# Dynamical Systems of Correspondences on the Projective Line II: Degrees of
Multiplier Maps

Rin Gotou  Rin Gotou Department of Mathematics, Graduate School of Osaka
University<EMAIL_ADDRESS>
###### Abstract

This paper is a sequel of [Got23]. In this paper, we consider a kind of
inverse problem of multipliers. The problem is to count number of isospectral
correspondences, correspondences which has the same combination of
multipliers. We give a primitive explicit upper bound. In particular, for a
generic rational map of degree $d$, there are at most $O(d^{10d})$ rational
maps with the same combination of multipliers for the fixed points and the
3-periodic points. This paper also includes two proofs of a correction
[Hut21], which states that the multipliers of the fixed and 2-periodic points
determines generic cubic morphism uniquely. One is done by proceeding the
computation in [HT13]. The other is done by more explicit computation with the
help of invariant theory given in [Wes15].

##  1 Introduction

This paper is a sequel of [Got23]. In [Got23], the author defined the moduli
spaces $\operatorname{Dyn}_{d,e}$ of dynamical systems of self-correspondences
on the projective line $\mathbb{P}^{1}$. The moduli space
$\operatorname{Dyn}_{d,e}$ parametrizes self-correspondences
$C\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ of degree $(d,e)$ with only mild
(i.e. of multiplicity $\leq\frac{d+e}{2}$) singularities on the diagonal of
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ ([Got23, Theorem 1.1]), up to conjugation
by $\operatorname{Aut}(\mathbb{P}^{1})=\operatorname{PGL}_{2}$. As essential
structures of the moduli space of dynamical system, iteration maps
$\Psi_{n}:\operatorname{Dyn}_{d,e}\dashrightarrow\operatorname{Dyn}_{d^{n},e^{n}}$
and fixed point multiplier map
$\lambda_{1,(d,e)}:\operatorname{Dyn}_{d,e}\dashrightarrow\mathbb{P}^{d+e}$
were introduced. These respectively indicates $n$-th iteration $C\mapsto
C\circ\cdots\circ C$ and the fixed point multipliers

$C:(f=0)\mapsto\left\\{\left.\lambda_{z}(f):=-\frac{\partial_{y}f(z,z)}{\partial_{x}f(z,z)}\right|z\in\mathbb{P}^{1}:f(z,z)=0\right\\}$

(under the isomorphism $\\{(d+e)\text{ (possibly multiple) points in
}\mathbb{P}^{1}\\}=\operatorname{Sym}_{d+e}\mathbb{P}^{1}\simeq\mathbb{P}^{d+e}$).
Multiplier maps $\lambda_{n,(d,e)}:=\lambda_{1,(d^{n},e^{n})}\circ\Psi_{n}$
were also introduced, but well-definedness of $\lambda_{n,(d,e)}$ for general
$(n,d,e)$ are remaining as a problem ([Got23, Problem 1.9, Remark 7.2]). If
the $n$-th multiplier map $\lambda_{n,(d,e)}$ is well-defined, then it
indicates

$C:(f=0)\mapsto\left\\{\left.\lambda_{(z_{i})}(f):=\prod_{i=0}^{n-1}\left(-\frac{\partial_{y}f(z_{i},z_{i+1})}{\partial_{x}f(z_{i},z_{i+1})}\right)\right|\genfrac{}{}{0.0pt}{}{(z_{i})_{i=0}^{n}\in(\mathbb{P}^{1})^{n+1}:}{f(z_{i},z_{i+1})=0,z_{0}=z_{n}}\right\\}.$

A purpose to define morphisms $\lambda_{n}:=\lambda_{n,(d,e)}$ is to consider
inverse problem of multiplier, that is, how extant information of multipliers
determines morphisms. The results about inverse problem of multiplier are
rephrased as the properties of multiplier maps to their images. Let

$\displaystyle\Lambda_{n}$
$\displaystyle:=\prod_{m:m|n}\lambda_{m}:\operatorname{Dyn}_{d,e}\dashrightarrow\prod_{m:m|n}\mathbb{P}^{d^{m}+e^{m}}\text{
and }$ $\displaystyle\Lambda$ $\displaystyle:=\prod_{m\geq
1}\lambda_{m}:\operatorname{Dyn}_{d,e}\dashrightarrow\prod_{m\geq
1}\mathbb{P}^{d^{m}+e^{m}},$

where $m|n$ means that $m$ divides $n$. Let
$\operatorname{Poly}_{d}\subset\operatorname{Dyn}_{1,d}$ be the locus of the
points indicates a conjugation class including some polynomial morphisms.
Table 1 is a brief review of known results about inverse problem of
multipliers, as the form of degrees of multiplier maps to their images. All
known results are about cases of usual morphisms, that is, the degree as self-
correspondence is $(1,d)$.

Reference | locus $X$ | degree of | map $F$ | degree of
---|---|---|---|---
|  | morphisms |  | $F:X\dashrightarrow F(X)$
[McM87] | $\operatorname{Dyn}_{1,d}$ | $d\geq 2$ | $\Lambda$ | $<\infty$
[Gor15] |  |  | $\Lambda_{n}\ (n\geq 3)$ | $<\infty$
[Sch16] |  |  |  |  $<$ Recursive formula
[Gor15, Conjecture] |  |  | $\Lambda_{2}$ | ?
[JX23] |  | $d\geq 4$ | $\Lambda$ | 1
[Mil93],[Sil98] |  | $d=2$ | $\Lambda_{1}(=\lambda_{1})$ | 1
(From dimension) |  | $d\geq 3$ | $\Lambda_{1}$ | $\infty$
[HT13] |  | $d=3$ | $\Lambda_{2}$ | $a_{3,2}$
[Fuj06] | $\operatorname{Poly}_{d}$ | $d\geq 2$ | $\Lambda_{1}$ | $(d-2)!$
[HT13] |  | $d=4,5$ | $\Lambda_{2}$ | $1$
Table 1:  Results about degree of multiplier maps onto their images

######  Remark 1.1.

There are some more precise results about the degrees for the loci which the
degrees of multiplier maps changes from generic behaviour ([McM87], [Sil07],
[Fuj06], [Fuj07], [Sug17], [Sug20]).

In [HT13], $a_{3,2}=12$ was stated, but the author corrected it to $a_{3,2}=1$
([Hut21]). An aim of this paper is to give precise proofs of correction.

######  Theorem 1.2 (Theorem 5.1).

The rational map

$\Lambda_{2,(1,3)}=\lambda_{1,(1,3)}\times\lambda_{2,(1,3)}:\operatorname{Dyn}_{1,3}\dashrightarrow\mathbb{P}^{3}\times\mathbb{P}^{9}$

is birational to its image.

In Section 5, we give two proofs of this theorem. In Subsection 5.1, we prove
Theorem 5.1 by proceeding the computation done in [HT13]. This proof is the
proof mentioned in [Hut21] and independent from other parts (except the
programs in Subsection A.1 used for the proof) of this paper.

The other proof in Subsection 5.2 is by a direct computation on the invariant
ring given in [Wes15], which is the coordinate ring of
$\operatorname{Dyn}_{1,3}$. The computation is done by an interpolation method
in Subsection A.2, and some unexpectedly simple relations among the coordinate
functions of (Remark 5.2). A merit of this method is that the part without the
direct computation of structure of the coordinate rings can be used for more
general cases of $\Lambda_{n,(d,e)}$. The other aim of this paper is to give a
primitive estimation of degree of multiplier map of correspondence along this
method. This gives very rough as an upper bound, but gives a finite number.

######  Theorem 1.3.

Let $p$ be a prime number. If the $p$-th multiplier map is well-defined and

$\Lambda_{p}:=\lambda_{1,(d,e)}\times\lambda_{p,(d,e)}:\operatorname{Dyn}_{d,e}\dashrightarrow\Lambda_{p}(\operatorname{Dyn}_{d,e})\subset\mathbb{P}(D_{d+e})\times\mathbb{P}(D_{M})$

is generically finite to its image, then its degree is at most

$\frac{\gcd(d+e,2)N^{de+d+e-3}(d+e-3)!(de-3)!}{2(d+e)\cdot(de+d+e-3)!},$

where

$N:=2(d+e-1)+\frac{2((d^{p}-1)(d^{p}-d)-(e^{p}-1)(e^{p}-e))}{p(d-e)}.$

######  Remark 1.4.

We can give upper bounds in similar method for generic $\Lambda_{n}$ in the
similar assumption, but $N$ becomes slightly more complicated polynomial. We
also note that $\Lambda_{2}$ can be generically finite to its image only if
$(d-e)^{2}\geq d+e-2$ (Remark 4.13 and Remark 4.6).

Combining with the finiteness result ([Gor15] in Table 1), we can see the
following:

######  Corollary 1.5.

For $d\geq 2$, the degree of $\Lambda_{3,(1,d)}$ is at most

$\frac{\gcd(d+1,2)2^{2d-2}(d^{5}+d^{4}-d^{2}+2d)^{2d-2}\cdot(d-2)!(d-3)!}{2\cdot
3^{2d-2}\cdot(d+1)\cdot(2d-1)!}.$

For $d=3$, this only gives the evaluation $\deg\Lambda_{3,(1,d)}\leq 4369320$.

######  Remark 1.6.

In [Sch16], an algorithm to count the degree of multiplier maps using
equivariant Gromov-Witten invariant is given. The author have not completed
the evaluation of the order of the recursion formula.

This paper is organized as follows. In Section 2, we set up notation and
terminology. In Section 3, we give an evaluation of extension degree of
rational function field by using the Hilbert series. In Section 4, we evaluate
the degree of multiplier map using the evaluation in Section 3. In Section 5,
we give two proofs of Theorem 1.2.

### Acknowledgments

I express my deepest appreciation to Seidai Yasuda for providing MAGMA[BCP97]
emvironment and proofreading the original version of this paper, which is the
latter half of my master thesis consists of Section 3 and the
$\operatorname{Dyn}_{1,d}$ case of Section 4. I am also grateful to Takehiko
Yasuda for supervising me and giving useful discussion. I apologize to
Benjamin Hutz, Zhuchao Ji and Junyi Xie for the late uploading of the pricise
of the correction. I also appliciate to Zhuchao Ji and Junyi Xie for contacted
me about [Hut21].

The author is granted by Kakenhi DC1(22KJ2090(21J22197)).

##  2 Notation and Terminology

Throughout this paper, we refer [Liu06] for the terminology of algebraic
geometry.

We fix a field $k$ of characteristic zero. Unless otherwise stated, we
consider any scheme as a scheme over $k$.

For a ring $R$ and a free $R$-module $M$ of finite rank, we denote by $R[M]$
the polynomial ring generated by a basis of $M$ (with suitable identifications
between different choices of basis). If a group $G$ and a representation
$\rho:G\to\operatorname{Aut}_{R}(M)$ are also given, we write $R[M]^{G}$ for
the invariant ring.

##  3 Volume of Algebra and Rational Field of Projective Variety

To evaluate degrees of multiplier maps, we need to evaluate the degree of a
rational map to its image. In [Got23], the moduli space
$\operatorname{Dyn}_{d,e}$ of dynamiacal systems is given by the projective
scheme $\operatorname{Proj}I(V_{d}\otimes V_{e})$ of the naturally graded
invariant ring $I(V_{d}\otimes V_{e}):=k[V_{d}\otimes
V_{e}]^{\operatorname{SL}_{2}}$. Here we have a problem that the graded ring
is not fully generated by linear terms. Moreover, full generator (secondary
invariants) and relations (syzygies) are not known for generic cases
([Oli17]). Moreover, we only have little information about multiplier maps. So
we use an evaluation only using Hilbert series. We use a trivial evaluation
(Proposition 3.9), maybe well-known for experts.

######  Definition 3.1.

(Hilbert-Poincaré Series) For a graded $k$-algebra
$A=\bigoplus_{i=0}^{\infty}A_{i}$, the hilbert series of $A$ is the series

$H_{A}(t):=\sum_{i=0}^{\infty}\left(\dim_{k}A_{i}\right)t^{n}.$

######  Definition 3.2.

For a graded $k$-algebra $A=\bigoplus_{i=0}^{\infty}A_{i}$ of Krull dimension
$d$, the volume of $A$ is defined by

$\operatorname{Vol}(A):=\lim_{t\to 1}(1-t)^{d}H_{A}(t).$

######  Remark 3.3.

In [DK15], the degree of $A$ is used instead of the volume of $A$. We choose
the word “the volume of $A$” to avoid confusing with the extension degree of
algebras.

######  Definition 3.4.

Let $A=\bigoplus_{i=0}^{\infty}A_{i}$ be a graded $k$-algebra.

  1. (i)

the algebra $A$ is saturated if $\dim A_{i}\neq 0$ for any sufficiently large
$i$.

  2. (ii)

The saturator of $A$ is the minimal positive integer $n$ such that the algebra
$A^{[n]}:=\bigoplus_{i=0}^{\infty}A_{ni}$ is satulated by the grading which
$A_{ni}$ is degree $i$.

######  Proposition 3.5.

Let $A=\bigoplus_{i=0}^{\infty}A_{i}$ be a finitely generated saturated graded
$k$-algebra of Krull dimension $d$ which is an integral domain. Then we have

$\dim_{k}A_{i}=\frac{\operatorname{Vol}(A)i^{r-1}}{(r-1)!}+O(i^{r-2})\
(i\to\infty).$

###### Proof.

From the proof of [DK15, Proposition 1.4.5], it follows that if $B$ is a
finitely generated graded algebra, the Hilbert series $H_{B}(t)$ has the form

$H_{B}(t)=\frac{P_{B}(t)}{\prod_{i=1}^{d}(1-t^{a_{i}})}$ (1)

where $P_{B}(t)$ is a polynomial, $d$ is the Krull dimension of $B$ and
$a_{i}$’s are positive integers.

Let us take nonzero elements $f_{m}\in A_{m}$ and $f_{n}\in A_{n}$ for coprime
positive integers $m$ and $n$. We assumed that $A$ is an integral domain,
therefore $f_{n}$ and $f_{m}$ are both regular element of $A$. We have

$H_{A}(t)=(1-t^{m})H_{A/f_{m}A}(t)=(1-t^{n})H_{A/f_{n}A}(t)$ (2)

for the natural gradings on $A/f_{m}A$ and $A/f_{n}A$. By writing (2) in the
form of (1), we obtain that the order of the pole of $H_{A}(t)$ other than $1$
is at most $d-1$. Thus the Hilbert function $H_{A}(t)$ has the partial
fraction decomposition of the form

$H_{A}(t)=\frac{\operatorname{Vol}(A)}{(1-t)^{d}}+\sum_{i=1}^{d-1}\sum_{\zeta\in\mu_{\infty}}\frac{n_{\zeta,i}}{(1-\zeta
t)^{i}},$

where $\mu_{\infty}$ is the set of all roots of unity and $n_{\zeta,i}=0$
except for finitely many $\zeta\in\mu_{\infty}$. The Taylor expansion

$\frac{1}{(1-t)^{e}}=\sum_{i=0}^{\infty}\binom{e+i-1}{e-1}t^{n}$

leads to the assertion of proposition. ∎

######  Corollary 3.6.

Let $A=\bigoplus_{i=0}^{\infty}A_{i}$ be a finitely generated graded
$k$-algebra of Krull dimension $d$ which is an integral domain. Let $n$ be the
saturator of $A$. Then we have

$\dim_{k}A_{ni}=\frac{n\operatorname{Vol}(A)(ni)^{r-1}}{(r-1)!}+O((ni)^{r-2})\
(i\to\infty).$

###### Proof.

Let
$H_{A^{[n]}}(t)=\frac{\operatorname{Vol}(A^{[n]})}{(1-t)^{d}}+\sum_{i=1}^{d-1}\sum_{\zeta\in\mu_{\infty}}\frac{n_{\zeta,i}}{(1-\zeta
t)^{i}}$ be the partial fraction decomposition. Here we have
$H_{A}(t)=H_{A^{[n]}}(t^{n})$, therefore

$\displaystyle\operatorname{Vol}(A)$ $\displaystyle=\lim_{t\to
1}(1-t)^{d}\left(\frac{\operatorname{Vol}(A^{[n]})}{(1-t^{n})^{d}}+\sum_{i=1}^{d-1}\sum_{\zeta\in\mu_{\infty}}\frac{n_{\zeta,i}}{(1-\zeta
t)^{i}}\right)$ $\displaystyle=\operatorname{Vol}(A^{[n]})\cdot\lim_{t\to
1}\frac{(1-t)^{d}}{(1-t^{n})^{d}}$
$\displaystyle=\operatorname{Vol}(A^{[n]})\cdot\frac{1}{n^{d}}.$

By substituting this into Proposition 3.5 for $A^{[n]}$, we obtain the
assertion. ∎

From now on, we fix a graded algebra $A:=\bigoplus_{i=0}^{\infty}A_{i}$ which
is an integral domain. We also fix a graded subalgebra $B$ of $A$. For any
graded algebra $C$, we write $KP(C)$ for the rational function field
$K(\operatorname{Proj}C)$. We have

$KP(A)=\bigcup_{i=0}^{\infty}\left\\{\left.\frac{a_{i}}{a^{\prime}_{i}}\
\right|a_{i}\in A_{i},\ a^{\prime}_{i}\in A_{i}\setminus\\{0\\}\right\\}.$

We assume that $KP(A)$ is a finite extension of $KP(B)$ and write $D$ for the
degree of extension.

######  Proposition 3.7.

There exists a $KP(B)$-basis of $KP(A)$ of the following form:

$\left\\{\left.\frac{a_{i}}{b_{i}}\ \right|i=1,2,\ldots,D,\ b_{i}\in
B_{n_{i}},a_{i}\in A_{n_{i}}\right\\}.$

###### Proof.

We write $K$ for the field $KP(B)$. Let $\\{\frac{a_{i}}{a^{\prime}_{i}}\mid
i=1,2,\ldots,D,\ a_{i},a^{\prime}_{i}\in A_{n_{i}}\\}$ be a $K$-basis of
$KP(A)$. If $B_{n_{i}}=0$ we replace $(a_{i},a^{\prime}_{i})$ by
$(aa_{i},aa^{\prime}_{i})$ for $a\in A$ of sufficiently large degree, and then
we can assume $B_{n_{i}}\neq 0$ and take $b_{i}\in A_{n_{i}}\setminus\\{0\\}$.
We have

$K\left(\frac{a_{1}}{b_{1}},\frac{a^{\prime}_{1}}{b_{1}},\ldots,\frac{a_{D}}{b_{D}},\frac{a^{\prime}_{D}}{b_{D}}\right)=KP(A)$

and each $\frac{a_{i}}{b_{i}}$ or $\frac{a^{\prime}_{i}}{b_{i}}$ is integral
over $K$. Therefore, for sufficiently large $N$,

$\left\\{\left.\frac{a_{1}^{e_{1}}\cdots a_{D}^{e_{D}}a_{1}^{\prime
e^{\prime}_{1}}\cdots a_{D}^{\prime
e^{\prime}_{D}}}{b_{1}^{e_{1}+e^{\prime}_{1}}\cdots
b_{D}^{e_{D}+e^{\prime}_{D}}}\ \right|\sum_{i}e_{i}+\sum_{i}e^{\prime}_{i}\leq
N\right\\}$

is a generator of $KP(A)$ as a $K$-vector space. ∎

By reducing to a common denominator, we obtain the following:

######  Corollary 3.8.

There exists a $KP(B)$-basis of $KP(A)$ which is the form

$\left\\{\left.\frac{a_{i}}{b_{0}}\ \right|i=1,2,\ldots,D,\ b_{0}\in
B_{n},a_{i}\in A_{n}\right\\}.$

######  Proposition 3.9.

We have

$\frac{\operatorname{Vol}(A)}{\operatorname{Vol}(B)}\geq\frac{s_{A}\operatorname{Vol}(A)}{s_{B}\operatorname{Vol}(B)}\geq[KP(A):KP(B)],$

where $s_{A}$ and $s_{B}$ are the satulators of $A$ and $B$ respectively.

###### Proof.

The first inequality is immediate from the inclusion $B\subset A$.

Let $\\{\frac{a_{i}}{b_{0}}\mid i=1,2,\ldots,D,\ b_{0}\in B_{m},a_{i}\in
A_{m}\\}$ be a $KP(B)$-basis of $KP(A)$ given by Corollary 3.8. Then, the
morphism

$B^{\oplus D}\ni(b_{i})\mapsto\sum_{i=1}^{D}a_{i}b_{i}\in A$

is injective. Therefore, we have $\dim A_{m+n}\geq D\dim B_{m}$ for an
arbitrary $m$. By applying Corollary 3.6, we have

$\frac{s_{A}\operatorname{Vol}(A)(m+n)^{r-1}}{(r-1)!}+O((m+n)^{r-2})\geq
D\frac{s_{B}\operatorname{Vol}(B)m^{r-1}}{(r-1)!}+O(m^{r-2})(m\to\infty).$

Therefore, we have

$s_{A}\operatorname{Vol}(A)\geq Ds_{B}\operatorname{Vol}(B).$

∎

##  4 Degree Bound of Multiplier Map

The volumes of the invariant algebras of irreducible representations of
$\operatorname{SL}_{2}$ are classically calculated by Hilbert and the
reducible cases are done in [dCCPHHS20].

###  4.1 Schur Polynomial

In [dCCPHHS20], Schur Polynomials are used to express the volumes of invariant
rings. We briefly introduce the polynomials in a form that we can instantly
give an explicit evaluation of the volumes.

######  Definition 4.1.

(Schur Polynomial) For a sequence of nonnegative integers $(d_{i})$ of length
$l$, the Schur polynomial $s_{(d_{i})}(x_{i})$ of $(d_{i})$ is the symmetric
polynomial of $l$ variables such that

$s_{(d_{i})}(x_{i})=\frac{\det(x_{i}^{j+d_{j}-1})_{i,j=1}^{l}}{\det(x_{i}^{j-1})_{i,j=1}^{l}}.$

######  Definition 4.2.

For a nonincreasing sequence of nonnegative integers $(d_{i})$, the Young
tableau of $(d_{i})$ is the set of lattice points

$T(d_{i}):=\\{(i,j)\in\mathbb{Z}^{2}\mid 1\leq i\leq d_{j}\\}.$

The set of $n$-semistandard tableau is the set

$\operatorname{SST}_{n}(d_{i}):=\\{f:T(d_{i})\to\\{1,\ldots,n\\}\mid
f(i,j)\leq f(i+1,j),f(i,j)<f(i,j+1)\\}.$

######  Theorem 4.3.

(Kostka’s Definition [Pra19, Corollary 12.5]) For a decreasing sequence of
nonnegative integers $(d_{1},\ldots,d_{l})$, we have

$s_{(d_{1},\ldots,d_{l},0,0,\ldots,0)}(x_{1},\ldots,x_{l+k})=\sum_{f\in\operatorname{SST}_{l+k}(d_{i})}\prod_{(i,j)\in
T(d_{i})}x_{f(i,j)}.$

###  4.2 Degrees of Linear Systems of Multiplier Maps

The graph cycle $\Gamma_{d,e}$ over the moduli of correspondence is the
irreducible hyperplane

$V_{+}\left(\sum_{i,j}a_{ij}x^{i}y^{j}\right)\subset\mathbb{P}(\langle
a_{ij}\rangle)\times\mathbb{P}^{1}_{x}\times\mathbb{P}^{1}_{y}=\operatorname{Corr}_{d,e}\times\mathbb{P}^{1}_{x}\times\mathbb{P}^{1}_{y}.$

As a hyperplane, this is of degree $(1,d,e)$. Let
$f(x,y):=\sum_{i,j}a_{ij}x^{i}y^{j}$ be the defining polynomial of $\Gamma$.
The $n$-th iteration of $f$, $\Psi_{n}(f)$ is given by

$\Psi_{n}(f)(z_{0},z_{n})=\operatorname{res}_{z_{1},\ldots,z_{n-1}}(f(z_{0},z_{1}),f(z_{1},z_{2}),\ldots,f(z_{n-1},z_{n})).$

Let $\Psi_{n}\Gamma$ be the graph of iteration morphism

$\Psi_{n}\Gamma:=V_{+}\left(\Psi_{n}(f)(x,y)\right)\subset\operatorname{Corr}_{d,e}\times\mathbb{P}^{1}_{x}\times\mathbb{P}^{1}_{y}.$

By using the expression of composition using the resultant [Got23, Section 5]
and the Sylvester formula, $\Psi_{n}\Gamma$ is of degree
$(\frac{d^{n}-e^{n}}{d-e},d^{n},e^{n})$. The cycle of periodic points
$\operatorname{Per}_{n}$ of period $n$ is given by

$\operatorname{Per}_{n}:=V_{+}(\Psi_{n}(f)(z,z))\subset\operatorname{Corr}_{d,e}\times\mathbb{P}^{1}_{z}.$

From the degree of $\Psi_{n}\Gamma$, the degree of $\operatorname{Per}_{n}$ is
$(\frac{d^{n}-e^{n}}{d-e},d^{n}+e^{n})$.

We remark that the cycle of periodic points $\operatorname{Per}_{n}$ includes
the cycles of fixed points, and moreover the cycles $\operatorname{Per}_{m}$
for $m|n$. We define the scheme $\operatorname{Per}_{n}^{*}$ of periodic
points of formal period $n$ by extracting the periodic points of shorter
periods. More explicitly, we define effective divisors
$\operatorname{Per}^{*}_{n}$ inductively as

$\operatorname{Per}^{*}_{1}:=\operatorname{Per}_{1},\operatorname{Per}_{n}^{*}:=\operatorname{Per}_{n}-\sum_{m<n,m|n}\operatorname{Per}_{m}.$

Let $\nu_{n}(x)$ be the family of polynomials, asymptotically defined by

$\nu_{1}(x)=x,\nu_{n}(x)=x^{n}-\sum_{m<n,m|n}\nu_{m}(x).$

In a closed form, $\nu_{n}$ is written by using the Möbius function $\mu$,

$\nu_{n}(x)=\sum_{m|n}\mu\left(n/m\right)x^{n}.$

Then the degree of $\operatorname{Per}^{*}_{n}$ is given by

$\left(\frac{\nu_{n}(d)-\nu_{n}(e)}{d-e},\nu_{n}(d)+\nu_{n}(e)\right).$

We write $\Pi_{n}^{*}f(z)$ for a defining form of the divisor
$\operatorname{Per}^{*}_{n}$.

######  Remark 4.4.

We have $\nu_{n}(1)=0$ for $n>1$. For the cases only considering rational
maps, $\nu_{n}(d)+\nu_{n}(1)$ is sometimes used instead of $\nu_{n}(d)$ (for
instance, $\nu_{n}(d)$ in [Sil07, Remark 4.3] and $N_{n}(d)$ in [DM06, Chapter
4]).

######  Proposition 4.5.

If the multiplier map

$\lambda_{n,(d,e)}:\operatorname{Corr}_{d,e}\dashrightarrow\operatorname{Corr}_{d^{n},e^{n}}\dashrightarrow\mathbb{P}^{d^{n}+e^{n}}$

is well-defined, then it is given by a linear system of degree

$2(d^{n}+e^{n}-1)\frac{d^{n}-e^{n}}{d-e}.$

Moreover, in this case, we can define the multiplier map of the periodic
orbits of period $n$,

$\lambda^{\circ}_{n,(d,e)}:\operatorname{Corr}_{d,e}\dashrightarrow\mathbb{P}^{(\nu_{n}(d)+\nu_{n}(e))/n}$

and it is given by a linear system of degree at most

$\frac{2((d^{n}-1)\nu_{n}(d)-(e^{n}-1)\nu_{n}(e))}{n(d-e)}.$

###### Proof.

By [Got23, Section 7] and the Sylvester formula, the degree of fixed point
multiplier map is $2(d+e-1)$. Since the morphism
$\Psi_{n}:\operatorname{Corr}_{d,e}\dashrightarrow\operatorname{Corr}_{d^{n},e^{n}}$
is given by a linear system of degree $\frac{d^{n}-e^{n}}{d-e}$, we obtain the
first assertion. Moreover, the morphism $\lambda_{n}:=\lambda_{n,(d,e)}$ is
given by

$\displaystyle\lambda_{n}([f])$ $\displaystyle=\lambda_{1,(d,e)}([\Psi_{n}f])$
$\displaystyle=\left[\prod_{z:\Psi_{n}f(z,z)=0}\left(\partial_{x}\Psi_{n}f(z,z)dx+\partial_{y}\Psi_{n}f(z,z)dy\right)\right]\in\mathbb{P}(D_{N}),$
(3)

where $N=\deg_{z}\Psi_{n}f(z,z)=d^{n}+e^{n}$. The well-definedness of
$\lambda_{n}$ implies that the factors of (3) are not zero. Therefore, we can
define $\lambda^{\bullet}_{n}([f])$ as

$\lambda^{\bullet}_{n}([f]):=\left[\prod_{z:\Pi_{n}^{*}f(z)=0}\left(\partial_{x}\Psi_{n}f(z,z)dx+\partial_{y}\Psi_{n}f(z,z)dy\right)\right]\in\mathbb{P}(D_{M}),$
(4)

where $M=\deg_{z}\Pi_{n}^{*}f(z)=\nu_{n}(d)+\nu_{n}(e)$. By [Got23, Remark
7.6], we can write this multiplier map as

$\displaystyle\lambda^{\bullet}_{n}([f])$
$\displaystyle=\left[\operatorname{res}_{z}\left(\Pi_{n}^{*}f(z),\partial_{x}\Psi_{n}f(z,z)dx+\partial_{y}\Psi_{n}f(z,z)dy\right)\right]$
$\displaystyle=\left[\operatorname{res}_{z}\left(\Pi_{n}^{*}f(z),d_{z}\Psi_{n}f(z,z)dz_{0}+z_{0}z_{1}\Omega^{1}\Psi_{n}f(z)dz_{1}\right)/A_{n,0}A_{n,1}\right],$
(5)

where $A_{n,0}$ and $A_{n,1}$ are the coefficients of respectively $z_{0}^{M}$
and $z_{1}^{M}$ of $\Pi_{n}^{*}f(z)$. Therefore, by the Sylvester formula, the
rational map $\lambda^{\circ}_{n}([f])$ is given by the linear system given by
the coefficients of $dz_{0}^{i}dz_{1}^{M-i}$ of

$\operatorname{res}_{z}\left(\Pi_{n}^{*}f(z),d_{z}\Psi_{n}f(z,z)dz_{0}+z_{0}z_{1}\Omega^{1}\Psi_{n}f(z)dz_{1}\right)/A_{n,0}A_{n,1},$
(6)

and their degree is

$(d^{n}+e^{n})\frac{\nu_{n}(d)-\nu_{n}(e)}{d-e}+(\nu_{n}(d)+\nu_{n}(e))\frac{d^{n}-e^{n}}{d-e}-2\frac{\nu_{n}(d)-\nu_{n}(e)}{d-e}.$
(7)

Any periodic orbit of formal period $n$, of a correspondence defined by
$f(x,y)$ is given by a tuple of points $(z_{0},z_{1},\ldots,z_{n}=z_{0})$ such
that $f(z_{i},z_{i+1})=0$. From the differential of composite functions, for
the periodic points of the same periodic orbits, the factor in (4) takes the
same value, that is,

$\lambda^{\bullet}_{n}([f])=\left[\prod_{\begin{subarray}{c}(z_{0},\ldots,z_{n-1}):\\\
\text{Periodic orbits of
}f(x,y)\end{subarray}}\left(dx+\frac{\partial_{y}\Psi_{n}f(z_{0},z_{0})}{\partial_{x}\Psi_{n}f(z_{0},z_{0})}dy\right)^{n}\right].$
(8)

This leads that the map $\lambda^{\bullet}_{n,(d,e)}([f])$ is given by an
$n$-th power of some rational function. Therefore, we can define
$\lambda^{\circ}_{n}$ as an $n$-th root of some quotient of (6). ∎

######  Remark 4.6.

By definition, for the Veronese embedding

$\nu_{n}:\mathbb{P}^{M/n}\simeq\mathbb{P}(k[x,y]_{M/n})\to\mathbb{P}(k[x,y]_{M})\simeq\mathbb{P}^{M}:f\mapsto
f^{n},$

we have $\lambda^{\bullet}_{n}=\nu_{n}\circ\lambda^{\circ}_{n}$. Moreover,
since for any periodic orbit $(z_{0},\ldots,z_{n-1},z_{n}=z_{0})$ we have

$\frac{\partial_{y}\Psi_{mn}f(z_{0},z_{0})}{\partial_{x}\Psi_{mn}f(z_{0},z_{0})}=\left(\frac{\partial_{y}\Psi_{n}f(z_{0},z_{0})}{\partial_{x}\Psi_{n}f(z_{0},z_{0})}\right)^{m},$

we can see that

$\operatorname{Im}\Lambda_{n}\simeq\operatorname{Im}\Lambda^{\bullet}_{n}\simeq\operatorname{Im}\Lambda^{\circ}_{n},$

where

$\Lambda^{\alpha}_{n}:=\prod_{m:m|n}\lambda^{\alpha}_{m}\text{ for
}\alpha\in\\{\circ,\bullet\\}.$

######  Remark 4.7.

Despite the form in (5) is given by an $n$-th power of some polynomial, it is
difficult to obtain more explicit form of the $n$-th root
$\lambda^{\circ}_{n}$. This phenomenon happens in computing resultant by
Cayley’s formula ([ESW03]). If $n=2$, Pfaffian is sometimes used to compute
Cayley’s formula. Whether analogous method exist for the second multiplier map
is a problem. As we see in (16), if we have a method to choose a specified
branch of the roots, we can compute the root directly by interpolation.

###  4.3 Evaluation

We use the following result to calculate the volume.

######  Theorem 4.8.

([dCCPHHS20]) Let $V\in\operatorname{Rep}_{k}(V_{1})$ be a representation of
$\operatorname{SL}_{2}(k)$ and $\operatorname{dim}_{k}V=n$. Then the Taylor
expansion of the Hilbert function of the invariant ring
$I(V):=k[V]^{\operatorname{SL}_{2}(k)}$ at $t=1$ has the form

$H_{I(V)}(t)=(1-t)^{-d+3}\cdot\sum_{i=0}^{\infty}\gamma_{i}(1-t)^{i}.$

Let $(a_{i})$ be the positive weights of $V$ and $l$ be the length of the
sequence. Then we have

$\gamma_{0}=\gcd(2,a_{1},\ldots,a_{l})\frac{s_{(l-3,l-3,l-3,l-4,l-5,\ldots,2,1,0)}(a_{1},a_{2},\ldots,a_{l})}{s_{(l-1,l-2,l-3,l-4,l-5,\ldots,2,1,0)}(a_{1},a_{2},\ldots,a_{l})}.$

######  Remark 4.9.

In [dCCPHHS20], higher terms ($\gamma_{1},\gamma_{2},\gamma_{3}$) are also
computed.

Throughout this subsection, we put $n:=d+e$. We give a rough estimate of
$\gamma_{0}$ for the case $V=V_{d}\otimes V_{e}$ of the moduli space of
dynamical systems.

######  Lemma 4.10.

We have

$\operatorname{Vol}I(V_{d}\otimes V_{e})\leq\frac{\gcd(n,2)}{2(n-2)(n-1)n}.$

###### Proof.

We put $\alpha:=(l-3,l-3,l-3,l-4,l-5,\ldots,1)$ and
$\delta:=(l-1,l-2,\ldots,2,1)$. Let
$\Phi_{k}:\operatorname{SST}_{l}(\alpha)\to\operatorname{SST}_{l}(\delta)$ for
$k=1,2$ be the map such that for any $f\in\operatorname{SST}_{l}(\alpha)$

$\Phi_{k}(f)(i,j):=\begin{cases}f(i,j)&((i,j)\in T(\alpha))\\\
l-1&((i,j)=(1,l-2))\\\ l&((i,j)=(2,l-2))\\\ l-1&((i,j)=(1,l-1),\ k=1)\\\
l&((i,j)=(1,l-1),\ k=2)\end{cases}.$

The maps $\Phi_{1}$ and $\Phi_{2}$ are both injective and the images are
disjoint. Therefore by Theorem 4.3 we have

$\displaystyle s_{\delta}(x_{1},\ldots,x_{l})=$
$\displaystyle(x_{l-1}x_{l}^{2}+x_{l-1}^{2}x_{l})s_{\alpha}(x_{1},\ldots,x_{l})$
$\displaystyle+(\text{polynomial with nonnegative coefficients}).$ (9)

The three largest among the weights of the representation $V_{d}\otimes V_{e}$
are $(n-2,n-2,n)$ and other weights are smaller than $n-2$. By substituting
the weights into (9), we obtain

$s_{\delta}(\nu,n-2,n-2,n)\geq 2(n-2)(n-1)n\cdot s_{\alpha}(\nu,n-2,n-2,n),$

where we put the sequence of positive weights smaller than $n-2$ by $\nu$. By
Theorem 4.8, we have

$\displaystyle\operatorname{Vol}(I(V_{d}\otimes V_{e}))$
$\displaystyle=\gcd(2,n)\frac{s_{\alpha}(\nu,n-2,n-2,n)}{s_{\delta}(\nu,n-2,n-2,n)}$
$\displaystyle\leq\gcd(2,n)\frac{1}{2(n-2)(n-1)n}.$

∎

######  Theorem 4.11.

Let $p$ be a prime number. If the first and the $p$-th multiplier map to the
image

$\Lambda^{\circ}_{p}:=\lambda^{\circ}_{1,(d,e)}\times\lambda^{\circ}_{p,(d,e)}:\operatorname{Dyn}_{d,e}\dashrightarrow\Lambda(\operatorname{Dyn}_{d,e})\subset\mathbb{P}(D_{d+e})\times\mathbb{P}(D_{M})$

is finite, then its degree is at most

$\frac{\gcd(n,2)N^{de+n-3}(n-3)!(de-3)!}{2n\cdot(de+n-3)!},$

where

$N:=2(d+e-1)+\frac{2((d^{p}-1)(d^{p}-d)-(e^{p}-1)(e^{p}-e))}{p(d-e)}.$

###### Proof.

We put $A:=I(V_{d}\otimes V_{e})$. Let $L_{1}$ and $L_{p}$ be the linear
systems which gives $\lambda^{\circ}_{1}$ and $\lambda^{\circ}_{p}$. By
Proposition 4.5, we can take the linear systems in $A$ such that whose degrees
are respectively at most

$2(d+e-1)\text{ and }\frac{2((d^{p}-1)(d^{p}-d)-(e^{p}-1)(e^{p}-e))}{p(d-e)}.$

By assumption and algebraic independence of discriminant-resultant [Gor15], we
can take $n$ elements $f_{1},\ldots,f_{n}$ in $L_{1}$ and
$\dim\operatorname{Dyn}_{d,e}-(n-1)=de-2$ elements $g_{1},\ldots,g_{de-2}$ in
$L_{m}$ to be algebraically independent.

Let $k[L_{1}\otimes L_{p}]\to A$ be the morphism of graded $k$-algebras
defined by $L_{1}\otimes L_{p}\ni f\otimes g\mapsto fg\in A$ and $B_{(1,p)}$
be its image. Then we have the degree of the morphism $\Lambda$ is the
extension degree $[KP(A):KP(B_{(1,p)})]$ of rational function fields.

The graded subalgebra $B_{(1,p)}\supset B:=k[f_{i}g_{j}\mid 1\leq i\leq n,\
1\leq j\leq de-2]$ of $A$ has the Hilbert series

$H_{B}(t)=\sum_{i=0}^{\infty}\binom{i+n-1}{n-1}\binom{i+de-3}{de-3}t^{iN}.$

By Corollary 3.6, we have

$\displaystyle\operatorname{Vol}(B)=\frac{1}{N^{de+n-3}}\cdot\frac{(de+n-3)!}{(de-3)!(n-1)!}.$

Therefore we have

$\displaystyle[KP(A):KP(B_{(1,p)})]$ $\displaystyle\leq[KP(A):KP(B)]$
$\displaystyle\leq\frac{\operatorname{Vol}(A)}{\operatorname{Vol}(B)}$
$\displaystyle\leq\frac{\gcd(n,2)}{2n(n-1)(n-2)}\cdot\frac{N^{de+n-3}(n-1)!(de-3)!}{(de+n-3)!}$
(10) $\displaystyle=\frac{\gcd(n,2)N^{de+n-3}(n-3)!(de-3)!}{2n\cdot(de+n-3)!}$

from Proposition 3.9. ∎

######  Remark 4.12.

By skipping Lemma 4.10, we can use

$\frac{s_{\delta}(\text{positive weights of }V_{d}\otimes
V_{e})}{s_{\alpha}(\text{positive weights of }V_{d}\otimes V_{e})}$ (11)

instead of $1/2n(n-1)(n-2)$ in (10). Experimentally (11) looks like of order
$O(n^{-(4+O(1))})$, but this difference of orders may be very small comparing
to $N^{de+d+e-3}$.

######  Remark 4.13.

From number of periodic orbit, and Holomorphic Lefschetz formula ([Ill77],
[Got23]) for $C$ and $\Psi_{2}C$, the dimension of fiber of
$\Lambda_{2}^{\circ}$ is at least

$\displaystyle(d+e-1)+\frac{(d^{2}-d)+(e^{2}-e)}{2}-1-(de+d+e-3)$
$\displaystyle=\frac{(d-e)^{2}-(d+e)+2}{2}.$ (12)

In particular, the map $\Lambda_{2}^{\circ}$ can be generically finite to its
image only if $(d-e)^{2}\geq d+e-2$.

##  5 Birationality of the Second Multiplier Map of Cubic Maps

In this section, we give two proofs of the following theorem.

######  Theorem 5.1.

The multiplier map $\Lambda_{2,(1,3)}$ is birational to its image.

###  5.1 Finite field reduction

The counting of the degree of

$\Lambda_{2}:=\lambda_{1,(1,3)}\times\lambda_{2,(1,3)}:\operatorname{Dyn}_{1,3}\to\mathbb{P}^{3}\times\mathbb{P}^{9}$

to its image is done in [HT13] by the following method. First, we fix a point
$P\in\lambda_{1}(\operatorname{Dyn}_{1,3})$ and consider the inverse image
$l:=\lambda_{1}^{-1}(P)$. An explicit morphism
$\phi_{P}:\mathbb{P}^{1}\to\operatorname{Corr}_{1,3}$ such that $l$ is
birational to the image of
$\mathbb{P}^{1}\xrightarrow{\phi_{P}}\operatorname{Corr}_{1,3}\xrightarrow{\pi}\operatorname{Dyn}_{1,3}$
is given in [HT13]. We denote the endomorphism on $\mathbb{P}^{1}$ indicated
by the point $\phi_{P}(a)$ by $\phi_{P,a}$. Then we will solve the equations
in the two variables $a$ and $b$,

$\displaystyle\phi_{P,a}^{2}(b)$ $\displaystyle=b$ (13)
$\displaystyle(\phi_{P,a}^{2})^{\prime}(b)$ $\displaystyle=\lambda$ (14)
$\displaystyle\phi_{P,a}(b)$ $\displaystyle\neq b$ (15)

for a given $P$ and $\lambda$. The equations (13) and (14) are of degree 9 and
16 respectively, in variables $a$ and $b$. By a MAGMA computation over a
finite field, we obtain the solutions as a 0-dimensional closed subscheme $Z$
of degree 144 on $\mathbb{P}^{2}$. Under a base-change to the algebraically
closed field, the support of $Z$ consists of 18 points. Six of them are non-
reduced and does not satisfy the inequality (15). Remaining 12 points
satisfies (15), moreover the MAGMA computation shows that they are reduced.

That was the computation done in [HT13]. We proceed computation from here. At
first, we note that for a solution $(a,b)$ of (13),(14) and (15), the points
$(a,\phi_{P,a}(b))$ is also a solution of equation. Therefore, we obtain 6
rational maps $\phi_{P,a}$ with periodic points of period two. For a value
$\lambda$, the solutions are given by explicit values of $(a,b)$. For the 6
rational maps, we compute other multipliers of periodic points of period 2.
Then we obtain that the values of other multipliers are mutually different, so
we obtain that $\Lambda_{3,2}$ is injective.

###  5.2 Direct computation with Graded-decomposition and interpolation

In this section, we show Theorem 5.1 by computing the full formula of the
second multiplier map $\Lambda^{\circ}_{2,(1,3)}$.

In the computational process of the explicit expression, we use the
information that the resulting polynomials are
$\operatorname{SL}_{2}$-invariant. Our algorithm (Subsection A.2) of graded-
piece-wise computation is applied for limited cases, but this method makes the
computation much faster.

We refer the expression of generators of the invariant ring $A:=I(V_{1}\otimes
V_{3})=I(V_{4}\oplus V_{2})$ given in [Wes15]. The invariant ring is given by

$A\simeq k[d,i,j,a,b,c]/r,$

where $d,i,j,a,b,c$ are generators of degree respectively $2,2,3,3,4,6$ and
$r$ is the relation

$2c^{2}=\frac{1}{54}d^{3}i^{3}-\frac{1}{9}d^{3}j^{2}-\frac{1}{12}di^{2}a^{2}-\frac{1}{3}ja^{3}+djab+\frac{1}{2}ia^{2}b-\frac{1}{2}dib^{2}-b^{3}.$

Let $f_{4}(z)$ and $f_{2}(z)$ be the fundamental covariants of $V_{4}$ and
$V_{2}$ respectively (denoted by $\mathbf{f}$ and $\mathbf{g}$ in [Wes15]
respectively). By [Got23, Remark 7.6], the first multiplier map $\lambda_{1}$
is given by the linear system consists of discriminant-resultants (named in
[Got22])

$\sigma_{r}:=DR_{4,r}(f_{4},f_{2})\ (r=0,2,3,4)\text{ of degree }(6-r,r),$

which are defined by

$\sum_{r=0}^{4}DR_{4,r}(f_{4},f_{2})t^{r}=\operatorname{res}_{z}(f_{4},\partial_{z}f_{4}+zf_{2}t).$

We put

$\Sigma_{\pm}:=\sigma_{0}+\sigma_{2}\pm\sigma_{3}+\sigma_{4}.$

By Proposition 4.5, we can have a linear system of $\lambda_{2}^{\circ}$ of
degree at most 24. Let

$L_{2}(t):=\operatorname{res}_{z}(\Pi_{2}^{*}f(z),d_{z}\Omega^{0}(\Psi_{2}f)(z)+t\cdot\Omega^{1}(\Psi_{2}f)(z)).$

In this case, $L_{2}(t)$ has a divisor $\Sigma_{-}^{4}$ and we can take a
square root of $L_{2}(t)/\Sigma_{-}^{4}$. We put the square root of $L_{2}(t)$
as

$\sqrt{L_{2}(t)}=:\delta(t)=:\delta_{0}+\delta_{1}t+\delta_{2}t^{2}+\delta_{3}t^{3}.$

The forms $\delta_{i}$’s are invariants in $A$ of degree $(48-4\cdot 6)/2=12$.
We have

$\displaystyle\delta_{0}^{2}\cdot\Sigma_{-}^{4}$
$\displaystyle=\operatorname{res}_{z}(\Pi_{2}^{*}f(z),d_{z}\Omega^{0}(\Psi_{2}f)(z))$
$\displaystyle=\Delta_{z}(\Pi_{2}^{*}f(z))\cdot\operatorname{res}_{z}(\Pi_{2}^{*}f(z),\Pi_{1}^{*}f(z))$

and by a direct computation we obtain

$\displaystyle\Delta_{z}(\Pi_{2}^{*}f(z))$
$\displaystyle=\Sigma_{+}\cdot\Sigma_{-}^{2}\cdot\phi^{2}\text{ and }$
$\displaystyle\operatorname{res}_{z}(\Pi_{2}^{*}f(z),\Pi_{1}^{*}f(z))$
$\displaystyle=\Sigma_{+}\cdot\Sigma_{-}^{2}\text{, where}$
$\displaystyle\phi$
$\displaystyle=2^{-27}\cdot(d^{3}-12d^{2}i+48di^{2}-64i^{3}+384j^{2}+288ja+54a^{2}),$

so we have $\delta_{0}=\pm\Sigma_{+}\cdot\phi$. By fixing the sign to be $+$,
we can compute $\delta(t)$ and obtain

$\displaystyle\delta_{1}=-\frac{1}{2}(-9\sigma_{0}+\sigma_{2}+6\sigma_{3}+11\sigma_{4})\cdot\phi,\delta_{2},\delta_{3}\in
A_{12}.$ (16)

Here we remark that

$K_{2}:=K(\Lambda_{2}(\operatorname{Dyn}_{1,3}))=k\left(\frac{\sigma_{i}}{\sigma_{j}},\frac{\delta_{i}}{\delta_{j}}\right)\subset
KP(A).$

So we start from the algebra

$B_{1}:=k[\sigma_{i}\cdot\phi(i=0,2,3,4),\delta_{2},\delta_{3}](\subset A)$

to larger sub-graded-algebras of $A$ with keeping the condition
$KP(B_{i})=K_{2}$. By seeking factorizable linear combinations of the
generators of $B_{1}$, we find

$\displaystyle\delta_{2}+\phi(-10\sigma_{0}+\sigma_{2}+10\sigma_{3})=\Sigma_{-}\cdot\psi,$
$\displaystyle\text{where
}\psi=\frac{1}{2^{31}}(-26048i^{3}+9936i^{2}d-884id^{2}+7d^{3}$
$\displaystyle\hskip 80.0pt-102912j^{2}-38784ja-72a^{2}-9600ib+2400db)\text{
and}$
$\displaystyle(\delta_{2}+\delta_{3})-\frac{1}{2}(11\sigma_{0}+\sigma_{2}-4\sigma_{3}-9\sigma_{4})\phi=\frac{5^{3}}{2^{19}}(\Sigma_{-}+\sqrt{2}\sigma_{3})(\Sigma_{-}-\sqrt{2}\sigma_{3}).$

Therefore, we can replace $B_{1}$ by

$B_{2}:=k[\sigma_{i},\phi,\psi].$

By computing the elimination ideal of generators of $B_{2}$, we obtain that
the only relation among the generators is only a relation of degree $60$
(degree $10$ polynomial of $\sigma_{i},\phi,\psi$), so we can see that the
Hilbert series of $B_{2}$ is given by

$H_{B_{2}}(t)=\frac{1-t^{60}}{(1-t^{6})^{6}}.$

Here we have

$\displaystyle H_{A}(t)$
$\displaystyle=\frac{1+t^{6}}{(1-t^{2})^{2}(1-t^{3})^{2}(1-t^{4})}.$

We recall that for any graded algebra $C$, $C^{[n]}:=\bigoplus_{i\geq
0}C_{in}$ with $\deg C_{in}=i$. By a direct computation, we have

$\displaystyle H_{A^{[6]}}(t)$
$\displaystyle=\frac{(1+t)(1+5t+9t^{2}+4t^{3})}{(1-t)^{4}(1-t^{2})}\text{ and
}H_{B_{2}^{[6]}}(t)=\frac{1-t^{10}}{(1-t)^{6}}.$

By Proposition 3.9, we have

$\deg(\Lambda_{2})=[KP(A^{[6]}):KP(B_{2}^{[6]})]\leq\frac{\operatorname{Vol}(A^{[6]})}{\operatorname{Vol}(B_{2}^{[6]})}=\frac{9}{5},$

this shows that $\deg(\Lambda_{2})=1$.

######  Remark 5.2.

Throughout this ad hoc proof, there are three steps completed unexpectedly
easily. The first is that there are factorizable linear combinations including
$\delta_{2}$ and $\delta_{3}$. The second is that a linear combination
moreover belongs to $k[\sigma_{i}]_{12}$. The third is that the relation among
$\sigma_{i},\phi,\psi$ was of degree 60. Because of this small degree (the
expected degree from the Hilbert series is 108), we can obtain the result in a
few minutes by simply computing the elimination ideal. Moreover, this degree
is also the lower bound to determine the extension degree to be $1$.

##  Appendix A Appendix: Programs

###  A.1 Programs used in Subsection 5.1

The MAGMA program run in [HT13] were the following.

  * l0:=3;
l1:=2;
l8:=4;
lB:=-5;
R<a,B,z>:=ProjectiveSpace(GF(101),2);
function h(P,d) Q:=0;
for i:=0 to d do for j:=0 to d-i do Q:=Q+ Term(Term(P,a,i),B,j)*z^(d-i-j);
end for;
end for;
return(Q);
end function;
function f(x,y) return((((l0 - 1)*l1 + (-l0 + 1))*x^3 + ((a*l0*l1 + (-l0 + (-a
+ 1)))*l8 + (((-a - 1)*l0 +1)*l1 + (2*l0 + (a - 2))))*x^2*y + ((-a*l0*l1 +
a*l0)*l8 + (a*l0*l1 - a*l0))*x*y^2));
end function;
function g(x,y) return((((l0 - 1)*l1 + (-l0 + 1))*l8*x^2*y + (((-l0 + (a +
1))*l1 + (a*l0 - 2*a))*l8 + (-a*l1 + ((-a + 1)*l0 +(2*a - 1))))*x*y^2 +
((-a*l1 + a)*l8 + (a*l1 - a))*y^3));
end function;
f1:=f(f(B,1),g(B,1));
g1:=g(f(B,1),g(B,1));
F1:=f1-B*g1;
F2:=g1*Derivative(f1,B) - f1*Derivative(g1,B) - lB*g1*g1;
G1:=h(F1,9);
G2:=h(F2,16);
C:=Scheme(R,[G1,G2]);
D:=ReducedSubscheme(C);
Degree(D);

After running this computation, we firstly compute the coordinates of the
points of $D$.

  * PointsOverSplittingField(D);
Output:
{@ (0 : 0 : 1), (0 : 4 : 1), (1 : 1 : 1), (1 : 49 : 1), (4 : 93*$.1^7 +
44*$.1^6 \+ 24*$.1^5 + 23*$.1^4 + 48*$.1^3 + 26*$.1^2 + 65*$.1 + 90 : 1), (4 :
8*$.1^7 \+ 57*$.1^6 + 77*$.1^5 + 78*$.1^4 + 53*$.1^3 + 75*$.1^2 + 36*$.1 + 79
: 1), (96 : 27 : 1), (96 : 6 : 1), (18*$.1^7 + 50*$.1^6 + 68*$.1^5 + 24*$.1^4
+ 59*$.1^3 + 22*$.1^2 + 93*$.1 + 93 : 55*$.1^7 + 55*$.1^6 + 80*$.1^5 +
72*$.1^4 + 5*$.1^3 + 89*$.1^2 + 52*$.1 + 10 : 1), (18*$.1^7 + 50*$.1^6 +
68*$.1^5 + 24*$.1^4 + 59*$.1^3 + 22*$.1^2 + 93*$.1 + 93 : 15*$.1^7 + 56*$.1^6
+ 83*$.1^5 + 3*$.1^4 + 62*$.1^3 + 95*$.1^2 + 21*$.1 + 80 : 1), (14*$.1^7 +
52*$.1^6 + 48*$.1^5 + 18*$.1^4 + 53*$.1^3 + 62*$.1^2 + 42*$.1 + 50 : 70*$.1^7
+ 7*$.1^6 + 3*$.1^5 + 27*$.1^4 + 47*$.1^3 + 32*$.1^2 + 64*$.1 + 28 : 1),
(14*$.1^7 + 52*$.1^6 + 48*$.1^5 + 18*$.1^4 + 53*$.1^3 + 62*$.1^2 + 42*$.1 + 50
: 88*$.1^7 + 85*$.1^6 + 73*$.1^5 + 50*$.1^4 + 65*$.1^3 + 59*$.1^2 + 96*$.1 +
70 : 1), (75*$.1^7 + 95*$.1^6 + 57*$.1^5 + 100*$.1^4 + 90*$.1^3 + 4*$.1^2 +
73*$.1 + 55 : 5*$.1^7 + 72*$.1^6 + 70*$.1^5 + 39*$.1^4 \+ 32*$.1^3 + 31*$.1^2
+ 74*$.1 + 26 : 1), (75*$.1^7 + 95*$.1^6 + 57*$.1^5 + 100*$.1^4 + 90*$.1^3 +
4*$.1^2 + 73*$.1 + 55 : 86*$.1^7 + 93*$.1^6 + 92*$.1^5 + 67*$.1^4 + 46*$.1^3 +
95*$.1^2 + 22*$.1 + 78 : 1), (95*$.1^7 + 5*$.1^6 + 29*$.1^5 + 60*$.1^4 +
13*$.1^2 + 95*$.1 + 87 : 66*$.1^7 + 33*$.1^6 \+ 62*$.1^5 + 50*$.1^4 + 83*$.1^3
+ 87*$.1^2 + 29*$.1 + 25 : 1), (95*$.1^7 + 5*$.1^6 + 29*$.1^5 + 60*$.1^4 +
13*$.1^2 + 95*$.1 + 87 : 19*$.1^7 + 3*$.1^6 \+ 42*$.1^5 + 96*$.1^4 + 64*$.1^3
+ 17*$.1^2 + 46*$.1 + 2 : 1), (47 : 1 : 0), (1 : 0 : 0) @}
Finite field of size 101^8

These are the coordinates $(a:b:z)$ of the solutions of (14) and (13) on
$\mathbb{P}^{2}$, with the homogenizing variable $z$. The parameters $a$ of 12
reduced points, consisted of 6 values as expected are the following.

  * {4,96,18*$.1^7 + 50*$.1^6 + 68*$.1^5 + 24*$.1^4 + 59*$.1^3 + 22*$.1^2 + 93*$.1 + 93, 14*$.1^7 + 52*$.1^6 + 48*$.1^5 + 18*$.1^4 + 53*$.1^3 + 62*$.1^2 + 42*$.1 + 50, 75*$.1^7 + 95*$.1^6 + 57*$.1^5 + 100*$.1^4 + 90*$.1^3 + 4*$.1^2 + 73*$.1 + 55, 95*$.1^7 + 5*$.1^6 + 29*$.1^5 + 60*$.1^4 + 13*$.1^2 + 95*$.1 + 87},

The multipliers are given by:

  * l0:=3;
l1:=2;
l8:=4;
lB:=-5;

F<w>:= GF(101,8);
R<a,B,z>:=PolynomialRing(F,3);

function f(x,y)
return((((l0 - 1)*l1 + (-l0 + 1))*x^3 + ((a*l0*l1 + (-l0 + (-a + 1)))*l8 +
(((-a - 1)*l0 +1)*l1 + (2*l0 + (a - 2))))*x^2*y + ((-a*l0*l1 + a*l0)*l8 +
(a*l0*l1 - a*l0))*x*y^2));
end function;
function g(x,y)
return((((l0 - 1)*l1 + (-l0 + 1))*l8*x^2*y + (((-l0 + (a + 1))*l1 + (a*l0 -
2*a))*l8 + (-a*l1 + ((-a + 1)*l0 +(2*a - 1))))*x*y^2 + ((-a*l1 + a)*l8 + (a*l1
- a))*y^3));
end function;
f2:=f(f(B,1),g(B,1));
g2:=g(f(B,1),g(B,1));
F2:=f2-B*g2;
redF2:=R!(F2/(f(B,1) - B * g(B,1)));
dF2:=g2*Derivative(f2,B) - f2*Derivative(g2,B) - z*g2*g2;

function mult(c);
return(Resultant(Evaluate(redF2,a,c),Evaluate(dF2,a,c),B));
end function;

result:
> Factorization(mult(4));
[ <z + 5, 2>, <z + 50, 2>, <z + 90, 2> ]
> Factorization(mult(96));
[ <z + 5, 2>, <z + 26, 2>, <z + 66, 2> ]
> Factorization(mult(18*w^7 + 50*w^6 + 68*w^5 + 24*w^4 + 59*w^3 + 22*w^2 +
> 93*w + 93));
[ <z + 5, 2>, <z + 78*w^7 + 93*w^6 + 47*w^5 + 57*w^4 + 23*w^3 + 80*w^2 + 73*w
+ 52, 2>, <z + 70*w^7 + 79*w^6 + 53*w^5 + 6*w^4 + 42*w^3 + 86*w^2 + 96*w + 53,
2> ]
> Factorization(mult(14*w^7 + 52*w^6 + 48*w^5 + 18*w^4 + 53*w^3 + 62*w^2 +
> 42*w + 50));
[ <z + 5, 2>, <z + 60*w^7 + 98*w^6 + 74*w^4 + 4*w^3 + 24*w^2 + 24*w + 21, 2>,
<z + 27*w^7 + 14*w^6 + 59*w^5 + 73*w^4 + 55*w^3 + 12*w^2 + 37*w + 11, 2> ]
> Factorization(mult(75*w^7 + 95*w^6 + 57*w^5 + 100*w^4 + 90*w^3 + 4*w^2 +
> 73*w + 55));
[ <z + 5, 2>, <z + 100*w^7 + 39*w^6 + 25*w^5 + 12*w^4 + 20*w^3 + 99*w^2 + 21*w
+ 73, 2>, <z + 84*w^7 + 84*w^6 + 91*w^5 + 31*w^4 + 44*w^3 + 70*w^2 + 92*w +
13, 2> ]
> Factorization(mult(95*w^7 + 5*w^6 + 29*w^5 + 60*w^4 + 13*w^2 + 95*w + 87));
[ <z + 5, 2>, <z + 89*w^7 + 42*w^6 + 58*w^5 + 91*w^4 + 11*w^3 + 22*w^2 + 91*w
+ 80, 2>, <z + 98*w^7 + 56*w^6 + 71*w^5 + 60*w^4 + 3*w^3 + 11*w^2 + 71*w + 81,
2> ]

This computation shows the other multipliers of period two orbits are mutually
different for the six solutions of (13).

###  A.2 An algorithm for Subsection 5.2

In order to make up block-decomposed interpolation matrix, we used the
following algorithm. The program file written by SAGE[The22] is attached, or
at [Got].

Algorithm 1 Degree-wise random-sampling interpolation method (probabilistic)

0:

Algorithms to compute $g_{1},g_{2},\ldots,g_{\beta}$ and
$H=h(g_{1},\ldots,g_{\beta})$,

the set of monomials
$M=\left\\{\left.\mathbf{y}^{\mathbf{d}}\right|\mathbf{d}\in\\{\mathbf{d}_{1},\ldots,\mathbf{d}_{l}\\}\right\\}$

such that $h=\sum c_{\mathbf{d}}\mathbf{y}^{\mathbf{d}}\
(c_{\mathbf{d}}\in\mathbb{Q}^{\beta})$

A map $\sigma:[n]\to[m]$ such that
$g_{i}(a_{1}x_{\sigma(1)},\ldots,a_{n}x_{\sigma(n)})$ is a monomial with
coefficient for each $g_{i}$ and $\mathbf{a}\in\mathbb{Q}^{n}$.

[Probably] The polynomial $h(y_{1},\ldots,y_{\beta})$

Separate $M$ by the degree of
$\left(\mathbf{g}(\mathbf{x_{\sigma}})\right)^{\mathbf{d}}$ into
$M_{1},\ldots,M_{p}$

$l_{p}:=\max\\#M_{i}$

for $j$ from 1 to $l_{p}+l^{\prime}$ do

Take a random vector $\mathbf{a}_{j}\in\mathbb{Q}^{n}$

Compute $H(\mathbf{a}_{j}\mathbf{x_{\sigma}})$

Compute $g_{i}(\mathbf{a}_{j})$’s and
$\mathbf{g}(\mathbf{a}_{j})^{\mathbf{d}}$

end for

for $k$ from 1 to $p$ do

Let $H_{j,k}$ be the coefficient of the term of degree
$\mathbf{g}(\mathbf{x}_{\sigma})^{\mathbf{d}}$ of
$H(\mathbf{a}_{j}\mathbf{x}_{\sigma})$ for $\mathbf{d}$ in $M_{k}$

Solve the system of linear equations $H_{j,k}=\sum_{\mathbf{d}\in
M_{k}}c_{\mathbf{d}}\mathbf{g}(\mathbf{a}_{j})^{\mathbf{d}}\
(j=1,\ldots,l_{p}+l^{\prime})$ for $c_{\mathbf{d}}$’s.

end for

$h(\mathbf{y})=\sum_{\mathbf{d}\in M}c_{\mathbf{d}}\mathbf{y}^{\mathbf{d}}$.

In our case, we only use addition and multiplication of the polynomials degree
less than $H$ to compute $H$, so the numbers of terms appears in the
computation are $O(p)$, thus it costs $O(p^{2}t_{H})$ to compute
$H(\mathbf{a}_{j}\mathbf{x_{\sigma}})$ par once. Therefore, the computational
complexity of we have
$O(l_{p}p^{2}t_{H}+l_{p}^{c}p)=O(N(pt_{H}+l_{p}^{c-1}))$. In our case
$l_{p}=70$ and the constants are sufficiently small. Moreover, we set
$l^{\prime}=5$ in the computation.

###  A.3 Data for Subsection 5.2

Explicit formula of $\delta_{2}$ and $\delta_{3}$ are

$\displaystyle\delta_{2}=$ $\displaystyle\frac{-1}{2^{31}\cdot
3}(306d^{5}i-2072d^{4}i^{2}-20544d^{3}i^{3}+300800d^{2}i^{4}-1691136di^{5}$
$\displaystyle+3483648i^{6}+96192d^{3}j^{2}-1286400d^{2}ij^{2}+10146816di^{2}j^{2}-16920576i^{3}j^{2}$
$\displaystyle+1008d^{3}ja+42432d^{2}ija+1430784di^{2}ja-4451328i^{3}ja+459d^{3}a^{2}$
$\displaystyle+14316d^{2}ia^{2}-6768di^{2}a^{2}-775104i^{3}a^{2}-1836d^{4}b+6624d^{3}ib$
$\displaystyle+173568d^{2}i^{2}b-1850880di^{3}b+4672512i^{4}b-23887872j^{4}-10616832j^{3}a$
$\displaystyle-156672j^{2}a^{2}+244224ja^{3}+26136a^{4}-589824dj^{2}b+9289728ij^{2}b$
$\displaystyle-672768djab+4202496ijab-111744da^{2}b+39168ia^{2}b+28800d^{2}b^{2}$
$\displaystyle-460800dib^{2}+1382400i^{2}b^{2}-2208d^{3}c+103296d^{2}ic-1027584di^{2}c$
$\displaystyle+2598912i^{3}c+9289728j^{2}c+3280896jac-76032a^{2}c-230400dbc+921600ibc),$
$\displaystyle\delta_{3}=$ $\displaystyle\frac{1}{2^{31}\cdot
3^{3}}(1458d^{5}i+2904d^{4}i^{2}-43072d^{3}i^{3}+2453760d^{2}i^{4}-11570688di^{5}$
$\displaystyle+40310784i^{6}-358464d^{3}j^{2}-10056960d^{2}ij^{2}+69424128di^{2}j^{2}-259780608i^{3}j^{2}$
$\displaystyle-1296d^{3}ja-730944d^{2}ija+12379392di^{2}ja-58973184i^{3}ja+2187d^{3}a^{2}$
$\displaystyle+100188d^{2}ia^{2}-730224di^{2}a^{2}-1881792i^{3}a^{2}-8748d^{4}b-95328d^{3}ib$
$\displaystyle+2674944d^{2}i^{2}b-20113920di^{3}b+82861056i^{4}b+107495424j^{4}+17915904j^{3}a$
$\displaystyle-3856896j^{2}a^{2}-2521728ja^{3}+143748a^{4}+18413568dj^{2}b-161243136ij^{2}b$
$\displaystyle+8280576djab-21399552ijab-693792da^{2}b+5664384ia^{2}b+475200d^{2}b^{2}$
$\displaystyle-12787200dib^{2}+43545600i^{2}b^{2}+7776d^{3}c+1173888d^{2}ic-7921152di^{2}c$
$\displaystyle+49268736i^{3}c-6912000b^{3}-71663616j^{2}c-3981312jac+1672704a^{2}c$
$\displaystyle-5529600dbc+49766400ibc).$

The relation among $\sigma_{i},\phi,\psi$ has 1261 terms. The data is
attached, or at [Got].

## References

  * [BCP97] Wieb Bosma, John Cannon, and Catherine Playoust, _The magma algebra system i: The user language_ , Journal of Symbolic Computation 24 (1997), no. 3-4, 235–265.
  * [dCCPHHS20] Pedro de Carvalho Cayres Pinto, Hans-Christian Herbig, Daniel Herden, and Christopher Seaton, _The hilbert series of sl 2-invariants_ , Communications in Contemporary Mathematics 22 (2020), no. 07, 1950017\.
  * [DK15] Harm Derksen and Gregor Kemper, _Computational invariant theory_ , Encyclopaedia of Mathematical Sciences 130. Invariant Theory and Algebraic Transformation Groups VIII, Springer, 2015, Second enlarged edition. With two appendices by Vladimir L. Popov, and an addendum by Norbert A’Campo and Popov.
  * [DM06] Valery Dolotin and Alexei Yurievich Morozov, _The universal mandelbrot set: Beginning of the story_ , World Scientific, 2006.
  * [ESW03] David Eisenbud, Frank-Olaf Schreyer, and Jerzy Weyman, _Resultants and chow forms via exterior syzygies_ , Journal of the American Mathematical Society 16 (2003), no. 3, 537–579.
  * [Fuj06] Masayo Fujimura, _Projective moduli space for the polynomials_ , Dynamics of Continuous, Discrete and Impulsive Systems Series A 13 (2006), no. 6, 787.
  * [Fuj07] , _The moduli space of rational maps and surjectivity of multiplier representation_ , Computational Methods and Function Theory 7 (2007), 345–360.
  * [Gor15] Igors Gorbovickis, _Algebraic independence of multipliers of periodic orbits in the space of rational maps of the riemann sphere_ , Moscow Mathematical Journal 15 (2015), no. 1, 73–87.
  * [Got] Rin Gotou, _Program files of “dynamical systems of correspondences on the projective line ii: Number of isospectral maps”_ , https://github.com/Rin-Gotou/dyn_corr_p1_II_multiplier_map.
  * [Got22] , _Bracket polynomial expression of discriminant-resultants as $sl_{2}$-invariant_, arXiv preprint arXiv:2211.15862 (2022).
  * [Got23] , _Dynamical systems of correspondences on the projective line i: Moduli spaces and multiplier maps_ , Conformal Geometry and Dynamics of the American Mathematical Society 27 (2023), no. 08, 294–321.
  * [HT13] Benjamin Hutz and Michael Tepper, _Multiplier spectra and the moduli space of degree 3 morphisms on $\mathbb{P}^{1}$_, JP Journal of Algebra, Number Theory and Applications 29 (2013), no. 2, 189–206.
  * [Hut21] Benjamin Hutz, _Errata for multiplier spectra and the moduli space of degree 3 morphisms on $\mathbb{P}^{1}$_, 2021, https://mathstat.slu.edu/~hutzba/multipliers_errata.pdf.
  * [Ill77] Luc Illusie, _Séminaire de géométrie algébrique du bois-marie 1965–66 - cohomologie l-adique et fonctions l - (sga 5)_ , Lecture Notes in Mathematics (1977), 252.
  * [JX23] Zhuchao Ji and Junyi Xie, _The multiplier morphism is generically injective_ , 2023.
  * [Liu06] Qing Liu, _Algebraic geometry and arithmetic curves_ , Translated from the French by Reinie Erné. Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, 2006.
  * [McM87] Curt McMullen, _Families of rational maps and iterative root-finding algorithms_ , Annals of Mathematics 125 (1987), no. 1, 467–493.
  * [Mil93] John Milnor, _Geometry and dynamics of quadratic rational maps_ , Experimental mathematics 2 (1993), no. 1, 37–83, With an appendix by the author and Lei Tan.
  * [Oli17] Marc Olive, _About gordan’s algorithm for binary forms_ , Foundations of Computational Mathematics 17 (2017), 1407–1466.
  * [Pra19] Amritanshu Prasad, _An introduction to schur polynomials_ , Graduate Journal of Mathematics 4 (2019), 62–84.
  * [Sch16] Johannes Schmitt, _A compactification of the moduli space of self-maps of $\mathbb{CP}^{1}$ via stable maps_, arXiv preprint arXiv:1604.08917v1 (2016).
  * [Sil98] Joseph H Silverman, _The space of rational maps on $\mathbb{P}^{1}$_, Duke mathematical journal 94 (1998), 41–77.
  * [Sil07] , _The arithmetic of dynamical systems_ , Graduate Texts in Mathematics 241 (2007).
  * [Sug17] Toshi Sugiyama, _The moduli space of polynomial maps and their fixed-point multipliers_ , Advances in Mathematics 322 (2017), 132–185.
  * [Sug20] , _The moduli space of polynomial maps and their holomorphic indices: I. generic properties in the case of having multiple fixed points_ , arXiv preprint arXiv:2009.11815 (2020).
  * [The22] The Sage Developers, _SageMath, the Sage Mathematics Software System_ , 2022, DOI 10.5281/zenodo.6259615.
  * [Wes15] Lloyd William West, _The moduli space of rational maps_ , Ph.D. thesis, City University of New York, 2015, p. 88.