{3},K_{4},K_{5},K_{6},1]=$
$\displaystyle\qquad[\Phi_{2}\,\Phi_{3}^{3}\,\Phi_{5},\Phi_{2}^{2}\,\Phi_{3}\,\Phi_{4}\,\Phi_{5},\Phi_{2}^{3}\,\Phi_{5}^{2},\Phi_{2}\,\Phi_{3}\,\Phi_{5}\,\Phi_{6},\Phi_{2}^{3}\,\Phi_{3}\,\Phi_{7},\Phi_{3}\,\Phi_{5}\,\Phi_{8},\Phi_{2}^{4}\,\Phi_{3}\,\Phi_{5}]$

on $\mathbf{CP}^{6}\setminus\\{\Phi_{2}=\Phi_{3}=\Phi_{5}=0\\}$. The generic
fiber over points in $\mathbf{CP}^{6}$ is a $\mathcal{G}_{8!}$ orbit given by

$\displaystyle\\{\Phi_{3}^{2}=K_{1}\,\Phi_{2}^{3}\\}\cap\\{\Phi_{4}=K_{2}\,\Phi_{2}^{2}\\}\cap\\{\Phi_{5}=K_{3}\,\Phi_{2}\,\Phi_{3}\\}$
$\displaystyle\cap\\{\Phi_{6}=K_{4}\,\Phi_{2}^{3}\\}\cap\\{\Phi_{7}=K_{5}\,\Phi_{2}\,\Phi_{5}\\}\cap\\{\Phi_{8}=K_{6}\,\Phi_{2}^{4}\\}.$

Exceptional locations are

$[0,0,1,0,0,0,0],\ [0,0,0,0,1,0,0],\ [0,0,0,0,0,1,0]$

where the respective fibers are the hypersurfaces $\\{\Phi_{3}=0\\}$,
$\\{\Phi_{5}=0\\}$, and $\\{\Phi_{2}=0\\}$. The parameters $K_{i}$ forge a
link between octic equations and $\mathcal{G}_{8!}$ actions. The connection
consists in $K$-parametrizations of each regime.

###  4.2. A family of $\mathcal{S}_{8}$ quintics

Let $\mathcal{G}_{v}$ be a version of $\mathcal{G}_{8!}$ that acts on a
$v$-coordinatized $\mathbf{CP}^{6}_{v}$. This will be a parameter space—the
coordinate $v$ merely stands in for $u$. The linear polynomials

(4.4) $X_{k}(x)=-7\,x_{k}+\sum_{i\neq k}x_{i}$

form an orbit of size eight. Let

$L_{k}(u)=X_{k}(\overline{H^{T}}u).$

Then the rational functions

$\sigma_{k}(v)=\frac{\Phi_{2}(v)\,L_{k}(v)}{\Phi_{3}(v)}$

also give an $8$-orbit. Taking the $\sigma_{k}$ as roots of a polynomial

$R_{v}(s)=\prod_{k=1}^{8}\bigl{(}s-\sigma_{k}(v)\bigr{)}=\sum_{k=0}^{8}C_{k}(v)\,s^{8-k}$

creates a family of octics whose members generically have $\mathcal{S}_{8}$
symmetry. Since $\mathcal{G}_{v}$ permutes the $\sigma_{k}(v)$, each
coefficient $C_{k}(v)$ is $\mathcal{G}_{v}$-invariant and hence, expressible
in the basic forms $\Phi_{k}(v)$. Ultimately, we can express each $C_{k}$ in
terms of the $K_{i}$. By direct calculation,

$\displaystyle C_{0}$ $\displaystyle=1$ $\displaystyle C_{1}$
$\displaystyle=0$ $\displaystyle C_{2}$ $\displaystyle=-\frac{32}{K_{1}}$
$\displaystyle C_{3}$ $\displaystyle=\frac{512}{3\,K_{1}^{2}}$ $\displaystyle
C_{4}$ $\displaystyle=\frac{512\,\bigl{(}1-2\,K_{2}\bigr{)}}{8\,K_{1}^{2}}$
$\displaystyle C_{5}$
$\displaystyle=\frac{16384\,(-5+6\,K_{3})}{15\,K_{1}^{2}}$ $\displaystyle
C_{6}$
$\displaystyle=\frac{-16384\,\bigl{(}-3+8\,K_{1}+18\,K_{2}-24\,K_{4}\bigr{)}}{9\,K_{1}^{3}}$
$\displaystyle C_{7}$
$\displaystyle=\frac{262144\,(35-70\,K_{2}-84\,K_{3}-120\,K_{3}\,K_{5})}{105\,K_{1}^{3}}$
$\displaystyle C_{8}$
$\displaystyle=\frac{131072\,\bigl{(}15-160\,K_{1}+180\,K_{2}+180\,K_{2}^{2}+384\,K_{1}\,K_{3}+480\,K_{4}-720\,K_{6}\bigr{)}}{45\,K_{1}^{4}}.$

Members of the family of octic _resolvents_

$R_{K}(s)=\sum_{n=0}^{8}C_{n}\,s^{8-n}$

parametrized by $K=(K_{1},\dots,K_{6})$ are particularly well-suited for an
iterative solution that employs $g_{4}$. For chosen values of the $K_{i}$, a
solution to the resulting form problem yields a root of $R_{K}$. We can use
$\mathcal{G}_{8!}$ symmetry to find such a solution without explicitly
inverting the $K_{i}$ equations. Our attention will turn to this issue after
we connect the general octic to the special family $R_{K}$.

###  4.3. Reduction of the general octic to the $\mathcal{G}_{8!}$ resolvent

By means of a linear _Tschirnhaus_ transformation

$x\longrightarrow y-\frac{a_{1}}{8}$

the general octic

$p(x)=x^{8}+a_{1}\,x^{7}+a_{2}\,x^{6}+a_{3}\,x^{5}+a_{4}\,x^{4}+a_{5}\,x^{3}+a_{6}\,x^{2}+a_{7}\,x+a_{8}$

becomes the _standard_ $7$-parameter _resolvent_

$q(y)=y^{8}+b_{2}\,y^{6}+b_{3}\,y^{5}+b_{4}\,y^{4}+b_{5}\,y^{3}+b_{6}\,y^{2}+b_{7}\,y+b_{8}$

where

$\displaystyle b_{2}=$ $\displaystyle\ \frac{-7\,a_{1}^{2}+16\,a_{2}}{16}$
$\displaystyle b_{3}=$ $\displaystyle\
\frac{7\,a_{1}^{3}-24\,a_{1}\,a_{2}+32\,a_{3}}{32}$ $\displaystyle b_{4}=$
$\displaystyle\
\frac{-105\,a_{1}^{4}+480\,a_{1}^{2}\,a_{2}-1280\,a_{1}\,a_{3}+2048\,a_{4}}{2048}$
$\displaystyle b_{5}=$ $\displaystyle\
\frac{7\,a_{1}^{5}-40\,a_{1}^{3}\,a_{2}+160\,a_{1}^{2}\,a_{3}-512\,a_{1}\,a_{4}+1024\,a_{5}}{1024}$
$\displaystyle b_{6}=$ $\displaystyle\
\frac{-35\,a_{1}^{6}+240\,a_{1}^{4}\,a_{2}-1280\,a_{1}^{3}\,a_{3}+6144\,a_{1}^{2}\,a_{4}-24576\,a_{1}\,a_{5}+65536\,a_{6}}{65536}$
$\displaystyle b_{7}=$ $\displaystyle\
\frac{3\,a_{1}^{7}-24\,a_{1}^{5}\,a_{2}+160\,a_{1}^{4}\,a_{3}-1024\,a_{1}^{3}\,a_{4}+6144\,a_{1}^{2}\,a_{5}-32768\,a_{1}\,a_{6}+131072\,a_{7}}{131072}$
$\displaystyle b_{8}=$ $\displaystyle\
\frac{-7}{16777216}\,\bigl{(}a_{1}^{8}+64\,a_{1}^{6}\,a_{2}-512\,a_{1}^{5}\,a_{3}+4096\,a_{1}^{4}\,a_{4}-32768\,a_{1}^{3}\,a_{5}+262144\,a_{1}^{2}\,a_{6}$
$\displaystyle-2097152\,a_{1}\,a_{7}+16777216\,a_{8}\bigr{)}.$

Application of another linear _Tschirnhaus_ transformation

$s\longrightarrow\frac{y}{\lambda}$

converts the $6$-parameter family $R_{K}(s)$ into a $\mathcal{G}_{8!}$
resolvent

$\displaystyle\Sigma_{K,\lambda}(y)$
$\displaystyle=\lambda^{8}\,R_{K}\Bigl{(}\frac{y}{\lambda}\Bigr{)}=\sum_{n=0}^{8}\lambda^{n}\,C_{n}\,y^{8-n}$

in the seven parameters $K_{1},\dots,K_{6}$, and the _auxiliary_ $\lambda$.

The functions

$b_{n}=\lambda^{n}\,C_{n}(K)$

relate the coefficients of $q$ and $\Sigma_{K,\lambda}$. The $b_{n}$ invert to

$\displaystyle K_{1}$ $\displaystyle=\frac{-9\,b_{3}^{2}}{8\,b_{2}^{3}}$
$\displaystyle K_{2}$ $\displaystyle=\frac{b_{2}^{2}-2\,b_{4}}{2\,b_{2}^{2}}$
$\displaystyle K_{3}$
$\displaystyle=\frac{5\,\bigl{(}b_{2}\,b_{3}-b_{5}\bigr{)}}{6\,b_{2}\,b_{3}}$
$\displaystyle K_{4}$
$\displaystyle=\frac{2\,b_{2}^{3}-3\,b_{3}^{2}-6\,b_{2}\,b_{4}+6\,b_{6}}{8\,b_{2}^{3}}$
$\displaystyle K_{5}$
$\displaystyle=\frac{7\,\bigl{(}b_{2}^{2}\,b_{3}-b_{3}\,b_{4}-b_{2}\,b_{5}+b_{7}\bigr{)}}{10\,b_{2}\,\bigl{(}b_{2}\,b_{3}-b_{5}\bigr{)}}$
$\displaystyle K_{6}$
$\displaystyle=\frac{b_{2}^{4}-4\,b_{2}\,b_{3}^{2}-4\,b_{2}^{2}\,b_{4}+2\,b_{4}^{2}+4\,b_{3}\,b_{5}+4\,b_{2}\,b_{6}-4\,b_{8}}{8\,b_{2}^{4}}$
$\displaystyle\lambda$ $\displaystyle=\frac{-3\,b_{3}}{16\,b_{2}}.$

Thus, almost any octic descends to a member of $R_{K}$. The reduction fails
when

$-7\,a_{1}^{2}+16\,a_{2}=16\,b_{2}=0\qquad\text{or}\qquad
7\,a_{1}^{3}-24\,a_{1}\,a_{2}+32\,a_{3}=32\,b_{3}=0$

or

$\displaystyle-105\,a_{1}^{5}+600\,a_{1}^{3}\,a_{2}-768\,a_{1}\,a_{2}^{2}-608\,a_{1}^{2}\,a_{3}+1024\,a_{2}\,a_{3}+512\,a_{1}\,a_{4}-1024\,a_{5}$
$\displaystyle\qquad=1024\,(b_{2}\,b_{3}-b_{5})=0.$

A solution to the special resolvent $R_{K}$ then ascends to a solution to the
general quintic.

###  4.4. A family of $\mathcal{S}_{8}$ actions

With the basic $\mathcal{G}_{v}$ forms and maps, we can define the
_parametrized change of coordinates_

$u=\tau_{v}w=\sum_{k=1}^{7}\bigl{(}\Phi_{9-k}(v)\,\phi_{k}(v)\bigr{)}w_{k}.$

For a choice of parameter $v$,

$\tau_{v}:\mathbf{CP}^{6}_{w}\longrightarrow\mathbf{CP}^{6}_{u}$

is linear in $w$ and provides a parametrized family of $\mathcal{G}_{8!}$
groups

$\mathcal{G}_{w}^{v}=\tau_{v}^{-1}\mathcal{G}_{u}\tau_{v}.$

A matrix form results from taking the $\phi_{k}(v)$ as column vectors:

$\begin{pmatrix}u_{1}\\\ \vdots\\\ u_{7}\end{pmatrix}=\begin{pmatrix}&&\\\
\Phi_{8}(v)\,\phi_{1}(v)&\dots&\Phi_{2}(v)\,\phi_{7}(v)\\\
&&\end{pmatrix}\begin{pmatrix}w_{1}\\\ \vdots\\\ w_{7}\end{pmatrix}.$

The setup here is as follows.

  * •

$\mathcal{G}_{u}$ is a version of $\mathcal{G}_{8!}$ that acts on a _reference
space_ $\mathbf{CP}^{6}_{u}$.

  * •

$\mathcal{G}_{v}$ is a version of $\mathcal{G}_{8!}$ that acts on a _parameter
space_ $\mathbf{CP}^{6}_{v}$.

  * •

$\mathcal{G}_{u}$ and $\mathcal{G}_{v}$ have identical expressions in their
respective coordinates.

  * •

$\mathcal{G}_{w}^{v}$ are versions of $\mathcal{G}_{8!}$ that act on _v-
parametrized spaces_ $\mathbf{CP}^{6}_{w}$.

  * •

The iteration that solves octics $R_{K}$ takes place in $\mathbf{CP}^{6}_{w}$.

Each $\mathcal{G}_{w}^{v}$ has its system of invariants and equivariants. From
this point of view we can see, in the resolvents $R_{v}$ and
$\mathcal{G}_{w}^{v}$ equivariants, a connection between octics and dynamical
systems. Furthermore, each $\mathcal{G}_{w}^{v}$ invariant and equivariant is
expressible in the $K_{i}$. This circumstance connects a resolvent $R_{K}$
with $\mathcal{G}_{w}^{v}$-symmetric maps.

By construction, $\tau_{v}w$ possesses an equivariance property:

$\tau_{Av}w=A\,\tau_{v}w\quad\text{for}\ A\in\mathcal{G}_{v},\mathcal{G}_{u}.$

The determinant of $\tau_{v}$ will enter into upcoming calculations and so,
demands some attention. Since

$\det{\tau_{Av}}=\det{A}\,\det{\tau_{v}},$

$\det{\tau_{v}}$ is invariant under the $\mathcal{A}_{8}$ subgroup
$\mathcal{G}_{8!/2}$ of $\mathcal{G}_{v}$ but only relatively invariant under
the full $\mathcal{S}_{8}$ group $\mathcal{G}_{8!}$. (The “even
transformations” have determinant $1$ while the odd elements have determinant
$-1$.) Furthermore,

$\displaystyle\det{\tau_{v}}$
$\displaystyle=\Bigl{(}\prod_{k=2}^{8}\Phi_{k}(v)\Bigr{)}\,\det{\begin{pmatrix}\phi_{1}(v)\
\cdots\ \phi_{7}(v)\end{pmatrix}}$
$\displaystyle=\Bigl{(}\prod_{k=2}^{8}\Phi_{k}(v)\Bigr{)}\,\Psi_{28}(v)$

where $\Psi_{28}$ is, according to a basic result in reflection group theory,
a scalar multiple of the product of the $28$ linear forms associated with the
$28$ hyperplanes that are fixed by the reflections that generate
$\mathcal{G}_{v}$. Furthermore, $\Psi_{28}$ is invariant under the group
$\mathcal{G}_{8!/2}$ (isomorphic to the alternating group $\mathcal{A}_{8}$)
but is relatively invariant under $\mathcal{G}_{8!}$. Consequently, the
degree-$126$ square of $\det{\tau_{v}}$ is $\mathcal{G}_{8!}$-invariant with
$K$-expression

$(\det{\tau_{v}})^{2}=\Phi_{2}^{63}(v)\,t_{K}.$

The explicit form of $t_{K}$ appears at [Crass 2000].

###  4.5. A family of $\mathcal{S}_{8}$ invariants

The equivariance in $v$ of $\tau_{v}w$ implies that $\Phi_{2}(\tau_{v}w)$ is
$\mathcal{G}_{v}$-invariant. Thus, each $w$ coefficient of
$\Phi_{2}(\tau_{v}w)$ inherits the same invariance. Since

$\deg_{v}\Phi_{2}(\tau_{v}w)=\deg_{u}\Phi_{2}(u)\cdot\deg_{v}\tau_{v}w=2\cdot
9=18,$

the rational function

$\frac{\Phi_{2}(u)}{\Phi_{2}(v)^{9}}=\frac{\Phi_{2}(\tau_{v}w)}{\Phi_{2}(v)^{9}}$

is degree zero in $v$ and thereby, expressible in $K$. For each $w$ monomial,
we can solve a system of linear equations whose dimension is that of the
degree-18 $\mathcal{G}_{v}$ invariants. The result is an explicit expression
in $K$ for each $w$-coefficient of $\Phi_{2}(\tau_{v}w)$. Let

(4.5) $\Phi_{2}(v)^{9}\,\Phi_{2_{K}}(w)=\Phi_{2}(u)$

define the basic degree-2 $\mathcal{G}_{w}^{v}$ invariant $\Phi_{2_{K}}(w)$.
Similar considerations apply in degree three where

(4.6) $\Phi_{2}(v)^{12}\,\Phi_{3}(v)\,\Phi_{3_{K}}(w)=\Phi_{3}(u).$

The results appear at [Crass 2000].

By Proposition 2.2, the degree-$4$ and degree-$5$ invariants derive from those
in degrees two and three. The chain rule determines a transformation formula
for the bordered hessian. Let $|\cdot|$ represent the determinant and $A^{T}$
the transpose of $A$.

######  Proposition 4.1.

For $y=Ax$,

$BH_{x}\bigl{(}F(y),G(y),J(y)\bigr{)}=\begin{pmatrix}A^{T}&0\\\
0&1\end{pmatrix}BH_{y}\bigl{(}F(y),G(y),J(y)\bigr{)}\begin{pmatrix}A&0\\\
0&1\end{pmatrix}$

where the subscript indicates the variable of differentiation. Thus,

$\bigl{|}BH_{x}\bigl{(}F(y),G(y),J(y)\bigr{)}\bigr{|}=|A|^{2}\bigl{|}BH_{y}\bigl{(}F(y),G(y),J(y)\bigr{)}\bigr{|}.$

Applied to the parametrized change of variable $w=\tau_{v}^{-1}u$, this
formula along with (4.5) and (4.6) yields

$\displaystyle G_{4}(u)$
$\displaystyle=\bigl{|}BH_{u}\bigl{(}\Phi_{2}(u),\Phi_{3}(u),\Phi_{3}(u)\bigr{)}\bigr{|}$
$\displaystyle=\Bigl{|}BH_{u}\bigl{(}\Phi_{2}(v)^{9}\,\Phi_{2_{K}}(w),\Phi_{2}(v)^{12}\,\Phi_{3}(v)\,\Phi_{3_{K}}(w),\Phi_{2}(v)^{12}\,\Phi_{3}(v)\,\Phi_{3_{K}}(w)\bigr{)}\Bigr{|}$
$\displaystyle=\frac{\Phi_{2}(v)^{6\cdot 9+2\cdot
12}\,\Phi_{3}(v)^{2}}{|\tau_{v}|^{2}}\,\Bigl{|}BH_{w}\bigl{(}\Phi_{2_{K}}(w),\Phi_{3_{K}}(w),\Phi_{3_{K}}(w)\bigr{)}\Bigr{|}$
$\displaystyle=\frac{\Phi_{2}(v)^{78}\,\Phi_{3}(v)^{2}}{\Phi_{2}(v)^{63}\,t_{K}}\,\Bigl{|}BH_{w}\bigl{(}\Phi_{2_{K}}(w),\Phi_{3_{K}}(w),\Phi_{3_{K}}(w)\bigr{)}\Bigr{|}$
$\displaystyle=\frac{\Phi_{2}(v)^{15}\,\Phi_{3}(v)^{2}}{t_{K}}\,\Bigl{|}BH_{w}\bigl{(}\Phi_{2_{K}}(w),\Phi_{3_{K}}(w),\Phi_{3_{K}}(w)\bigr{)}\Bigr{|}$
$\displaystyle=\frac{\Phi_{2}(v)^{18}}{t_{K}}\,\frac{\Phi_{3}(v)^{2}}{\Phi_{2}(v)^{3}}\Bigl{|}BH_{w}\bigl{(}\Phi_{2_{K}}(w),\Phi_{3_{K}}(w),\Phi_{3_{K}}(w)\bigr{)}\Bigr{|}$
$\displaystyle=\Phi_{2}(v)^{18}\,\frac{K_{1}}{t_{K}}\Bigl{|}BH_{w}\bigl{(}\Phi_{2_{K}}(w),\Phi_{3_{K}}(w),\Phi_{3_{K}}(w)\bigr{)}\Bigr{|}$
$\displaystyle=\Phi_{2}(v)^{18}\,G_{4_{K}}(w).$

Using Proposition 2.2 we obtain

$\displaystyle\Phi_{4}(u)$
$\displaystyle=\frac{1}{576}\,\bigl{(}72\,\Phi_{2}(v)^{18}\,\Phi_{2_{K}}(w)^{2}+\Phi_{2}(v)^{18}\,G_{4_{K}}(w)\bigr{)}$
$\displaystyle=\Phi_{2}(v)^{18}\,\Phi_{4_{K}}(w)$

so that

$\displaystyle G_{5}(u)$
$\displaystyle=\bigl{|}BH_{u}\bigl{(}\Phi_{2}(u),\Phi_{3}(u),\Phi_{4}(u)\bigr{)}\bigr{|}$
$\displaystyle=\Bigl{|}BH_{u}\bigl{(}\Phi_{2}(v)^{9}\,\Phi_{2_{K}}(w),\Phi_{2}(v)^{12}\,\Phi_{3}(v)\,\Phi_{3_{K}}(w),\Phi_{2}(v)^{18}\,\Phi_{4_{K}}(w)\bigr{)}\Bigr{|}$
$\displaystyle=\Phi_{2}(v)^{21}\,\Phi_{3}(v)\,G_{5_{K}}(w)$

and

$\displaystyle\Phi_{5}(u)$
$\displaystyle=\frac{1}{768}\,\bigl{(}96\,\Phi_{2}(v)^{9}\,\Phi_{2_{K}}(w)\,\Phi_{2}(v)^{12}\,\Phi_{3}(v)\,\Phi_{3_{K}}(w)+\Phi_{2}(v)^{21}\,\Phi_{3}(v)\,G_{5_{K}}(w)\bigr{)}$
$\displaystyle=\Phi_{2}(v)^{21}\,\Phi_{3}(v)\,\Phi_{5_{K}}(w).$

Employed here are the obvious definitions

$\displaystyle G_{4_{K}}(w)$
$\displaystyle=\frac{K_{1}}{t_{K}}\Bigl{|}BH_{w}\bigl{(}\Phi_{2_{K}}(w),\Phi_{3_{K}}(w),\Phi_{3_{K}}(w)\bigr{)}\Bigr{|}$
$\displaystyle G_{5_{K}}(w)$
$\displaystyle=\frac{1}{t_{K}}\Bigl{|}BH_{w}\bigl{(}\Phi_{2_{K}}(w),\Phi_{3_{K}}(w),\Phi_{4_{K}}(w)\bigr{)}\Bigr{|}$

as well as natural definitions for $\Phi_{4_{K}}(w)$ and $\Phi_{5_{K}}(w)$.

###  4.6. A family of $\mathcal{S}_{8}$ equivariants

Emerging from each $\mathcal{G}_{w}^{v}$ action is a version
$\tau_{v}^{-1}\,g_{4}(\tau_{v}w)$ of $g_{4}(u)$. Being
$\mathcal{G}_{v}$-invariant, these maps also admit parametrization by $K$. In
this way, each octic $R_{K}$ enters into association with a dynamical system
$g_{K}$ on $\mathbf{CP}^{6}_{w}$.

The reversed identity $R$ and gradient $\nabla^{r}=R\,\nabla$ appeared in the
context of a change from eight $x$ coordinates to seven $u$ coordinates. (See
Definition 3.1.) In the present setting, a _reversed transpose_ arises.

######  Definition 4.1.

The _repose_ $A^{r}$ of an $n\times n$ matrix $A$ is its reflection through
the _reversed diagonal_ —the entries whose subscripts sum to $n+1$.
Alternatively,

$A^{r}=R\,A^{T}\,R.$

######  Proposition 4.2.

For a coordinate change $x=Ay$ and a function $F(y)=\tilde{F}(x)$, the
reversed gradient map transforms by

$\nabla_{y}^{r}F(y)=A^{r}\,\nabla_{x}^{r}\tilde{F}(x).$

###### Proof.

See [Crass 2001], Proposition 4.2, but note that the coordinate change there
should be $w=Au$.

∎

Using (3.2), the degree-$1$ $\mathcal{G}_{8!}$ map is

$\displaystyle\phi_{1}(u)$
$\displaystyle=-\frac{8}{2}\nabla_{u}^{r}\Phi_{2}(v)^{9}\,\Phi_{2_{K}}(w)$
$\displaystyle=-4\,\Phi_{2}(v)^{9}\,(\tau_{v}^{-1})^{r}\,\nabla_{w}^{r}\Phi_{2_{K}}(w)$
$\displaystyle=-4\,\Phi_{2}(v)^{9}\,\tau_{v}\,\tau_{v}^{-1}\,(\tau_{v}^{-1})^{r}\,\nabla_{w}^{r}\Phi_{2_{K}}(w)$
$\displaystyle=-4\,\tau_{v}\,\Phi_{2}(v)^{9}\,(\tau_{v}^{r}\,\tau_{v})^{-1}\,\nabla_{w}^{r}\Phi_{2_{K}}(w).$

Thus,

$\tau_{v}^{-1}\,\phi_{1}(\tau_{v}w)=-4\,\Phi_{2}(v)^{9}\,(\tau_{v}^{r}\,\tau_{v})^{-1}\,\nabla_{w}^{r}\Phi_{3_{K}}(w).$

Using the description on the left-hand side, a straightforward calculation
reveals this map to be invariant in $v$ so that the matrix
$\tau_{v}^{r}\,\tau_{v}$ has entries that are degree-$18$ $\mathcal{G}_{v}$
invariants. Hence, the matrix product has the $K$-expression

(4.7)
$\tau_{v}^{r}\,\tau_{v}=\Phi_{2}(v)^{9}\,T_{K}\quad\text{or}\quad(\tau_{v}^{r}\,\tau_{v})^{-1}=\frac{T_{K}^{-1}}{\Phi_{2}(v)^{9}}.$

(See [Crass 2000] for the explicit form.) Also, note that

$\det{T_{K}}=\frac{\det{(\tau_{v}^{r}\,\tau_{v})}}{\Phi_{2}(v)^{9\cdot
7}}=\frac{(\det{\tau_{v}})^{2}}{\Phi_{2}(v)^{63}}=t_{K}.$

Making use of (4.7) to express the transformation of basic equivariants yields

$\displaystyle\phi_{1}(u)$
$\displaystyle=-4\,\Phi_{2}(v)^{9}\,\tau_{v}\,\frac{T_{K}^{-1}}{\Phi_{2}(v)^{9}}\,\nabla_{w}^{r}\Phi_{2_{K}}(w)$
$\displaystyle=-4\,\tau_{v}\,T_{K}^{-1}\,\nabla_{w}^{r}\Phi_{2_{K}}(w).$

As for the other relevant maps, similar calculations give

$\displaystyle\phi_{2}(u)$
$\displaystyle=-\frac{8}{3}\,\Phi_{2}(v)^{12}\,\Phi_{3}(v)\,\tau_{v}\,\frac{T_{K}^{-1}}{\Phi_{2}(v)^{9}}\,\nabla_{w}^{r}\Phi_{3_{K}}(w)$
$\displaystyle=-\frac{8}{3}\,\Phi_{2}(v)^{3}\,\Phi_{3}(v)\,\tau_{v}\,T_{K}^{-1}\,\nabla_{w}^{r}\Phi_{3_{K}}(w)$
$\displaystyle\phi_{4}(u)$
$\displaystyle=-\frac{8}{4}\,\Phi_{2}(v)^{21}\,\Phi_{3}(v)\,\tau_{v}\,\frac{T_{K}^{-1}}{\Phi_{2}(v)^{9}}\,\nabla_{w}^{r}\Phi_{5_{K}}(w)$
$\displaystyle=-2\,\Phi_{2}(v)^{12}\,\Phi_{3}(v)\,\tau_{v}\,T_{K}^{-1}\,\nabla_{w}^{r}\Phi_{5_{K}}(w).$

Finally, we can identify a $K$-parametrized $4$-map $g_{K}(w)$ that is
conjugate to $g_{4}(u)$. The map’s expression in basic terms appears after
substitution into the formula found in Section 3.4:

$\displaystyle g_{4}(u)=$ $\displaystyle\
84\,\phi_{4}(u)-9\,\Phi_{2}(u)\,\phi_{2}(u)-2\,\Phi_{3}(u)\,\phi_{1}(u)$
$\displaystyle=$ $\displaystyle\
-8\,\Phi_{2}^{12}(v)\,\Phi_{3}(v)\,\tau_{v}\,T_{K}^{-1}\,\Bigl{(}21\,\nabla_{w}^{r}\Phi_{5_{K}}(w)$
$\displaystyle\
-3\,\Phi_{2_{K}}(w)\,\nabla_{w}^{r}\Phi_{3_{K}}(w)-\Phi_{3_{K}}(w)\,\nabla_{w}^{r}\Phi_{2_{K}}(w)\Bigr{)}.$

Thus, we have a $K$-parametrized family of $4$-maps on $\mathbf{CP}^{6}_{w}$:

$g_{K}(w)=T_{K}^{-1}\,\Bigl{(}21\,\nabla_{w}^{r}\Phi_{5_{K}}(w)-3\,\Phi_{2_{K}}(w)\,\nabla_{w}^{r}\Phi_{3_{K}}(w)-\Phi_{3_{K}}(w)\,\nabla_{w}^{r}\Phi_{2_{K}}(w)\Bigr{)}$

whose relation to the reference $4$-map is

$g_{4}(u)=-8\,\Phi_{2}^{12}(v)\,\Phi_{3}(v)\,\tau_{v}\,g_{K}(w).$

###  4.7. Root selection

Being conjugate to $g_{4}(u)$ each $g_{K}(w)$ shares the former’s conjectured
reliable dynamics. Accordingly, the attractor for each choice of $K$ is the
$8$-point orbit in the corresponding $\mathbf{CP}^{6}_{w}$ so that for almost
every $w_{0}\in\mathbf{CP}^{6}_{w}$,

$g_{K}^{n}(w_{0})\longrightarrow\tau_{v}^{-1}p^{8}_{\ell}\qquad\text{for some
$8$-point}\ p^{8}_{\ell}\in\mathbf{CP}^{6}_{u}.$

To solve the resolvent $R_{K}$, the output of the iteration must link with the
roots of $R_{K}$. From here, we see that solving $R_{K}$ amounts to inverting
$\tau_{v}$—the form problem in another guise. This is effectively what the
dynamics of $g_{K}$ accomplishes with the assistance of a $\mathcal{G}_{8!}$
tool that I now describe.

The quadratic $\mathcal{S}_{7}$-invariants

$X^{2}_{k}(x)=-7\,x_{k}^{2}+\sum_{i\neq k}x_{i}^{2}$

form a $\mathcal{G}_{8!}$ orbit of size eight. Recall that

$L_{k}(u)=X_{k}(\overline{H^{T}}u)$

and let

$Q_{k}(u)=X^{2}_{k}(\overline{H^{T}}u)$

be the $\mathcal{H}_{u}$ expression for $X_{k}^{2}$. Furthermore, each of the
eight forms

$G_{k}(u)=Q_{k}(u)-\frac{3}{4}\,L_{k}(u)^{2}\quad k=1,\dots,8$

vanish at the $8$-points $p^{8}_{\ell}$ with $\ell\neq k$ but not at
$p^{8}_{k}$.

Now, consider the rational function

$J_{v}(w)=\alpha\,\sum_{k=1}^{8}\frac{G_{k}(\tau_{v}w)}{\Phi_{2}(\tau_{v}w)}\,\frac{\Phi_{2}(v)\,L_{k}(v)}{\Phi_{3}(v)}=\alpha\,\sum_{k=1}^{8}\frac{G_{k}(\tau_{v}w)}{\Phi_{2}(\tau_{v}w)}\,\sigma_{k}(v)$

where $\alpha$ is a constant to be determined. Since the $v$-degree of the
numerator and denominator is $21=2\cdot 9+3$ while the $w$-degree is $2$, the
function is rationally degree zero in both variables. At an $8$-point
$\tau_{v}^{-1}p^{8}_{\ell}$ in $\mathbf{CP}^{6}_{w}$ seven of the eight terms
in $J_{v}$ vanish; this leaves

$\alpha\,\frac{G_{\ell}(p^{8}_{\ell})}{\Phi_{2}(p^{8}_{\ell})}\,\sigma_{\ell}(v).$

Choosing

$\alpha=\frac{\Phi_{2}(p^{8}_{k})}{G_{k}(p^{8}_{k})}=\frac{1}{48}\qquad(k=1,\dots,8)$

“selects” the root $\sigma_{\ell}(v)$ of $R_{K}(s)$. Since the iterative
“output” of $g_{K}(w)$ is a single $8$-point in $\mathbf{CP}^{6}_{w}$, the
dynamics produces one root.

To obtain a usable form of the root-selector $J_{v}(w)$, let

$\Gamma_{v}(w)=\sum_{k=1}^{8}G_{k}(\tau_{v}w)\,L_{k}(v).$

Since $\mathcal{G}_{v}$ permutes its terms, $\Gamma_{v}$ is invariant under
the action and hence, expressible in $K$:

$\Gamma_{v}(w)=\Phi_{2}(v)^{8}\,\Phi_{3}(v)\,\Gamma_{K}(w).$

(The explicit form of $\Gamma_{K}$ appears at [Crass 2000].) Application of
(4.5) yields

$\displaystyle J_{v}(w)$
$\displaystyle=\frac{\Phi_{2}(v)\,\Gamma_{v}(w)}{48\,\Phi_{3}(v)\,\Phi_{2}(\tau_{v}w)}$
$\displaystyle J_{K}(w)$
$\displaystyle=\frac{\Gamma_{K}(w)}{48\,\Phi_{2_{K}}(w)}.$

###  4.8. The procedure summarized

At [Crass 2000], there are _Mathematica_ data files and a notebook that
implement the iterative solution to the octic.

  1. (1)

Select a general $8$-parameter octic $p(x)$.

  2. (2)

Tschirnhaus transform $p(x)$ into a member $R_{K}(s)$ of the $6$-parameter
family of $\mathcal{G}_{8!}$ octics—this determines values for
$K_{1},\dots,K_{6}$ as well as the auxiliary parameter $\lambda$.

  3. (3)

For the selected $K$ values, compute the matrix $T_{K}$, the invariants
$\Phi_{2_{K}}(w)$ and $\Phi_{3_{K}}(w)$, the $4$-map $g_{K}(w)$, the form
$\Gamma_{K}(w)$, and the root-selector $J_{K}(w)$.

  4. (4)

From an arbitrary initial point $w_{0}$ iterate $g_{K}$ until convergence:

$g_{K}^{n}(w_{0})\longrightarrow w_{\infty}.$

Conjecturally, the output $w_{\infty}$ is an $8$-point in
$\mathbf{CP}^{6}_{w}$.

  5. (5)

Compute a root $\sigma=J_{K}(w_{\infty})$ of $R_{K}$.

  6. (6)

Transform $\sigma$ into a root of $p(x)$.

##  5\. Beyond the octic

###  5.1. The general case

In the following discussion the $\mathcal{S}_{n}$ actions under consideration
derive from permutation of coordinates on the $\mathcal{S}_{n}$-invariant
hyperplanes

$\mathcal{H}^{n-1}=\Biggl{\\{}\sum_{k=1}^{n}x_{k}=0\Biggr{\\}}.$

These irreducible representations of $\mathcal{S}_{n}$ project to actions
$\mathcal{G}_{n!}$ on $\mathbf{P}\mathcal{H}^{n-1}\simeq\mathbf{CP}^{n-2}$.

For equations of degree $n\neq 8$, does the analogue of the octic-solving
algorithm exist? Evidently, the reduction of the $n$th degree polynomial to an
$(n-2)$-parameter family of $\mathcal{G}_{n!}$ resolvents is general.

######  Query 5.1.

Is there an $\mathcal{S}_{n}$-equivariant $4$-map on
$\mathbf{P}\mathcal{H}^{n-1}$ that, on a $\binom{n}{2}$-line $\mathcal{L}^{1}$
(where all but two coordinates are equal):

  * •

superattracts in the off-line directions

  * •

restricts to a reliable map whose attractor consists of the pair of $n$-points
on $\mathcal{L}^{1}$?

######  Query 5.2.

If so, is the map expressible as

$z\longrightarrow z^{3}\,\frac{a\,z+b}{b\,z+c}\qquad
a,b\in\mathbf{R}\quad\text{and}\ |b|<|a|$

when restricted to $\mathcal{L}^{1}$?

The affine space $\\{x_{n}\neq 0\\}$ parametrized by

$\Bigl{[}x_{1},\dots,x_{n-2},-\sum_{k=1}^{n-2}x_{k}-1,1\Bigr{]}$

is tangent to $\mathbf{P}\mathcal{H}^{n-1}$ at the affine part of
$\mathcal{L}^{1}$ given by

$[\zeta,\dots,\zeta,(2-n)\,\zeta-1,1].$

We can identify these spaces respectively with

$\\{(x_{1},\dots,x_{n-2})\\}\simeq\mathbf{C}^{n-2}\quad\text{and}\quad\\{(\zeta,\dots,\zeta)\\}\simeq\mathbf{C}.$

######  Definition 5.1.

Abusing notation, let $\mathcal{L}^{1}$ be the affine part of an
$\binom{n}{2}$-line. Use $\mathcal{T}_{\mathcal{L}^{1}}$ to denote the tangent
space to $\mathbf{P}\mathcal{H}^{n-1}$ along $\mathcal{L}^{1}$. Also,

$\mathcal{L}^{1}_{\perp}=\Biggl{\\{}\sum_{k=1}^{n-2}x_{k}=0\Biggr{\\}}\simeq\mathbf{C}^{n-3}$

is the (euclidean) orthogonal complement in $\mathcal{T}_{\mathcal{L}^{1}}$ to
$\mathcal{L}^{1}$.

The subgroup $S_{\mathcal{L}^{1}}$ of $\mathcal{G}_{n!}$ that stabilizes
$\mathcal{T}_{\mathcal{L}^{1}}$ is isomorphic to $\mathcal{S}_{n-2}$ and acts
by permutations on $\mathbf{C}^{n-2}$. The action of $S_{\mathcal{L}^{1}}$
fixes $\mathcal{T}_{\mathcal{L}^{1}}$ and $\mathcal{L}^{1}_{\perp}$ set-wise
and fixes $\mathcal{L}^{1}$ in a point-wise manner.

######  Remark.

In the treatment of $\mathcal{L}^{1}$ below, we need not worry about the point
at infinity

$[1,\dots,1,2-n,0],$

since this point is in the same $\mathcal{G}_{n!}$-orbit as the affine point

$[1,\dots,1,0,2-n]$

which we have identified with

$\frac{1}{2-n}\,(1,\dots,1).$

For $n\geq 5$, the family of $4$-maps on $\mathbf{P}\mathcal{H}^{n-1}$ is, in
homogeneous parameters,

$g_{\alpha}=\alpha_{1}\,f_{4}+\alpha_{2}\,F_{2}\,f_{2}+\alpha_{3}\,F_{3}\,f_{1}$

where definitions of $F_{k}$ and $f_{k}$ are obvious extensions from the
$\mathcal{G}_{8!}$ case. (For $n<5$, the family of $4$-maps is not
$3$-dimensional.) Do three parameters suffice to obtain a map $g_{\alpha}$
with jacobian matrix $g_{\alpha}^{\prime}(z)$ at $z$ such that

$g_{\alpha}^{\prime}(z)\,\mathcal{L}^{1}_{\perp}=0\qquad\text{for all}\
z\in\mathcal{L}^{1}?$

First of all, symmetry demands that, for each $g_{\alpha}$ and all
$z\in\mathcal{L}^{1}$, there is only one “off-line” eigenvalue of
$g_{\alpha}^{\prime}(z)$.

######  Lemma 5.1.

For an action $\mathcal{G}$ on $\mathbf{CP}^{n}$ and a
$\mathcal{G}$-equivariant $f$ that is holomorphic at $a$, the jacobian
$f^{\prime}(a)$ is equivariant on the tangent space
$\mathcal{T}_{a}\simeq\mathbf{C}^{n}$ under the stabilizer $\mathcal{S}_{a}$
of $a$.

###### Proof.

After treating several technical matters, the proof amounts to a simple
calculation.

  1. (1)

Take $a$ to be $[0,\dots,0,1]$ and $\mathcal{T}_{a}$ to be $\\{x_{n+1}\neq
0\\}$ and lift them to

$\hat{a}=(0,\dots,0,1)\qquad\text{and}\qquad\widehat{\mathcal{T}{a}}=\\{x_{n+1}=1\\}.$

  2. (2)

Strictly speaking, $\mathcal{S}_{a}$ is a group of projective transformations.
Here, we choose linear representatives $T$ of $\mathcal{S}_{a}$ that, as maps
on $\mathbf{C}^{n+1}$, satisfy $Ta=a$. This group of linear transformations
acts on $\widehat{\mathcal{T}_{a}}$.

  3. (3)

For a homogeneous polynomial $G(x)$ of degree $r$,

$r\,G(x)=\bigl{(}\nabla_{x}G(x)\bigr{)}^{T}x$

is a familiar identity. Generalized to a rational map $g(x)$, the result is

$r\,g(x)=g^{\prime}(x)\,x.$

Given $T\in\mathcal{G}$, let $\hat{T}$ be a lift to $\mathbf{C}^{n+1}$. Then
for all $x\in\mathbf{C}^{n+1}$,

$\displaystyle\bigl{(}f^{\prime}(Tx)\,T\bigr{)}x$
$\displaystyle=\bigl{(}f^{\prime}(Tx)\bigr{)}Tx$ $\displaystyle=r\,f(Tx)$
$\displaystyle=T(r\,f(x))$ $\displaystyle=T(f^{\prime}(x)\,x)$
$\displaystyle=\bigl{(}Tf^{\prime}(x)\bigr{)}x.$

Thus,

$f^{\prime}(Tx)\,T=Tf^{\prime}(x).$

In particular, when $x=a$ and $T\in\mathcal{S}_{a}$,

$f^{\prime}(a)\,T=f^{\prime}(Ta)\,T=T\,f^{\prime}(a).$

∎

######  Lemma 5.2.

A linear $S_{\mathcal{L}^{1}}$-equivariant $T$ preserves
$\mathcal{L}^{1}_{\perp}$.

###### Proof.

The transformation

$A:\mathcal{T}_{\mathcal{L}^{1}}\longrightarrow\mathcal{T}_{\mathcal{L}^{1}}$

that cyclically permutes coordinates according to $(12\dots(n-3)(n-2))$ has
$n-2$ eigenspaces, namely, the lines $\mathcal{L}^{1}_{k}$ given by

$\mathcal{L}^{1}_{k}=\Biggl{\\{}t\,w_{k}\ |\ t\in\mathbf{C},\
w_{k}=\begin{pmatrix}1\\\ \mathbf{\omega}^{k}\\\ \mathbf{\omega}^{2\,k}\\\
\vdots\\\ \mathbf{\omega}^{(n-3)\,k}\end{pmatrix},\
\mathbf{\omega}=e^{2\,\pi\,i/(n-2)}\Biggr{\\}}\quad k=1,\dots,n-2.$

Note that $\mathcal{L}^{1}$=$\mathcal{L}^{1}_{n-2}$ and $\\{w_{k}\ |\
k=1,\dots,n-3\\}$ is a basis for $\mathcal{L}^{1}_{\perp}$. Moreover, each
line $\mathcal{L}^{1}_{k}$ has the eigenvalue $\mathbf{\omega}^{(n-3)\,k}$.

Since $A\in S_{\mathcal{L}^{1}}$ and

$A(Tw_{k})=T(Aw_{k})=\mathbf{\omega}^{(n-3)\,k}\,Tw_{k},$

$Tw_{k}$ is also an $\mathbf{\omega}^{(n-3)\,k}$-eigenvector of $A$. Thus,
each $\mathcal{L}^{1}_{k}$ is an eigenspace of $T$ so that
$\mathcal{L}^{1}_{\perp}$ is $T$-invariant.

∎

######  Proposition 5.1.

For all $z\in\mathcal{L}^{1}$, the eigenvectors of the jacobian
$g_{\alpha}^{\prime}(z)$ span $\mathcal{L}^{1}_{\perp}$. Moreover, all
associated eigenvalues are equal, making $\mathcal{L}^{1}_{\perp}$ an
eigenspace of $g_{\alpha}^{\prime}(z)$.

###### Proof.

Recall that the stabilizer $\mathcal{S}_{\mathcal{L}^{1}}$ is isomorphic to
$\mathcal{S}_{n-2}$. (The $n-2$ things that $\mathcal{S}_{\mathcal{L}^{1}}$
permutes are the vectors $w_{k}$ defined above.) By Lemmas 5.1 and 5.2, let
$v\in\mathcal{L}^{1}_{\perp}$ be an eigenvector of $g_{\alpha}^{\prime}(z)$
with eigenvalue $\lambda$. For $A\in S_{\mathcal{L}^{1}}$, Lemma 5.1 gives

$g_{\alpha}^{\prime}(z)Av=A\,g_{\alpha}^{\prime}(z)\,v=\lambda\,Av.$

Hence, $Av$ is also a $\lambda$-eigenvector. Clearly, $\\{Av\,|\,A\in
S_{\mathcal{L}^{1}}\\}$ spans $\mathcal{L}^{1}_{\perp}$.

∎

The question now is whether there is always some parameter-choice for which
the eigenvalue of $\mathcal{L}^{1}_{\perp}$ vanishes for all $z$.

######  Proposition 5.2.

For $n\geq 5$, there is a $4$-map $g$ whose critical set includes the
$\binom{n}{2}$-lines. Moreover, at each point on an $\binom{n}{2}$-line, $g$
is critical in every direction away from the line.

###### Proof.

To facilitate exposition, we work in the linear space $\mathcal{H}^{n-1}$. For
an arbitrary member of the family $g_{\alpha}$, select a lift
$\widehat{g_{\alpha}}$ to $\mathcal{H}^{n-1}$. The line $\mathcal{L}^{1}$
lifts to the plane $\widehat{\mathcal{L}^{1}}$ parametrized by

(5.8) $(x+y,\ \dots,\ ,x+y,(1-n)\,x+y,x+(1-n)\,y).$

Furthermore, the orthogonal complement $\widehat{\mathcal{L}^{1}}_{\perp}$ in
$\mathcal{H}^{n-1}$ is

$\Biggl{\\{}\sum_{k=1}^{n-2}x_{k}=0,x_{n-1}=x_{n}=0\Biggr{\\}}.$

By symmetry, we can consider a single line. Using the parametrization above,
the pair of $n$-points on $\mathcal{L}^{1}$ correspond to the lines $x=0$ and
$y=0$. Meanwhile, the line specified by

$x+y=0$

determines an element in one of the special orbits of $\binom{n}{2}$-points.

Proposition 5.1 implies that the characteristic polynomial for the jacobian of
$\widehat{g_{\alpha}}$, when restricted to $\widehat{\mathcal{L}^{1}}$, has
the form

$\chi_{\widehat{g_{\alpha}}^{\prime}}|_{\widehat{\mathcal{L}^{1}}}=\det{(t\,I_{n}-\widehat{g_{\alpha}}^{\prime})}|_{\widehat{\mathcal{L}^{1}}}=\
\bigl{(}t-A(x,y)\bigr{)}^{n-3}\bigl{(}t^{2}-B(x,y)\,t+C(x,y)\bigr{)}$

Note that the factor

$t^{2}+B\,t+C$

is the characteristic polynomial of the jacobian of the map
$\widehat{g_{\alpha}}|_{\widehat{\mathcal{L}^{1}}}$. Hence,

$B(x,y)=B_{1}\,x^{3}+B_{2}\,x^{2}y+B_{3}\,xy^{2}+B_{4}\,y^{3}$

where the $B_{i}$ are linear in the parameters $\alpha_{j}$. It follows that

$A(x,y)=A_{1}\,x^{3}+A_{2}\,x^{2}y+A_{3}\,xy^{2}+A_{4}\,y^{3}$

where the $A_{i}$ are linear in the $\alpha_{j}$.

The polynomial $A$ gives the eigenvalue in the off-line directions in
$\mathbf{P}\mathcal{H}^{n-1}$. The remaining factor corresponds to behavior
along $\mathcal{L}^{1}$. Our interest here is $A$. In particular, we want to
force it to vanish identically in $x$ and $y$.

Since $\chi_{\widehat{g_{\alpha}}^{\prime}}$ is invariant under permutation of
coordinates, its restriction to $\widehat{\mathcal{L}^{1}}$ is invariant under
the interchange $x\leftrightarrow y$ (which corresponds to the transposition
$((n-1)\,n)$ in (5.8)). Accordingly, $A_{1}=A_{4}$ and $A_{2}=A_{3}$ so that

$\displaystyle A$ $\displaystyle=\ A_{1}\,(x^{3}+y^{3})+A_{2}\,xy(x+y)$ (5.9)
$\displaystyle=\ (x+y)(A_{1}\,(x^{2}-xy+y^{2})+A_{2}\,xy).$

Thus, the $\binom{n}{2}$-points are automatically critical away from
$\mathcal{L}^{1}$.

We want to solve the linear equations $A_{1}=0$ and $A_{2}=0$ in the three
parameters $\alpha_{1},\alpha_{2},\alpha_{3}$. Such a system has non-trivial
solutions that give $A=0$ for all $x,y$.

∎

Remarks

  1. (1)

The price of $A=0$ is at most two parameters. With the third parameter, we can
only normalize the map. In the discussion that follows, we will discover that
the cost is two parameters so that the resulting map is unique.

  2. (2)

Forcing $A_{1}=0$ is tantamount to making the pair of $n$-points on
$\mathcal{L}^{1}$ critical. This “two-birds-with-one-stone” effect is what
makes the procedure successful.

  3. (3)

Ostensibly, this argument is consistent with our obtaining a map that blows up
at the $n$-points. At an $n$-point such a map would be critical in the
“radial” direction in which $\mathcal{H}^{n-1}$ projects to
$\mathbf{P}\mathcal{H}^{n-1}$. Such a circumstance would force $C=0$ when
$xy=0$. Can the maps be critical in other directions as well? The preceding
results and proofs provide for explicit calculation of the special $4$-map. As
a consequence, we see that the map does not blow up at the $n$-points.
Furthermore, we derive the form of the map on an $\binom{n}{2}$-line.

At first, we use the parametrization

$[x,\dots,x,y,(2-n)\,x-y]$

for $\mathcal{L}^{1}$. Restricting the basic invariants to $\mathcal{L}^{1}$
gives

(5.10) $\widetilde{F_{k}}=\ F_{k}|_{\mathcal{L}^{1}}=\
(n-2)\,x^{k}+y^{k}+((2-n)\,x-y)^{k}.$

As for basic maps, note that

(5.11) $f_{k}=[(f_{k})_{1},\dots,(f_{k})_{n}]\ \text{with}\
(f_{k})_{\ell}=F_{k}-n\,x_{\ell}^{k}.$

Thus,

$\widetilde{f_{k}}=\ f_{k}|_{\mathcal{L}^{1}}=\
\Bigl{[}\widetilde{F_{k}}-n\,x^{k},\dots,\widetilde{F_{k}}-n\,x^{k},\widetilde{F_{k}}-n\,y^{k},\widetilde{F_{k}}-n\,((2-n)\,x-y)^{k}\Bigr{]}$

and we can express the homogeneous map _on_ $\mathcal{L}^{1}$ as

$\phi_{k}:[x,y]\longrightarrow\Bigl{[}\widetilde{F_{k}}-n\,x^{k},\widetilde{F_{k}}-n\,y^{k}\Bigr{]}.$

Restricting the family of $4$-maps to $\mathcal{L}^{1}$ gives

$\widetilde{g_{\alpha}}=\alpha_{1}\,\widetilde{f_{4}}+\alpha_{2}\,\widetilde{F_{2}}\,\widetilde{f_{2}}+\alpha_{3}\,\widetilde{F_{3}}\,\widetilde{f_{1}}$

which, as a map on the line, is

$\displaystyle\gamma_{\alpha}[x,y]=$ $\displaystyle\
\Bigl{[}\alpha_{1}\,(\widetilde{F_{4}}-n\,x^{4})+\alpha_{2}\,\widetilde{F_{2}}\,(\widetilde{F_{2}}-n\,x^{2})-n\,\alpha_{3}\,x\,\widetilde{F_{3}},$
$\displaystyle\
\alpha_{1}\,(\widetilde{F_{4}}-n\,y^{4})+\alpha_{2}\,\widetilde{F_{2}}\,(\widetilde{F_{2}}-n\,y^{2})-n\,\alpha_{3}\,y\,\widetilde{F_{3}}\Bigr{]}.$

We can now determine three linear conditions on the $\alpha_{i}$ that
correspond to

  1. (1)

normalizing the map so that the $n$-points are fixed in the affine
sense—hence, they are not blown up

  2. (2)

making the map critical in every direction at the $n$-points

  3. (3)

making the map critical in every off-line direction along the
$\binom{n}{2}$-lines.

Consider the cases in turn.

  1. (1)

Since $[x,y]=[1,1]$ corresponds to an $n$-point, specify that

(5.12)
$1=(\gamma_{\alpha}[1,1])_{1}=\bigl{(}n^{2}-2\,n+2)\,\alpha_{1}+(n-1)\,n\,\alpha_{2}+(n-2)\,n\,\bigl{(}(n-1)\,n\,\alpha_{3}\bigr{)}.$

  2. (2)

For the dehomogenized map

$f(z)=\frac{(\gamma_{\alpha}[z,1])_{1}}{(\gamma_{\alpha}[z,1])_{2}},$

set

$f^{\prime}(1)=0.$

This requires

(5.13) $-4\,\alpha_{1}-2\,(n-1)\,n\,\alpha_{2}+(n-2)\,(n-1)\,n\,\alpha_{3}=0$

which amounts to $A_{1}=0$ in the proof of Proposition 5.2.

  3. (3)

To arrive at $A_{2}=0$, evaluate the jacobian
$\widehat{g_{\alpha}}^{\prime}(x)$ at a point $P$ on $\mathcal{L}^{1}$ other
than an $n$-point or a $\binom{n}{2}$-point—say, the point corresponding to
$[x,y]=[1,0]$. Now, apply $g_{\alpha}^{\prime}(P)$ to the _eigenvector_

$v=(1,-1,0,\dots,0)$

in $\widehat{\mathcal{L}^{1}}_{\perp}$. Since $v$ is an eigenvector, we can
consider just the first component of $g_{\alpha}^{\prime}(P)v$. This component
is

$g_{\alpha}^{\prime}(P)_{1,1}-g_{\alpha}^{\prime}(P)_{1,2}$

where

$\displaystyle g_{\alpha}^{\prime}(P)_{1,1}$ $\displaystyle=\
-4\,(n-1)\,x^{3}\,\alpha_{1}-2\,x\,(n\,x^{2}+(n-2)\,\widetilde{F_{2}}(1,0))\,\alpha_{2}-n\,(3\,x^{3}+\widetilde{F_{3}}(1,0))\,\alpha_{3}$
$\displaystyle g_{\alpha}^{\prime}(P)_{1,2}$ $\displaystyle=\
4\,x^{3}\,\alpha_{1}-2\,x\,(n\,x^{2}-2\,\widetilde{F_{2}}(1,0))\,\alpha_{2}-3\,n\,x^{3}\,\alpha_{3}.$

Thus, the condition for $A_{2}=0$ is

(5.14)
$n\,\bigl{(}-4\,\alpha_{1}-2\,(n-2)\,(n-1)\,\alpha_{2}+(n-3)\,(n-2)\,(n-1)\,\alpha_{3}\bigr{)}=(g_{\alpha}^{\prime}(P)v)_{1}=0.$

Solving (5.12), (5.13), and (5.14) yields a unique $4$-map $g=g_{\alpha}$ with

(5.15) $\displaystyle\alpha_{1}$ $\displaystyle=\ \frac{1}{(n-4)\,n^{3}}$
$\displaystyle\alpha_{2}$ $\displaystyle=\ \frac{-6}{(n-4)(n-1)n^{4}}$
$\displaystyle\alpha_{3}$ $\displaystyle=\ \frac{-8}{(n-4)(n-2)(n-1)n^{4}}.$

Natural coordinates in which to express $g$ as a map on $\mathcal{L}^{1}$are
those of (5.8) where $x$ and $y$ are symmetrical. The result is

$\displaystyle[x,y]\longrightarrow$ $\displaystyle\
[(\gamma_{\alpha}[x+y,(1-n)\,x+y])_{1}-(\gamma_{\alpha}[x+y,(1-n)\,x+y])_{2},$
$\displaystyle\
(n-1)\,(\gamma_{\alpha}[x+y,(1-n)\,x+y])_{1}-(\gamma_{\alpha}[x+y,(1-n)\,x+y])_{2}].$

Using the inhomogeneous coordinate $z=\tfrac{x}{y}$ the map becomes

$z\longrightarrow-\frac{(n-1)\,z-4}{4\,z-(n-1)}$

in agreement with Section 3.4. As for dynamics the respective pair of
$n$-points are $0$ and $\infty$. Since $n-1\geq~{}4$, the restricted map has
$0$ and $\infty$ as its only attractor. The respective basins are
$\\{|z|<1\\}$ and $\\{|z|>1\\}$.

###  5.2. Another description

Using $n-1$ homogeneous coordinates, generalize to $\mathbf{CP}^{n-1}$ the $v$
coordinates that describe the $\mathcal{G}_{8!}$-equivariant $g_{4}$. These
place the $n$-points at

$p_{1}=[1,0,\dots,0],\ \dots\ ,\ p_{n-1}=[0,\dots,0,1],\ p_{n}=[1,\dots,1].$

The coordinate change is given by

$x=Pv\qquad v=Qx$

where

$P=(p_{ij})=\begin{cases}1-n&i=j\\\ 1&i\neq j\end{cases}\qquad\text{for}\
1\leq i\leq n,\ 1\leq j\leq n-1$

and $Q$ is the “inverse” of $P$:

$Q=(q_{ij})=\begin{cases}-1&i=j\\\ 0&i\neq j,\ j<n\\\
1&j=n\end{cases}\qquad\text{for}\ 1\leq i\leq n-1,\ 1\leq j\leq n.$

Using $[v_{1},\dots,v_{n-1}]$ for this system, we define $S_{k}$ to be the
$k$th elementary symmetric function in $n-2$ variables and the coordinates

$\widehat{v_{k}}=(\dots,v_{k-1},v_{k+1},\dots)$

complementary to $v_{k}$. The stabilizer $\mathcal{G}_{(n-1)!}$ of $p_{n}$ is
the $\mathcal{S}_{n-1}$ group of permutations of the $v_{k}$. The order-$2$
transformation $Z_{n}$ that exchanges $p_{1}$ and $p_{n}$ while fixing the
remaining $p_{k}$ generates $\mathcal{G}_{n!}$ over $\mathcal{G}_{(n-1)!}$.
Note that $\binom{n-1}{2}$ of the $\binom{n}{2}$-lines consist of points for
which all but two coordinates vanish, while the remaining $n$ lines have
points with all but one coordinate equal.

Let

$g(x)=[g_{1}(x),\dots,g_{n}(x)].$

To compute the special $4$-map

$\gamma(v)=Q(g(Pv))=[g_{n}(Pv)-g_{1}(Pv),\dots,g_{n}(Pv)-g_{n-1}(Pv)]$

in $v$ coordinates, we need to find only the first component of $\gamma$.
Permutation symmetry in $v$ tends to the remaining components. Note that

$Pv=\begin{pmatrix}(1-n)\,v_{1}+S_{1}(\widehat{v_{1}})\\\
v_{1}+S_{1}(\widehat{v_{1}})-n\,v_{2}\\\ \vdots\\\
v_{1}+S_{1}(\widehat{v_{1}})-n\,v_{n-1}\\\
v_{1}+S_{1}(\widehat{v_{1}})\end{pmatrix}=\begin{pmatrix}u_{1}-n\,v_{1}\\\
u_{1}-n\,v_{2}\\\ \vdots\\\ u_{1}-n\,v_{n-1}\\\ u_{1}\end{pmatrix}$

where $u_{1}=v_{1}+S_{1}(\widehat{v_{1}})$. Application of (5.11) gives

$\displaystyle\gamma_{1}(v)=$ $\displaystyle\ g_{n}(Pv)-g_{1}(Pv)$
$\displaystyle=$ $\displaystyle\
\Bigl{(}\alpha_{1}\,\bigl{(}F_{4}(Pv)-n\,u_{1}^{4}\bigr{)}+\alpha_{2}\,\bigl{(}F_{2}(Pv)^{2}-n\,F_{2}(Pv)u_{1}^{2}\bigr{)}+n\,\alpha_{3}\,F_{3}(Pv)u_{1}\Bigr{)}$
$\displaystyle-\Bigl{(}\alpha_{1}\,\bigl{(}F_{4}(Pv)-n\,(u_{1}-n\,v_{1})^{4}\bigr{)}+\alpha_{2}\,\bigl{(}F_{2}(Pv)^{2}-n\,F_{2}(Pv)(u_{1}-n\,v_{1})^{2}\bigr{)}$
$\displaystyle\ +n\,\alpha_{3}\,F_{3}(Pv)(u_{1}-n\,v_{1})\Bigr{)}$
$\displaystyle=$ $\displaystyle\
n\,\Bigl{(}\alpha_{1}\,\bigl{(}(u_{1}-n\,v_{1})^{4}-u_{1}^{4}\bigr{)}+\alpha_{2}\,F_{2}(Pv)\bigl{(}(u_{1}-n\,v_{1})^{2}-u_{1}^{2}\bigr{)}$
$\displaystyle\
+\alpha_{3}\,F_{3}(Pv)\bigl{(}(u_{1}-n\,v_{1})-u_{1}\bigr{)}\Bigr{)}$
$\displaystyle=$ $\displaystyle\
-n^{2}\,v_{1}\,\Bigl{(}\alpha_{1}\,\bigl{(}(u_{1}-n\,v_{1})^{3}+(u_{1}-n\,v_{1})^{2}u_{1}+(u_{1}-n\,v_{1})u_{1}^{2}+u_{1}^{3}\bigr{)}$
$\displaystyle\
+\alpha_{2}\,F_{2}(Pv)(2\,u_{1}-n\,v_{1})+\alpha_{3}\,F_{3}(Pv)\Bigr{)}.$

Straightforward calculation yields

$\displaystyle F_{2}(Pv)=$ $\displaystyle\
u_{1}^{2}-n\,v_{1}^{2}-n\,S_{1}(\widehat{v_{1}})^{2}+2\,n\,S_{2}(\widehat{v_{1}})$
$\displaystyle F_{3}(Pv)=$ $\displaystyle\
2\,u_{1}^{3}-3\,n\,u_{1}\,v_{1}^{2}+n^{2}\,v_{1}^{3}-3\,n\,u_{1}\,S_{1}(\widehat{v_{1}})^{2}+n^{2}\,S_{1}(\widehat{v_{1}})^{3}$
$\displaystyle\
+6\,n\,u_{1}\,S_{2}(\widehat{v_{1}})-3\,n^{2}\,S_{1}(\widehat{v_{1}})\,S_{2}(\widehat{v_{1}})+3\,n^{2}\,S_{3}(\widehat{v_{1}}).$

Using (5.15), the substitution for $u_{1}$, and permutation in the $v_{i}$,
the map takes the form

(5.16) $\gamma(v)=[v_{1}\,T_{1}(v),\dots,v_{n-1}\,T_{n-1}(v)]$

where

$T_{k}(v)=v_{k}^{3}-a_{2}\,v_{k}^{2}\,S_{1}(\widehat{v_{k}})+a_{3}\,v_{k}\,S_{2}(\widehat{v_{k}})-a_{4}\,S_{3}(\widehat{v_{k}})$

and

$a_{2}=\frac{4}{n-1}\qquad a_{3}=\frac{12}{(n-1)(n-2)}\qquad
a_{4}=\frac{24}{(n-1)(n-2)(n-4)}.$

Evidently, the maps in (5.16) are $\mathcal{G}_{(n-1)!}$-equivariant. They
also satisfy

$Z_{n}\circ g=g\circ Z_{n}\qquad\text{for all $n\geq 5$}.$

###  5.3. Revisiting the quintic

In solving the quintic, [Crass 2001] harnesses the dynamics of a degree-$6$
map whose behavior is similar to the maps treated in the present paper. The
$\mathcal{G}_{5!}$ equivariant $4$-map is also a good candidate for inclusion
in a quintic-solving algorithm. Having lower degree, its global dynamics might
be more tractable. For instance, this map has a kind of _critical finiteness_
that other $\mathcal{G}_{n!}$ $4$-maps do not share. I plan to examine this
map in more dynamical detail in an upcoming paper.

##  6\. Basin portraits

The plots that follow are productions of the program _Dynamics 2_ running on a
Dell Dimension XPS with a Pentium II processor. Its BA and BAS routines
produced the pictures. (See the manual [Nusse and Yorke 1998].) Briefly, each
procedure divides the screen into a grid of cells and then colors each cell
according to which attracting point its trajectory approaches. If it finds no
such attractor after a specified number of iterations—usually $60$, the cell
is black. The BA algorithm looks for an attractor whereas BAS requires the
user to specify a candidate attracting set of points. Each portrait exhibits
the highest resolution available—a $720\times 720$ grid.

All of the images show $g_{4}$ restricted to either a $\mathbf{CP}^{1}$ or
$\mathbf{RP}^{2}$ that it preserves. Some restricted maps have attracting
sites that are not the $8$-points. However, none of the detected “restricted
attractors” other than the $8$-points themselves are _overall_ attractors with
$6$-dimensional basins.

### Figure 2

When restricted to the line $\mathcal{L}^{1}_{168}$, $g_{4}$ has only trivial
symmetry. Consequently, there is no natural choice of coordinates in which to
express the restriction to the line. With an $8$-point at $0$ and a $28$-point
$q^{28}_{ij}$ at $\infty$, the map takes the form

$z\longrightarrow-\frac{z^{3}\,(3\,z+4)}{4\,(z^{2}+8\,z+14)}$

The two basins that appear are associated with the $8$-point and $28$-point.
The latter is an “equatorial” saddle point on the $28$-line
$\mathcal{L}^{1}_{28_{ij}}$. The basin in $\mathcal{L}^{1}_{168}$ is a slice
of its $5$-dimensional stable manifold. The reflective symmetry that appears
is due to the map’s anti-holomorphic equivariance. (See Section 3.4.)

Not pictured is the portrait for $\mathcal{L}^{1}_{210}$ on which the map
takes the form

$z\longrightarrow-z^{2}\,\frac{3\,z+1}{z+3}.$

As in the case of a $28$-line, this map has two basins: $\\{|z|<1\\}$ and
$\\{|z|>1\\}$. The attracting points are of the types

$[1,1,-1,-1,0,0,0,0]\qquad\text{and}\qquad[1,1,1,1,-1,-1,-1,-1].$

Overall, these behave as saddles. In fact, at the latter point, $g_{4}$ blows
up onto the associated hyperplane

$\\{x_{1}+x_{2}+x_{3}+x_{4}=0\\}\cap\\{x_{5}+x_{6}+x_{7}+x_{8}=0\\}.$

### Figure 3

This shows $g_{4}$ on $\mathcal{L}^{1}_{280}$ in the form

$z\longrightarrow\frac{8\,z^{3}}{5\,z^{4}+6\,z^{2}-3}.$

The basin at $0$ is due to a $28$-point $p^{28}_{ij}$. This point belongs to
$\mathcal{L}^{1}_{28_{ij}}$ and is repelling on that line. Thus, the basin
pictured here is a $1$-dimensional slice of the $5$-dimensional stable
manifold attached to the point. At $\pm 1$ we see petals due to rationally
indifferent points of type

$[3,3,3,3,3,-5,-5,-5].$

The indifferent local behavior at these points is evident in this and several
subsequent images.

### Figure 4

On $\mathcal{M}^{1}_{280}$, $g_{4}$ is a polynomial map expressible by

$z\longrightarrow-z\,(7\,z^{2}-5\,z+1).$

Here, an $8$-point is at $\infty$ (the dark blue basin) and, once again, a
rationally indifferent point of type

$[3,3,3,3,3,-5,-5,-5]$

appears at $0$.

### Figure 5

We see the $\mathcal{S}_{3}$-symmetric restriction to the $\mathbf{RP}^{2}$
intersection of $\mathcal{L}^{2}_{56}$ and $\mathcal{R}$. The three basins
result from a triple of $8$-points arranged symmetrically on the unit circle.
A line of reflective symmetry passes through each of the $8$-points and
$(0,0)$—which corresponds to a point of type

$[3,3,3,3,3,-5,-5,-5].$

These lines are $\mathbf{RP}^{1}$ intersections of an $\mathcal{L}^{1}_{168}$
with the $\mathbf{RP}^{2}$. Each $\mathbf{RP}^{1}$ contains points in the
basin of an $8$-point and as well as points in the boundary between the basins
of the other two $8$-points. This interval lies in the basin of a $28$-point
$q^{28}_{ij}$ as seen in Figure 2.

### Figure 6

The image displays the basins of the $\mathcal{S}_{4}$-symmetric map on the
intersection of $\mathcal{L}^{2}_{105}$ and $\mathcal{R}$. Each
$\mathcal{L}^{2}_{105}$ is canonically associated with a $\mathbf{CP}^{3}$,
$\mathcal{L}^{3}_{105}$, whose points have coordinates that come in four
mutually negative pairs:

$\begin{array}[]{lcl}\\{x_{i}=x_{j}\\}\cap\\{x_{k}=x_{\ell}\\}&\longleftrightarrow&\\{x_{i}=-x_{j}\\}\cap\\{x_{k}=-x_{\ell}\\}\\\
\cap\\{x_{m}=x_{n}\\}\cap\\{x_{p}=x_{q}\\}&&\cap\\{x_{m}=-x_{n}\\}\cap\\{x_{p}=-x_{q}\\}.\end{array}$

Under $g_{4}$, $\mathcal{L}^{3}_{105}$ blows down to $\mathcal{L}^{2}_{105}$.
Furthermore, each point in the orbit of

$[1,-1,1,-1,1,-1,1,-1]$

belongs to $24$ of the $\mathcal{L}^{3}_{105}$ so that the blowing-down of
each $\mathbf{CP}^{3}$ forces all coordinates of the image to be equal. Hence,
they must all vanish.

To describe things explicitly, take the plane

$\mathcal{L}^{2}_{105}=\\{x_{1}=x_{2}\\}\cap\\{x_{3}=x_{4}\\}\cap\\{x_{5}=x_{6}\\}\cap\\{x_{7}=x_{8}\\}$

parametrized by

$[x,y,z]\longrightarrow[x+y+z,x+y+z,x-y-z,x-y-z,-x-y+z,-x-y+z,-x+y-z,-x+y-z].$

There is a $3$-point orbit at

$[1,0,0],\ [0,1,0],\ [0,0,1]$

corresponding to the points

$[1,1,1,1,-1,-1,-1,-1],\ [1,1,-1,-1,-1,-1,1,1],\ [1,1,-1,-1,1,1,-1,-1]$

and a $4$-point orbit at $[\pm 1,\pm 1,1]$ corresponding to the $28$-points

$\displaystyle q^{28}_{12}$ $\displaystyle=\ [-3,-3,1,1,1,1,1,1],\
q^{28}_{34}=[1,1,-3,-3,1,1,1,1]$ $\displaystyle q^{28}_{56}$ $\displaystyle=\
[1,1,1,1,-3,-3,1,1],\ q^{28}_{78}=[1,1,1,1,1,1,-3,-3].$

Associated with the $3$-points are the lines

$\\{x=0\\},\ \\{y=0\\},\ \\{z=0\\}$

corresponding to intersections with three of the $\mathcal{L}^{3}_{105}$:

$[u,u,-u,-u,v,v,-v,-v],\ [u,u,v,v,-v,-v,-u,-u],\ [u,u,v,v,-u,-u,-v,-v].$

In these coordinates, the map has the expression

$[x,y,z]\longrightarrow[yz(4\,x^{2}+y^{2}+z^{2}),xz(x^{2}+4\,y^{2}+z^{2}),xy(x^{2}+y^{2}+4\,z^{2})].$

The attractor here seems to consist of four $28$-points $q^{28}_{ij}$ at $(\pm
1,\pm 1)$. The points $q^{28}_{ij}$ are on $\mathcal{L}^{1}_{28_{ij}}$ and,
thus, repel in directions away from the picture-plane. Here we see off-line
directions relative to $\mathcal{L}^{1}_{28_{ij}}$. Each $q^{28}_{ij}$ lies on
$15$ of the $\mathcal{L}^{2}_{105}$ and their “restricted” basins in the
$\mathbf{CP}^{2}$ belong to their stable manifolds as well as to the _overall_
Julia set of $g_{4}$. Hence, these pieces of the Julia set have zero measure
in $\mathbf{CP}^{6}$. As for real dynamics, the basins appear to be the four
quadrants

$\\{x>0,y>0\\},\ \\{x<0,y>0\\},\ \\{x<0,y<0\\},\ \\{x>0,y<0\\}$

which are forward and, thus, totally invariant. Accordingly, the coordinate
axes form the basin boundaries.

Finally, each coordinate axis blows down to its companion point while a
$3$-point blows up onto its associated axis.

### Figure 7

For the $\mathbf{Z}_{2}^{2}$ map on the intersection of
$\mathcal{L}^{2}_{280}$ and $\mathcal{R}$ the two prominent basins belong to a
pair of $8$-points at $(\pm 1,0)$. The other two attracted regions are
associated with a pair of indifferent points (both eigenvalues are $-1$) of
type

$[3,3,3,3,3,-5,-5,-5].$

### Figure 8

On the $\mathbf{Z}_{2}$-symmetric restriction to the intersection of
$\mathcal{L}^{2}_{420}$ and $\mathcal{R}$, two $8$-points at $(\pm 1,0)$ and a
$q^{28}_{ij}$ at $(0,-1)$ account for the basins.

### Figure 9

As for the behavior of the $\mathbf{Z}_{2}$-symmetric map on the intersection
of $\mathcal{L}^{2}_{840}$ and $\mathcal{R}$, there is an $8$-point at $(0,0)$
and a pair of $28$-points $q^{28}_{ij}$, $q^{28}_{k\ell}$ at $(\pm 1,1)$.
Trajectories starting in the thin vertical black strip near the bottom middle
require too many iterations in order to converge to one of the three
attractors for the BA routine to detect attraction. Experimental results
suggest that this region is filled with rather small components of the three
basins.

Figure 2. Dynamics of $g_{4}$ on $\mathcal{L}^{1}_{168}$

Figure 3. Dynamics of $g_{4}$ on $\mathcal{L}^{1}_{280}$

Figure 4. Dynamics of $g_{4}$ on $\mathcal{M}^{1}_{280}$

Figure 5. Dynamics of $g_{4}$ on $\mathcal{L}^{2}_{56}\cap\mathcal{R}$

Figure 6. Dynamics of $g_{4}$ on $\mathcal{L}^{2}_{105}\cap\mathcal{R}{}$

Figure 7. Dynamics of $g_{4}$ on $\mathcal{L}^{2}_{280}\cap\mathcal{R}$

Figure 8. Dynamics of $g_{4}$ on $\mathcal{L}^{2}_{420}\cap\mathcal{R}$

Figure 9. Dynamics of $g_{4}$ on $\mathcal{L}^{2}_{840}\cap\mathcal{R}$

## References

  * [Crass 1999] S. Crass. _Solving the sextic by iteration: A study in complex geometry and dynamics._ Experiment. Math. 8 (1999) No. 3, 209-240. Preprint at arxiv.org/abs/math.DS/9903111.
  * [Crass 2001] S. Crass. _Solving the quintic by iteration in three dimensions._ Experiment. Math. 10 (2001) No.1, 1-24.
Preprint at arxiv.org/abs/math.DS/9903054.

  * [Crass 2000] S. Crass. _Mathematica_ notebooks and supporting files for the solution of the octic.
See csulb.edu/ scrass/math/octic/Solve.

  * [Klein 1913] F. Klein. _Lectures on the Icosahedron and the Solutions of Equations of the Fifth Degree_ , Kegan Paul, London, 1913. Reprint by Dover, New York. Translation by G. Morrice of _Vorlesungen über das Ikosaeder und die Auflösungen der Gleichungen vom fünften Grade_ , Teubner, Leipzig, 1884.
  * [Nusse and Yorke 1998] H. Nusse and J. Yorke. _Dynamics: Numerical Explorations, 2e_ Springer-Verlag, 1998. Computer program _Dynamics 2_ by B. Hunt and E. Kostelich.
  * [Orlik and Terao 1992] P. Orlik and H. Terao. _Arrangements of Hyperplanes_. Springer-Verlag, 1992.
  * [Shephard and Todd 1954] G. C. Shephard and T. A. Todd. _Finite unitary reflection groups._ Canad. J. Math. 6 (1954), 274-304.
  * [Wiman 1899] A. Wiman. _Ueber die Darstellung der symmetrischen und alternirenden Vertauschungsgruppen als Collineationsgruppen von möglichst geringer Dimensionzahl_. Mathematische Annalen 52 (1899), 243-270.


# ViMA – the spinning rotor gauge to measure the viscosity of tritium between
77 and 300 K

###### ?abstractname?

Experimental values for the viscosity of the radioactive hydrogen isotope
tritium (T2) are currently unavailable in literature. The value of this
material property over a wide temperature range is of interest for
applications in the field of fusion, neutrino physics, as well as to test ab
initio calculations. As a radioactive gas, tritium requires careful experiment
design to ensure safe and environmental contamination free measurements. In
this contribution, we present a spinning rotor gauge based, tritium compatible
design of a gas viscosity measurement apparatus (ViMA) capable of covering the
temperature range from $80\text{\,}\mathrm{K}300\text{\,}\mathrm{K}$.

Johanna Wydraa<EMAIL_ADDRESS>Alexander Marstellera,b Robin Größlea
Florian Priestera Michael Sturma

aInstitute for Astroparticle Physics, Tritium Laboratory Karlsruhe(IAP-TLK),
Karlsruhe Institute of Technology (KIT),
Hermann-von-Helmholtz-Platz 1
Eggenstein-Leopoldshafen
76344
Baden-Württemberg
Germany bCenter for Experimental Nuclear Physics and Astrophysics,
Department of Physics, University of Washington (UW),
Seattle
98195
Washington
United States of America

tritium viscosity TLK - Tritium Laboratory Karlsruhe spinning rotor gauge
cryogenic setup

##  1 Introduction

The viscosity is one of the fundamental material properties of gases, needed
for gas dynamic calculations. Therefore, the viscosity of many gases has been
measured over a wide temperature range [1, 2, 3, 4, 5, 6]. For many simple
systems like noble gases[7] or molecular hydrogen[8, 9], ab initio
calculations have been performed. Simple molecules which have not been covered
widely due to their radioactivity are the tritiated isotopologues of hydrogen.
Ab initio calculations exist [10], but since they have been carried out using
a classical approach and neglect quantum effects, they are only suitable for
temperatures of $300\text{\,}\mathrm{K}$ and above. To the best of the
authors’ knowledge, no experimental values for the viscosity of tritiated
hydrogen molecules exist in published literature.

The cryogenic region of viscosity covered by neither theoretical calculations
nor experiment is of interest for several applications such as the closed
tritium cycle design, development and operation for nuclear fusion and
experimental neutrino physics. The Karlsruhe Tritium Neutrino Experiment
(KATRIN) is one such experiment which aims to measure the electron
antineutrino mass, using the electron spectrum of the tritium $\upbeta$-decay.
The KATRIN collaboration has published a new experimental upper limit on the
neutrino mass of $0.8\text{\,}\mathrm{eV}$ ($90\text{\,}\mathrm{\char
37\relax}\mathrm{C}\,\mathrm{L}\,$) [11]. In this experiment, tritium is
circulated through a $10\text{\,}\mathrm{m}$ long windowless gaseous tritium
source (WGTS) at a temperature of $80\text{\,}\mathrm{K}$. Viscosity is the
central parameter to model the density profile of tritium inside the WGTS and
is therefore a systematic effect for the KATRIN Experiment. In fusion fuel
cycle the viscosity of tritium is needed for example to simulate the tokamak
exhaust process, isotope separation systems based on gas chromatography and
others.

In order to measure the viscosity of tritium down to temperatures relevant to
the KATRIN experiment, we have built a viscosity measurement apparatus (ViMA)
based on a spinning rotor gauge. In this paper, we describe an apparatus
design capable of measuring gas viscosities from
$77\text{\,}\mathrm{K}300\text{\,}\mathrm{K}$ in a manner that is compatible
with the material requirements for later use with tritium.

##  2 theory of the spinning rotor gauge (SRG)

Whereas the spinning rotor gauge (SRG) normally is used as a pressure gauge
for ultra high vacuum, where its measurements are independent of the viscosity
of the surrounding gas, it can also be used at higher pressures, between
$20\text{\,}\mathrm{Pa}2000\text{\,}\mathrm{Pa}$, to measure the viscosity of
the gas, if the pressure is measured separately. In both cases, the normalized
deceleration rate of the rotor is measured in dependence of the pressure and
the temperature. The underlying effect used here, is the damping of the rotor,
caused by the collisions of the surrounding gas with the SRG rotor. The torque
$D$ normalized to the frequency $\Omega$ of the rotor can be written as

$\frac{D}{\Omega}=I\cdot\frac{\nicefrac{{\mathrm{d}\Omega}}{{\mathrm{d}t}}}{\Omega}\,,$
(1)

with $I$ being the moment of inertia of a sphere. By inverting the normalized
torque, the formula for measuring the viscosity with the SRG, which is derived
in [12, 13, 14], reads as follows:

$\frac{1}{\nicefrac{{D}}{{\Omega}}}=\frac{1}{A\cdot C_{0}\mu}+\frac{1}{p}\cdot
B\,,$ (2)

where $A$ and $B$ are constants for a given temperature $T$, $p$ is the
pressure, $\mu$ is the viscosity and $C_{0}$ is a calibration factor (compare
to [15]). Experimentally, the viscosity is derived using Equation 2 by fitting
a linear function to the taken data and extracting $\mu$ from the y-axis
intersection. As for the measurement procedure, there are different methods to
acquire the necessary data for the fit, which are currently under
investigation in terms of reproducibility, systematic effects and handling.

##  3 design constraints

Since the setup is planned to operate with tritium, there are some
constraints. The whole system needs to be built according to the technical
terms of delivery and acceptance (TLA) and the best engineering practice [16].
The technical design pressure range is limited to $90\text{\,}\mathrm{kPa}$,
so that in case of leakage, tritium release against atmospheric pressure is
minimized. The whole primary system, which is in contact with tritium, needs
to be fully built out of metal including the sealing gaskets. In addition, the
system needs to fit into a glove box, leaving enough space to handle it using
the gloves. Furthermore, there are specific needs, to guarantee most accurate
measurements of the viscosity. The thermal stabilization has to be better than
$1\text{\,}\mathrm{K}\text{\,}{\mathrm{h}}^{-1}$, to exclude any thermal drift
inside the system during one measurement. The temperature gradient inside the
measurement cell should not exceed $1\text{\,}\mathrm{K}$. The pressure range
inside the measurement cell needs to run from
$100\text{\,}\mathrm{Pa}2000\text{\,}\mathrm{Pa}$ at room temperature and from
$20\text{\,}\mathrm{Pa}500\text{\,}\mathrm{Pa}$ at liquid nitrogen
temperature. The temperature is limited on the low end to
$77\text{\,}\mathrm{K}$ by the use of liquid nitrogen as a coolant and on the
high side to $330\text{\,}\mathrm{K}$ by the maximum operating temperature of
the SRG. Lower temperatures, down to
$10\text{\,}\mathrm{K}20\text{\,}\mathrm{K}$ where the saturation vapor
pressure becomes limiting, are desirable, but were not targeted in this setup
due to the associated increase in system complexity of going below liquid
nitrogen temperatures.

##  4 Design

?figurename? 1: Flow pattern of Cryo-ViMA. The primary system, meaning all
parts in contact with tritium, are colored in red. The cold gas is marked in
blue and the isolation vacuum is colored in green. ?figurename? 2: CAD model
and photograph of Cryo-ViMA. Shown are the SRG head ① connected to that sample
gas tube ② (bellow connection not shown in CAD), which also contains the
sample gas temperature sensor ③. Cold nitrogen gas from the cold gas system is
injected via a nozzle ④. A diffusion plate ⑤ improves temperature stability
and cooling efficency. Surrounding these components is a vacuum insulated
Dewar ⑥.

After proof-of-principle setups demonstrated the feasibility of the approach,
the new and refined Cryogenic viscosity measurement apparatus (Cryo-ViMA) will
enable us to measure the viscosity of tritium, while fulfilling all the
constraints listed above. An annotated CAD model and photograph of the setup
are shown in Figure 2. To have a thermal insulation against the environment,
the SRG, together with five temperature sensors (thermocouple type K), is
built up inside a cryostat, consisting of two stainless steel barrels with
insulation vacuum in between. This cryostat is cooled inside with nitrogen gas
by a dedicated cold gas system. The cold gas system can be controlled in the
whole temperature range between $80\text{\,}\mathrm{K}300\text{\,}\mathrm{K}$
and is stabilized to $2\text{\,}\mathrm{K}\text{\,}{\mathrm{h}}^{-1}$. At the
time of writing this paper, the thermal stabilization of the cryostat is not
sufficient enough for the planned measurement procedure, which is why the
procedure has been changed as described in section 5. Since the sample gas is
stored at room temperature, a pre-cooling is needed. This is realised with a
long, thin-walled bellows tube, which is installed as a loop through the whole
cryostat, before it connects to the SRG thimble. The tritium compatibility is
guaranteed by only using stainless steel in the primary system, which is
proved to hold pressures up to $500\text{\,}\mathrm{kPa}$. This qualification
for high pressures is necessary to ensure that even if the system is filled
with the maximum operating pressure of $90\text{\,}\mathrm{kPa}$ at
$77\text{\,}\mathrm{K}$, a warm-up to room temperature does not lead to
leakage or structural damage of the system. In this fashion, even in the case
of failure of the cooling system or operator mistakes, safe enclosure of
tritium in the primary system can be guaranteed.

##  5 Example commissioning measurements

?figurename? 3: Example commissioning measurement. Shown are the thermal cycle
measurements at different pressure setpoints. The color indicates the
temperature between $180\text{\,}\mathrm{K}300\text{\,}\mathrm{K}$. At low
pressures (right side of the graph) a sligt hysteresis can be seen,
indicating, that some adjustments still have to be made for the cycling-speed.

The measurements described in [15] are conducted at constant temperature by
changing the pressure inside the SRG. The temperature of Cryo-ViMA can only be
stabilized to $2\text{\,}\mathrm{K}\text{\,}{\mathrm{h}}^{-1}$, so
measurements with constant temperature are not feasible for high precision
results. With Cryo-ViMA, the measurements are conducted by fixing the absolute
particle number inside the relevant volume, by choosing a pressure set-point
between $10\text{\,}\mathrm{Pa}2000\text{\,}\mathrm{Pa}$ and closing the
volume by closing valves HV03 and HV04 (compare to Figure 1). This way, the
pressure inside the SRG is still monitored. Afterwards the temperature of the
cold gas system is ramped from room temperature down to
$77\text{\,}\mathrm{K}$ and up again. One such thermal cycle takes about one
day and is repeated for different pressure set-points. The result can be seen
in Figure 3, where every loop corresponds to one thermal cycle. From here the
viscosity can be calculated by picking measured values for one temperature and
fitting Equation 2.

##  6 Summary

The measurement of gas viscosity using rotating bodies is an established
method in literature [14, 17], and SRGs have been used for this at room
temperature in the past [14, 13, 12]. Our efforts have been focused on
extending these prior works with the goal of achieving measurements at
cryogenic temperatures and featuring tritium compatibility. In this
contribution, we have presented a setup that enables the viscosity measurement
of gases in the range from $77\text{\,}\mathrm{K}300\text{\,}\mathrm{K}$. We
have taken care in our design and material selection to achieve safe enclosure
of the sample gas, allowing us to use the setup to measure the viscosity of
the radioactive hydrogen isotope tritium. In general, the setup we have
presented in this contribution can be used to measure the viscosity of
hazardous (radioactive, corrosive, toxic, …) gases, making it applicable
outside of our particular field of research as well.

## Acknowledgments

We acknowledge the support of the Helmholtz Association (HGF), Germany, the
German Ministry for Education and Research BMBF (05A20VK3) and the DFG
graduate school KSETA, Germany (GSC 1085).

##  ?refname?

  * [1] M. J. Assael, A. Kalyva, S. Monogenidou, M. L. Huber, R. A. Perkins, D. G. Friend, and E. F. May, “Reference values and reference correlations for the thermal conductivity and viscosity of fluids,” _Journal of physical and chemical reference data_ , 47, _2_ , 021501 (2018); https://doi.org/10.1063/1.5036625.

  * [2] A. Rietveld and A. van Itterbeek, “Viscosity of mixtures of H2 and HD between 300 and 14 °K,” _Physica_ , 23, _6-10_ , 838 (1957); https://doi.org/10.1016/S0031-8914(57)94957-1.
  * [3] A. Van Itterbeek and A. Claes, “Measurements on the viscosity of hydrogen-and deuterium gas between 293 K and 14 K,” _Physica_ , 5, _10_ , 938 (1938); https://doi.org/10.1016/S0031-8914(38)80036-7.
  * [4] A. van Itterbeek and O. van Paemel, “Measurements on the viscosity of neon, hydrogen, deuterium and helium as a function of the temperature, between room temperature and liquid hydrogen temperatures,” _Physica_ , 7, _3_ , 265 (1940); 10.1016/s0031-8914(40)90115-x.
  * [5] J. Kestin and W. Leidenfrost, “An absolute determination of the viscosity of eleven gases over a range of pressures,” _Physica_ , 25, _7_ , 1033 (1959); https://doi.org/10.1016/0031-8914(59)90024-2.
  * [6] J. Kestin, R. Paul, A. Clifford, and W. Wakeham, “Absolute determination of the thermal conductivity of the noble gases at room temperature up to 35 MPa,” _Physica A: Statistical Mechanics and its Applications_ , 100, _2_ , 349 (1980); 10.1016/0378-4371(80)90125-9.
  * [7] John J. Hurly and Michael R. Moldover, “Ab Initio Values of the Thermophysical Properties of Helium as Standards,” _Journal of Research of the National Institute of Standards and Technology_ , 2000, _105_ , 667 (2000); 10.6028/jres.105.054.
  * [8] J. B. Mehl, M. L. Huber, and A. H. Harvey, “Ab initio transport coefficients of gaseous hydrogen,” _International Journal of Thermophysics_ , 31, _4_ , 740 (2010); 10.1007/s10765-009-0697-9.
  * [9] J. Schaefer, “Transport coefficients of dilute hydrogen gas, calculations and comparisons with experiments,” _Chemical Physics_ , 368, _1-2_ , 38 (2010); 10.1016/j.chemphys.2009.12.007.
  * [10] B. Song, K. Kang, Z. Zhang, X. Wang, and Z. Liu, “Ab Initio Values of the Gas Transport Properties of Hydrogen Isotopologues and Helium–Hydrogen Mixtures at Low Density,” _Journal of Chemical & Engineering Data_, 61, _5_ , 1910 (2016); 10.1021/acs.jced.6b00076.
  * [11] “Direct neutrino-mass measurement with sub-electronvolt sensitivity,” _Nature Physics_ , 18, _2_ , 160 (2022).
  * [12] P. Tekasakul, J. A. Bentz, R. V. Tompson, and S. K. Loyalka, “The spinning rotor gauge: Measurements of viscosity, velocity slip coefficients, and tangential momentum accommodation coefficients,” _Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films_, 14, _5_ , 2946 (1996); 10.1116/1.580249.
  * [13] J. Bentz, R. Tompson, and S. Loyalka, “The spinning rotor gauge: measurements of viscosity, velocity slip coefficients, and tangential momentum accommodation coefficients for N2 and CH4,” _Vacuum_ , 48, _10_ , 817 (1997); 10.1016/s0042-207x(97)00031-6.
  * [14] S. K. Loyalka, “Theory of the spinning rotor gauge in the slip regime,” _Journal of Vacuum Science & Technology A_, 14, _5_ , 2940 (1996); 10.1116/1.580248.
  * [15] J. Wydra, A. Marsteller, R. Größle, L. Schlee, F. Priester, and M. Sturm, “Towards the first direct measurement of the dynamic viscosity of gaseous tritium at cryogenic temperatures,” _Vacuum_ , 203, 111237 (2022); 10.1016/j.vacuum.2022.111237.
  * [16] S. Welte, U. Besserer, D. Osenberg, and J. Wendel, “Tritium Laboratory Karlsruhe: Administrative and Technical Framework for Isotope Laboratory Operation,” _Fusion Science and Technology_ , 67, _3_ , 635 (2015); 10.13182/fst14-t98.
  * [17] Y. Shimokawa, Y. Matsuura, T. Hirano, and K. Sakai, “Gas viscosity measurement with diamagnetic-levitation viscometer based on electromagnetically spinning system,” _Review of Scientific Instruments_ , 87, _12_ , 125105 (2016); https://doi.org/10.1063/1.4968026.