# Determining Weak Mixing Angle at $\mu$TRISTAN

Lisong Chen<EMAIL_ADDRESS>Institute for Theoretical Particle Physics
(TTP), Karlsruhe Institute of Technology (KIT), Wolfgang-Gaede-Str. 1, 76131
Karlsruhe, Germany  Syuhei Iguro<EMAIL_ADDRESS>Institute for
Theoretical Particle Physics (TTP), Karlsruhe Institute of Technology (KIT),
Wolfgang-Gaede-Str. 1, 76131 Karlsruhe, Germany  Institute for Astroparticle
Physics (IAP), KIT, Hermann-von-Helmholtz-Platz 1,
76344 Eggenstein-Leopoldshafen, Germany  Institute for Advanced Research
(IAR), Nagoya University, Nagoya 464–8601, Japan  Kobayashi-Maskawa Institute
(KMI) for the Origin of Particles and the Universe,
Nagoya University, Nagoya 464–8602, Japan  KEK Theory Center, IPNS, KEK,
Tsukuba 305–0801, Japan  Yu Hamada<EMAIL_ADDRESS>Deutsches Elektronen-
Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  Research and Education
Center for Natural Sciences, Keio University, 4-1-1 Hiyoshi, Yokohama,
Kanagawa 223-8521, Japan

###### Abstract

$\mu$TRISTAN is a realistic high energy lepton collider based on the existing
technology aiming at indirect and direct search for the physics beyond the
standard model (SM). We propose a measurement to determine one of the most
prominent parameters of the SM, weak mixing angle to test the SM and probe the
new physics effect in Moller scattering with a wide range of interaction
scales. We show that this experiment not only can determine the weak mixing
angle with percent to milli-level accuracy but also scan over a wide range of
interaction scales that have never been archived in a single experiment
elsewhere.

††preprint: P3H-24-0XX, TTP24-0XX, KEK–TH–2622, DESY-24-072

##  I Introduction

Precision measurements of weak neutral current have been very important to
verify the Standard Model of electroweak interactions, see Ref. [PDG2022] for
the recent review. The ultra-precise measurements of the weak mixing angle at
different interaction scales can test the electroweak theory and indirectly
probe the new physics effect at the TeV scale. Therefore, it provides a
complementary probe to high energy colliders, e.g., large hadron collider
(LHC).

The definition of a weak mixing angle depends on the renormalization scheme.
We discuss in the modified minimal subtraction scheme $\overline{\rm{MS}}$ in
which quantum corrections appear as the scale dependence of the angle. In this
scheme, $\sin^{2}$ of the angle is given as

$\displaystyle\sin^{2}\theta_{w}(\mu)=\frac{g^{\prime
2}(\mu)}{g^{2}(\mu)+g^{\prime 2}(\mu)},$ (1)

where $g$ and $g^{\prime}$ respectively corresponds to the coupling constant
of $SU(2)$ and $U(1)_{Y}$ of the SM.

At the very low energy $\simeq\mathcal{O}(10){\rm{MeV}}$, measuring nuclear
weak charge with atomic parity-violating transitions can precisely determine
$\sin^{2}\theta_{w}$ ($s_{W}^{2}$ for short) [Safronova:2017xyt]. The heavy
nuclei e.g. Cs, Ti, Pb, Ra, Yb, enjoy the large nuclear enhancement in parity
violating effect, and further precision is foreseen [Aubin:2013uwaa,
Antypas:2017vni]. The parity-violating asymmetry $A_{PV}$ and polarization
asymmetry $A_{LR}$ in t (and u)-channel topology e.g. $e^{-}e^{-}\to
e^{-}e^{-}$ (Møller scattering) and $e^{-}p\to e^{-}p$ determine $s_{W}^{2}$
at $\simeq\mathcal{O}(10^{2\sim 4}){\rm{MeV}}$ [Prescott:1979dh,
SLACE158:2005uay, Wang:2014guo, PVDIS:2014cmd, Qweak:2018tjf].

On the other hand, a high-energy collider is also a powerful tool for the
$s_{W}^{2}$ determination. The NuTeV experiment determined $s_{W}^{2}$ with
$\nu{\rm{N}}$ and $\overline{\nu}{\rm{N}}$ interactions using high energy
$\nu$ and $\overline{\nu}$ beams [NuTeV:2001whx]. The previous precision
measurements at the Z pole e.g. large electron positron (LEP) collider
[ALEPH:2005ab], Stanford lepton collider (SLC) [SLD:1996gjt, SLD:2000leq,
SLD:2000jop, SLD:2000ujp], and Tevatron [D0:2017ekd, CDF:2018cnj] determined
$s_{W}^{2}$ around $Q=m_{Z}$ based on forward-backward asymmetry $A_{FB}$.
Recently the ATLAS, CMS, and LHCb also measured leptonic $A_{FB}$
[LHCb:2015jyu, ATLAS:2015ihy, ATLAS:2018gqq, CMS:2018ktx, CMS:2024aps] and the
resulting precision is comparable with previous colliders. Also, the high
luminosity (HL)-LHC which extends pseudorapidity acceptance will improve the
statistics further and will halve the uncertainty [CMS:2017vxj, ATLAS:2018qvs,
LHCb:2018roe] where remaining uncertainty largely comes from PDFs. Moreover,
there is a discrepancy between SLC and LEP, and the NuTeV result is not
perfectly consistent with the SM prediction.#1#1#1 Regarding the NuTeV result,
the assumption of symmetric $s$-$\overline{s}$ parton distribution function,
$\nu_{e}$ contamination from $K$ decays would account for the NuTeV
discrepancy; however, the definitive answer remains elusive; see Ref.
[PDG2016] for detail. Therefore further independent measurement is of crucial
importance.

In this regard, a proposed $\mu^{+}e^{-}$ and $\mu^{+}\mu^{+}$ collider,
$\mu$TRISTAN could be interesting [Hamada:2022mua].#2#2#2 There are several
future measurements, DUNE [deGouvea:2019wav], Polarized Belle II
[USBelleIIGroup:2022qro], FCC-ee [FCC:2018evy], CEPC [Zhao:2022lyl], ILC
[ILC:2013jhg], Moller [MOLLER:2014iki], SoLID [JeffersonLabSoLID:2022iod] and
P2 [Becker:2018ggl] which will measure $s_{W}^{2}$ at different scale. The
clear advantage of this muon collider is that it can be readily realized based
on the existing $\mu^{+}$ ultra cooling technology [Hamada:2022mua], which can
not be applied with $\mu^{-}$. It can not only reach the same energy scales as
the future experiments listed in the footnote but also has better sensitivity
in testing Lepton Flavor Universality (LFU) since it uses $\mu^{+}$ beams in
both proposed scatterings. Although the primary targets of this collider are
Higgs precision and direct search for heavy particles, this multi-purpose
experiment#3#3#3See Refs. [Hamada:2022uyn, Das:2022mmh, Fridell:2023gjx,
Lichtenstein:2023iut, Dev:2023nha, Fukuda:2023yui, Okabe:2023esr,
Goudelis:2023yni, Das:2024gfg, Heeck:2024uiz] for recent studies. is also
useful for indirect searches for new physics since it is a lepton machine. For
instance, the Møller scatterings of $\mu^{+}\mu^{+}$ and $\mu^{+}e^{-}$
exhibit nice sensitivity to deviations from the SM as studied within the SMEFT
framework which would encode $\mathcal{O}(10^{1\sim 2})$ TeV new physics
effect [Hamada:2022uyn]. We point out that, thanks to the well-controlled
polarization of the $e^{-}$ beam and the large momentum of the $\mu^{+}$ beam,
this $\mu^{+}e^{-}$ Møller scattering is quite useful for the $s_{W}^{2}$
determination. Moreover, utilizing the parity asymmetric component of the
cross section, which is not taken in Ref. [Hamada:2022uyn], one can reduce
potential systematic uncertainty and determine $s_{W}^{2}$ precisely. We
evaluate the $\mu$TRISTAN sensitivity to $s_{W}^{2}$ in the following.

The outline of this letter is given below. We introduce the asymmetry in Sec.
II. In Sec. III we evaluate the $\mu$TRISTAN sensitivity of $s_{W}^{2}$. We
conclude our findings in Sec. IV and briefly discuss the $\mu^{+}\mu^{+}$
version of $\mu$TRISTAN.

##  II Polarization asymmetry

At the proposed $\mu^{+}e^{-}$ collider, t-channel gauge boson exchange allows
us to access $s_{W}^{2}$. We consider the collision in the center of mass
frame $p_{\mu,i}=(E,0,0,E)$, $p_{e,i}=(E,0,0,-E)$,
$p_{\mu,f}=(E,0,E\sin\theta,E\cos\theta)$ and
$p_{e,f}=(E,0,-E\sin\theta,-E\cos\theta)$ with $s=4E^{2}=4E_{\mu}E_{e}$,
$t=-2E^{2}(1-\cos\theta)$, $u=-s-t$. $\theta$ is the opening angle in the
center of mass (CM) frame, and $E_{\mu}$ and $E_{e}$ are initial beam energy
in the lab frame.#4#4#4See Appendix A for the relation between the CM frame
and lab frame. We neglected the charged lepton masses.

We consider the polarization asymmetries defined in terms of differential
cross sections as

$\displaystyle A_{PV}^{e}(P_{e^{-}},\cos\theta_{\rm min},\cos\theta_{\rm
max})$ $\displaystyle=\int_{\cos\theta_{\rm min}}^{\cos\theta_{\rm
max}}d\cos\theta\left(\frac{d\sigma_{0,P_{e^{-}}}}{d\cos\theta}-\frac{d\sigma_{0,-P_{e^{-}}}}{d\cos\theta}\right)\biggl{/}\left(\frac{d\sigma_{0,P_{e^{-}}}}{d\cos\theta}+\frac{d\sigma_{0,-P_{e^{-}}}}{d\cos\theta}\right),$
(2) $\displaystyle A_{PV}^{\mu}(P_{\mu^{+}},\cos\theta_{\rm
min},\cos\theta_{\rm max})$ $\displaystyle=\int_{\cos\theta_{\rm
min}}^{\cos\theta_{\rm
max}}d\cos\theta\left(\frac{d\sigma_{P_{\mu^{+}},0}}{d\cos\theta}-\frac{d\sigma_{0,0}}{d\cos\theta}\right)\biggl{/}\left(\frac{d\sigma_{P_{\mu^{+}},0}}{d\cos\theta}+\frac{d\sigma_{0,0}}{d\cos\theta}\right).$
(3)

The polarized cross section is subdivided as

$\displaystyle\frac{d\sigma_{P_{\mu^{+}},P_{e^{-}}}}{d\cos\theta}=\frac{1}{4}\biggl{\\{}$
$\displaystyle\left(1+P_{\mu^{+}}\right)\left(1+P_{e^{-}}\right)\frac{d\sigma_{LR}}{d\cos\theta}+\left(1-P_{\mu^{+}}\right)\left(1-P_{e^{-}}\right)\frac{d\sigma_{RL}}{d\cos\theta}$
$\displaystyle\left(1-P_{\mu^{+}}\right)\left(1+P_{e^{-}}\right)\frac{d\sigma_{RR}}{d\cos\theta}+\left(1+P_{\mu^{+}}\right)\left(1-P_{e^{-}}\right)\frac{d\sigma_{LL}}{d\cos\theta}\biggl{\\}},$
(4)

where the polarization of a right (left) handed fermion corresponds to
$P_{f}=+1\,(-1)$ and the polarization of a right (left) handed anti-fermion
corresponds to $P_{\overline{f}}=-1\,(+1)$. The subscript “$0$” means non-
polarized beam, $P_{e^{-}/\mu^{+}}=0$. Assuming the SM, each differential
polarized cross section at the tree level is given as

$\displaystyle\frac{d\sigma_{LL}}{d\cos\theta}=\frac{d\sigma_{RR}}{d\cos\theta}=\frac{s}{8\pi}\left|\sum_{V=\gamma,Z}\frac{g_{L}^{l,V}g_{R}^{l,V}}{t-m_{V}^{2}}\right|^{2},~{}~{}\frac{d\sigma_{LR}}{d\cos\theta}=\frac{u^{2}}{8\pi
s}\left|\sum_{V=\gamma,Z}\frac{g_{R}^{l,V}g_{R}^{l,V}}{t-m_{V}^{2}}\right|^{2},~{}~{}\frac{d\sigma_{RL}}{d\cos\theta}=\frac{u^{2}}{8\pi
s}\left|\sum_{V=\gamma,Z}\frac{g_{L}^{l,V}g_{L}^{l,V}}{t-m_{V}^{2}}\right|^{2},$
(5)

where $g_{L}^{l,\gamma}=g_{R}^{l,\gamma}=eQ_{f}$,
$g_{L}^{l,Z}=\frac{g}{c_{W}}\left(T^{3}_{l}-Q_{l}s_{W}^{2}\right)$ and
$g_{R}^{l,Z}=-\frac{g}{c_{W}}Q_{l}s_{W}^{2}$ with $Q_{l}=-1$, $T^{3}_{l}=-1/2$
are defined. Indices $i,\,j$ of $\sigma_{ij}$ denote the beam polarizations of
initial beams $\mu^{+}_{i}e^{-}_{j}$. The lepton flavor universal interaction
of the SM is assumed. The energy transfer in the t-channel propagator,
$Q^{2}\equiv-t$ is a function of $\cos\theta$ for given beam energy, and hence
scanning $\cos\theta$ corresponds to scanning $\mu$. With s-channel topology
e.g. LEP and LHC, the reconstructing Z-boson mass uniquely determined $Q$ and
other t-channel experiments e.g. SLAC E158 experiment worked with the far
finite opening angle detector which fixes $Q$.

It is noted that the higher-order up to NNLO EW radiative corrections shift
the parity-violating asymmetries by roughly $41.3\%$ mainly due to the
$\gamma-Z$ mixing via hadronic vacuum polarization [Czarnecki:1995fw,
Du:2019evk, Erler:2022ckm], resulting in approximately $3\%$ shift on
$s_{W}^{2}$. The MC tools such as SANC [Andonov:2004hi, Arbuzov:2021oxs] will
also help us have systematic uncertainty under better control. Certainly, such
a level of precision, although being out of the scope in this paper, is yet
indispensable, given the sensitivity projected by $\mu$TRISTAN, which will be
discussed in the following section.

##  III $\mu$TRISTAN sensitivity

Figure 1:  $\mu$TRISTAN sensitivity to $s_{W}^{2}$ with the polarized muon
beam (left) and electron beam (right). The blue contour shows the SM
prediction. Colored bins correspond to the $\mu$TRISTAN sensitivity for a
given set of initial beam energy assuming 1ab-1 of data with the polarized
$\mu^{+}$. The existing constraints are shown with 1$\sigma$ error. Results
from Tevatron, LEP, SLC, and LHC are all around $\mu=m_{Z}$, however, they are
slightly shifted for better readability. Strictly speaking, the reference
energy scale $Q_{ref}$ in our measurement is different from the scale $\mu$ on
which $s_{W}$ depends in the $\overline{MS}$ scheme. Nevertheless, we identify
them for the sake of illustration.

At this stage, there is no concrete detector design for $\mu$TRISTAN. The HL-
LHC detector is assumed [Kaji] as a benchmark study although $\mu$TRISTAN is a
lepton collider. We expect a better control of uncertainty at future lepton
collider.#5#5#5We note that the energy resolution of the future lepton
collider, e.g., ILC detector, is a factor 2 better for $\eta<2.5$ compared to
the HL-LHC detector, but unfortunately, it is not available for $\eta\geq 3.0$
currently. A more detailed experimental study is necessary to address this
issue. The HL-LHC detector covers up to $|\eta|=4$ approximately corresponding
to $\theta^{\prime}>2^{\circ}$ where $\theta^{\prime}$ is the opening angle
between initial and final $\mu$ in the lab frame. As shown in Appendix B, the
statistical uncertainty of $s_{W}^{2}$ is expressed by

$\displaystyle\delta s_{W}^{2}=\left(\frac{\delta A_{\rm LR}}{\delta
s_{W}^{2}}\right)^{-1}\frac{1}{\sqrt{N_{\rm{tot}}}}.$ (6)

We assume that the systematic error is small compared to the statistical one.
Unlike the LHC case, the asymmetry is reconstructed by the detector in the
common angle, which mostly cancels the uncertainty. The precise estimation of
the systematic errors needs the careful consideration of experimental setup
[OPAL:2004xxz, Delahaye:2019omf, ZurbanoFernandez:2020cco, ILC2022], and hence
we simply neglect this at this stage. We divide the opening angle into five to
six sub-segments. The width of $\cos\theta$ is set such that the detector
resolution is smaller than each width. The electron and muon tagging
efficiencies are assumed to be $100\%$.

Based on the event number for the given $\cos\theta$ range and assuming the SM
we can calculate the statistic uncertainty of $s_{W}^{2}$. We defined a
reference (event rate weighted) momentum scale $\langle
Q_{ref}\rangle\equiv\int^{{\cos\theta_{\rm max}}}_{{\cos\theta_{\rm
min}}}d\cos\theta\,Q\times\sqrt{\sum_{ij}\frac{d\sigma_{ij}}{d\cos\theta}}\biggl{/}\int^{{\cos\theta_{\rm
max}}}_{{\cos\theta_{\rm
min}}}d\cos\theta\sqrt{\sum_{ij}\frac{d\sigma_{ij}}{d\cos\theta}}$ to recast
the sensitivity on the $s_{W}^{2}$ vs $\mu$ plane. Regarding polarization we
fixed $P_{e^{-}}=\pm 0.7$ and $P_{\mu^{+}}=0,\,+0.8$ [Roney:2021pwz,
Hamada:2022mua].

At this stage of the $\mu$TRISTAN project, it would be interesting to consider
several energy beam combinations to see the potential. In Fig. 1 (left) we
considered four cases as a benchmark setup: $(E_{\mu},\,E_{e})=(1{\rm
TeV},\,100{\rm GeV})$, $(1{\rm TeV},\,30{\rm GeV})$, $(100{\rm GeV},\,30{\rm
GeV})$, and $(30{\rm GeV},\,10{\rm GeV})$ in light green, magenta, orange, and
cyan for each with $P_{\mu^{+}}=0,+0.8$ and $P_{e^{-}}=0$. The integrated
luminosity of 1 ab-1 is assumed for each. Fig. 1 (right) shows the projected
sensitivity with $P_{\mu^{+}}=0$ and $P_{e^{-}}=\pm 0.7$.

It is observed that for the wide range of $\mu$, these future colliders can
precisely determine $s_{W}^{2}$ with unprecedented accuracy. The larger $\mu$
bins suffer from the small event number which inflates the uncertainty.
Moreover, the lowest $\mu$ bin tends to have the larger uncertainty due to the
small $A_{PV}$. Interestingly, the high energy options (magenta and light
green) allow us to scan $s_{W}^{2}$ at $\mu\geq m_{Z}$. On the other hand, the
low energy ones (orange and cyan) can validate the intermediate region of
$\mu\simeq 10^{1\sim 2}\,$GeV. Therefore we conclude that those future lepton
colliders provide a crucial precision test of the SM and $\mathcal{O}(1-10)$
TeV scale new physics.

##  IV Conclusions and discussion

In this letter, we evaluated the $\mu$TRISTAN sensitivity to the prominent SM
parameter, weak mixing angle at different scales. The $\mu$TRISTAN is one of
the most promising future Higgs factories that can be readily built based on
the existing $\mu^{+}$ ultra cooling technology. The precisely determined
$s_{W}^{2}$ at Z pole, even though a more than $3\sigma$ discrepancy between
SLAC and LEP has been reported, can be used to indirectly probe the new
physics at TeV scale which affects the $s_{W}^{2}$ via RG running. One obvious
advantage given by $\mu$TRISTAN is that it gives a better constraint on non-
LFU models due to the nature of $e^{-}\,\mu^{+}$ scattering. Thanks to the
nature of the t-channel process and the large $\sqrt{s}$ of
$\mathcal{O}(10^{3}\sim 10^{4})$ GeV, we can scan $s_{W}^{2}$ over the wide
range of interaction scale by reconstructing the scattering angle. We found
that $\mu$TRISTAN can not only measure $s_{W}^{2}$ at $\mu\geq m_{Z}$ for the
first time but also allows us to scan $s_{W}^{2}$ down to
$\mathcal{O}(1)\,{\rm GeV}$ with unprecedented accuracy.

It should be repeated that since the study here is based on the simplified
analysis e.g., leading order process without initial/final state radiation and
without a concrete detector design, a more detailed study is necessary from
both the theoretical and experimental sides.

Although we focused on the $\mu^{+}e^{-}$ option of $\mu$TRISTAN in this
letter, the other option $\mu^{+}\mu^{+}$ collider is also interesting to
measure $s_{W}^{2}$ at a different scale. In this setup, the final state is
identical and hence indistinguishable. As a result, both t-channel and
u-channel prevent us from uniquely determining $Q$. Nevertheless, it would be
interesting to point out that at $\cos\theta=0$ the transfer energies in both
channels coincide. Therefore It would be possible to measure $s_{W}^{2}$ by
changing $\sqrt{s}$.

###### Acknowledgements.

We are very grateful to Yasuyuki Horii, Surabhi Tiwari and Xunwu Zuo for
enlightening discussions. We appreciate Robert Ziegler for successfully
organizing the BSM seminar at KIT which motivated this work. S. I. and L. C.
are supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under grant 396021762-TRR 257. S. I. is also supported by the JSPS
Core-to-Core Program (Grant No. JPJSCCA20200002). Y. H. is supported by JSPS
Grant-in-Aid for Scientific Research KAKENHI Grant No. JP22KJ3123 and the
Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy - EXC 2121
Quantum Universe - 390833306.

##  Appendix A Kinematics

In the main text, we consider the CM frame. Here we will show the relationship
between the CM frame and the lab frame. In the collider we consider energetic
$\mu^{+}$ injection with $p_{1}=(E_{\mu,i},\,0,\,0,\,E_{\mu,i})$ which
collides with electron beam of $p_{2}=(E_{e,i},\,0,\,0,\,-E_{e,i})$ resulting
in
$p_{3}=(E_{\mu,f},0,E_{\mu,f}\sin\theta^{\prime},E_{\mu,f}\cos\theta^{\prime})$
and
$p_{4}=(E_{\mu,i}+E_{e,i}-E_{\mu,f},0,-E_{\mu,f}\sin\theta^{\prime},E_{\mu,i}-E_{e,i}-E_{\mu,f}\cos\theta^{\prime})$
where masses are neglected and
$E_{\mu,f}=2E_{\mu,i}E_{e,i}/(E_{\mu,i}+E_{e,i}-(E_{\mu,i}-E_{e,i})\cos\theta^{\prime})$.
$\theta^{\prime}$ is the opening angle between initial and final $\mu$ in the
lab frame. However calculation of cross section is easy in the CM frame, which
we defined in the beginning of Sec. III. The connecting relation of angles is
given as

$\displaystyle\cos\theta^{\prime}=\frac{(E_{\mu,i}+E_{e,i})\cos\theta+E_{\mu,i}-E_{e,i}}{E_{\mu,i}+E_{e,i}+(E_{\mu,i}-E_{e,i})\cos\theta}.$
(7)

##  Appendix B Statistic uncertainty

The uncertainty in a generic function $F(x_{1},\,x_{2},\,x_{3},....x_{n})$
with $x_{i}$ being the argument of $F$ is given as

$\displaystyle\delta
F(x_{1},\,x_{2},\,x_{3},....x_{n})=\sqrt{\sum_{i}^{n}\left(\frac{dF}{dx_{i}}\right)^{2}(\delta
x_{i})^{2}}.$ (8)

We evaluate the statistic uncertainty of asymmetry $\mathcal{A}$ which is the
function of polarized cross sections, $\sigma_{L}$ and $\sigma_{R}$ as,

$\displaystyle\mathcal{A}=\frac{\sigma_{R}-\sigma_{L}}{\sigma_{R}+\sigma_{L}}=\frac{N_{R}-N_{L}}{N_{R}+N_{L}}$
(9)

where $\sigma$ and $N$, respectively, denote the cross section and the number
of events for a given polarized beam for a given opening angle
$\theta^{\prime}$. The derivatives respective to $N_{R}$ and $N_{L}$ are
respectively given as

$\displaystyle\frac{d\mathcal{A}}{dN_{R}}=\frac{2N_{L}}{(N_{L}+N_{R})^{2}},\,\,\frac{d\mathcal{A}}{dN_{R}}=-\frac{2N_{R}}{(N_{L}+N_{R})^{2}},$
(10)

and hence we obtain

$\displaystyle\delta\mathcal{A}=\frac{2}{(N_{R}+N_{L})^{2}}\sqrt{N_{L}^{2}(\delta
N_{R})^{2}+N_{R}^{2}(\delta N_{L})^{2}}.$ (11)

When $\mathcal{A}\ll 1$ holds we can simplify the expression as

$\displaystyle\delta\mathcal{A}\simeq\frac{\delta
N_{R}}{\sqrt{2}N_{R}}=\frac{\delta
N_{R}^{r}}{\sqrt{2}}=\frac{1}{\sqrt{2}}\frac{1}{\sqrt{N_{R}}}=\frac{1}{\sqrt{N_{\rm{tot}}}},$
(12)

where $N_{R}\simeq N_{L}\simeq N_{\rm{tot}}/2$ is used. Based on the chain
rule and assuming the other uncertainty is negligible, we evaluate the
uncertainty of $s_{W}^{2}$ as

$\displaystyle\delta s_{W}^{2}=\left(\frac{\delta\mathcal{A}}{\delta
s_{W}^{2}}\right)^{-1}\delta\mathcal{A}.$ (13)

We evaluate the uncertainty at $\langle Q_{ref}\rangle$ to overlay on the
plane.


#  On effective $\epsilon$-integrality in orbits of rational maps over
function fields and multiplicative dependence

Jorge Mello  University of New South Wales. mailing adress:
School of Mathematics and Statistics UNSW Sydney NSW, 2052 Australia.
<EMAIL_ADDRESS>
###### Abstract.

We give effective bounds for the set quasi-integral points in orbits of non-
isotrivial rational maps over function fields under some conditions,
generalizing previous work of Hsia and Silverman (2011) for orbits over
function fields of characteristic zero. We then use this to prove finiteness
results for algebraic functions whose orbit under a rational function has
multiplicative dependent elements modulo rings of $S$-integers, generalizing
recent results over number fields.

###### Key words and phrases:

Integral points on orbits, arithmetic dynamics, quantitative estimates.

##  1\. Introduction

Let $K$ be a function field of a smooth projective curve over an algebraically
closed field of characteristic $0$, endowed as usual with a set $M_{K}$ of
absolute values(places) satisfying the product formula, $S$ a finite subset of
places of $M_{K}$, and $\epsilon>0$. An element $x\in K$ is said to be
quasi-$(S,\epsilon)$-integral if

$\displaystyle\sum_{v\in S}\log(\max\\{|x|_{v},1\\})\geq\epsilon h([x,1])$,

where $h$ is the absolute logarithmic height in $\mathbb{P}^{1}(K)$ and
$[x,1]\in\mathbb{P}^{1}(K)$.

Let $\phi\in K(z)$ a of rational functions of degree at least $2$, let $P\in
K$ and let

$\mathcal{O}_{\phi}(P)=\\{\phi^{(n)}(P)|n\in\mathbb{N}\\}$

denote the forward orbit of $P$ under $\phi$. When $K$ is a number field and
$\phi^{2}=\phi\circ\phi\notin K[z]$, Hsia and Silverman proved [8] that the
number of quasi-$(S,\epsilon)$\- integral points in the orbit of a point $P$
with infinite orbit is bounded by a constant depending only on
$\phi,\hat{h}_{\phi}(P),\epsilon,S$, and $[K:\mathbb{Q}]$ ( see Section 2 for
the correspondent definitions). We also note that these results, according to
[8, Remark 1], have some applications as the existence of quantitative
estimates for the size of Zsigmondy sets for such orbits and their primitive
divisors, as well for quantitative versions of a dynamical local-global
principle in orbits on the projective line. This research was also used to
prove finiteness of multiplicatively dependent iterated values by rational
functions in [4].

In this present paper we generalize this bound for cases over function fields.
This is presented all over Section 2. Making use of such results, this work is
also place for the study of multiplicative dependence modulo $S$-integers for
elements in orbits of rational functions over function fields, generalizing
the results of [4] for this context, but now also with effective bounds. This
is done in Section 3. Sometimes it is only possible to bound, although
effectively, the height of algebraic functions studied, instead of their
cardinality. This is related with the fact that even the so called Northcott
finiteness property (which is a particular case of finiteness for elements
with multiplicative dependent orbit elements) fails over function fields. One
can see once again that the ambient of function fields is relevantly different
to number field one. For this, we made use of specific tools from the function
field case. Namely, a recent version of effective Roth type of theorem over
function fields due to Wang [16], a certain finiteness property for canonical
heights due to Baker [1], and some effective results for superellitic
equations over function fields in one variable.

##  2\. Effective bounds for quasiintegral points in orbits over function
fields

###  2.1. Canonical Heights, Distance and dynamics on the projective line

We always assume that $K$ is a fixed function field of a curve over an
algebraically closed field of characteristic $0$ and $K(z)$ is the field of
rational functions over $K$ for the rest of the paper. We identify
$K\cup\\{\infty\\}=\mathbb{P}^{1}(K)$ by fixing an affine coordinate $z$ on
$\mathbb{P}^{1}$, so $\alpha\in K$ is equal to
$[\alpha,1]\in\mathbb{P}^{1}(K)$, and the point at infinity is $[1,0]$. In
this way, we assume $z$ is the first left coordinate for points in
$\mathbb{P}^{1}$, and with respect to this affine coordinate, we identify
rational self-maps of $\mathbb{P}^{1}$ with rational functions in $K(z)$.

If $P=[x_{0},...,x_{N}]\in\mathbb{P}^{N}(K)$, the naive logarithmic height is
given by

$h(P)=\sum_{v\in M_{K}}\log(\max_{i}|x_{i}|_{v})$,

where, $M_{K}$ is the set of places of $K$, and for each $v\in M_{K}$,
$|.|_{v}$ denotes the corresponding absolute values on $K$ satisfying the
product formula, and can be extended to any algebraic closure and respective
completions of $K$, so that $h$ can be well defined on $\overline{K}$. Also,
we write $K_{v}$ for the completion of $K$ with respect to $|.|_{v}$, and we
let $\hat{K}_{v}$ denote the completion of an algebraic closure of $K_{v}$.
Initially, we also recall that one can define the convergent limit
$\hat{h}_{f}(\alpha)=\lim_{n\rightarrow\infty}\dfrac{h(f(\alpha))}{d^{n}}$ for
any $\alpha\in K$, called the canonical height associated with $f$, which also
satisfies $\hat{h}_{f}(f(\alpha))=d\hat{h}_{f}(\alpha)$, and that $\alpha$ has
infinite orbit (is not preperiodic) if and only if $\hat{h}_{f}(f(\alpha))>0$(
see [14, Theorem 3.20]).

For each $v\in M_{K}$, we let $\rho_{v}$ denote the chordal metric defined on
$\mathbb{P}^{1}(\hat{K}_{v})$, where we recall that for
$[x_{1},y_{1}],[x_{2},y_{2}]\in\mathbb{P}^{1}(\hat{K}_{v})$,

$\rho_{v}([x_{1},y_{1}],[x_{2},y_{2}])=\dfrac{|x_{1}y_{2}-x_{2}y_{1}|_{v}}{\max\\{|x_{1}|_{v},|y_{1}|_{v}\\}\max\\{|x_{2}|_{v},|y_{2}|_{v}\\}}.$

######  Definition 2.1.

The logarithmic chordal metric function

$\lambda_{v}:\mathbb{P}^{1}(\hat{K}_{v})\times\mathbb{P}^{1}(\hat{K}_{v})\rightarrow\mathbb{R}\cup\\{\infty\\}$

is defined by

$\lambda_{v}([x_{1},y_{1}],[x_{2},y_{2}])=-\log\rho_{v}([x_{1},y_{1}],[x_{2},y_{2}]).$

It is a matter of fact that $\lambda_{v}$ is a particular choice of an
arithmetic distance function as defined by Hsia and Silverman [8] over number
fields, which is a local height function
$\lambda_{\mathbb{P}^{1}\times\mathbb{P}^{1},\Delta}$, where $\Delta$ is the
diagonal of $\mathbb{P}^{1}\times\mathbb{P}^{1}$. The logarithmic chordal
metric and the usual metric can relate in the following way.

######  Lemma 2.2.

Let $v\in M_{K}$ and let $\lambda_{v}$ be the logarithmic chordal metric on
$\mathbb{P}^{1}(\mathbb{C}_{v})$. Then for $x,y\in\mathbb{C}_{v}$ the
inequality $\lambda_{v}(x,y)>\lambda_{v}(y,\infty)$ implies

$\lambda_{v}(y,\infty)\leq\lambda_{v}(x,y)+\log|x-y|_{v}\leq
2\lambda_{v}(y,\infty)$.

###### Proof.

The proof works in the same way as the proof over number fields appearing in
[8, Lemma 3]. ∎

Now, let $\phi:\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$ be a rational map of
degree $d\geq 2$ defined over $K$. In this situation we let

$\phi^{(n)}=\phi\circ...\circ\phi$

with $\phi^{(0)}=$Id. For simplicity $\phi^{(n)}=\phi^{n}$.

For a point $P\in\mathbb{P}^{1}$, the $\phi$-orbit of $P$ is defined as

$\mathcal{O}_{\phi}(P)=\\{\phi^{(n)}(P)|n\geq 0\\}.$

The point $P$ is called preperiodic for $\phi$ if $\mathcal{O}_{\phi}(P)$ is
finite.

We set $\text{Wander}_{K}(\phi)=\\{P\in\mathbb{P}^{1}(K):P\text{ is not
preperiodic for }\phi\\}$.

We recall that for $P=[x_{0},x_{1}]\in\mathbb{P}^{1}(K)$ the height of $P$ is

$h(P)=\sum_{v\in M_{K}}\log(\max\\{|x_{1}|_{v},|x_{1}|_{v}\\}$.

And using the definition of $\lambda_{v}$, we see that

$h(P)=\sum_{v\in M_{K}}\lambda_{v}(P,\infty)+O(1)$.

For a polynomial $f=\sum a_{i}z^{i}$ and an absolute value $v\in M_{K}$, we
define $|f|_{v}=\max_{i}\\{|a_{i}|_{v}\\}$ and

$h(f)=\sum_{v\in M_{K}}\log|f|_{v}$.

Given a rational function $\phi(z)=f(z)/g(z)\in K(z)$ of degree $d$ written in
normalized form, let us write $f(z)=\sum_{i\leq d}a_{i}z^{i},g(z)=\sum_{i\leq
d}b_{i}z^{i}$ with $a_{d}$ and $b_{d}$ different from zero, and $f$ and $g$
relatively prime in $K[z]$.

For $v\in M_{K}$, we set $|\phi|_{v}=\max\\{|f|_{v},|g|_{v}\\}$, and then the
height of $\phi$ is defined by

$h(\phi):=\sum_{v\in M_{K}}\log|\phi|_{v}$.

######  Proposition 2.3.

Let $\phi$ be a rational function with $\deg\phi=d\geq 2$. Then for all $n\geq
1$, we have

$h(\phi^{n})\leq\left(\dfrac{d^{n}-1}{d-1}\right)h(\mathcal{F})+d^{2}\left(\dfrac{d^{n-1}-1}{d-1}\right)\log
8$.

###### Proof.

[8, Proposition 5 (d)]. ∎

######  Lemma 2.4.

For a rational map $\phi:\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$ of degree
$d\geq 2$ defined over $K$ and $L=\mathcal{O}_{\mathbb{P}^{1}}(1)$, it is true
that

(a) $|h(\phi(P))-dh(P)|\leq c_{1}h(\phi)+c_{2}$.

(b) $\hat{h}_{\phi}(P)=\lim_{n}h(\phi^{(n)}(P))/d^{n}$.

(c) $|\hat{h}_{\phi}(P)-h(P)|\leq c_{3}h(\phi)+c_{4}$.

Where $c_{1},c_{2},c_{3}$ and $c_{4}$ above depend only on $d$.

###### Proof.

This is stated in [8, Proposition 6] over number fields. For (a), the same
procedure given in [5, Proposition A] works over the referred function fields.
For (b) and (c), the proof of the existence of the limit defining the
canonical height ([14, Theorem 3.20]) together with (a) yields the desired. ∎

######  Lemma 2.5.

[1, Theorem 1.6] Let $\varphi(z)\in K(z)$ of degree at least $2$, and assume
that $\varphi$ is not isotrivial, and $\hat{h}_{\varphi}$ is the canonical
height associated with $\varphi$. Then there exists $\varepsilon>0$ (
depending on $K$ and $\varphi$) such that the set

$\\{P\in\mathbb{P}^{1}(K):\hat{h}_{\varphi}(P)\leq\varepsilon\\}$

is finite.

###  2.2. A distance estimate and an effective version of Roth’s Theorem

We will state three results that will be needed to prove our main theorems.
The first one is a result that gives explicit estimates for the dependence on
local heights of points and function.

Let us recall that, for a rational function $f(z)$, $P\neq\infty$ and
$f(P)\neq\infty$, the ramification index of $f$ at $P$ is defined as the order
of $P$ as a zero of the rational function $f(z)-f(P)$, i.e.,

$e_{P}(f)=$ ord${}_{P}(f(z)-f(P))$.

If $P=\infty$, or $f(P)=\infty$, we change coordinates through a linear
fractional transformation $L$, such that
$L^{-1}(P)=\beta\neq\infty,L^{-1}(f(L(\beta)))\neq\infty$, and define
$e_{P}(f)=e_{\beta}(L^{-1}\circ f\circ L)$. It will not depend on the choice
of $L$. We say that $f$ is totally ramified at $P$ if $e_{P}(f)=\deg f$. It is
also an exercise to show that

$e_{P}(g\circ f)=e_{P}(f)e_{g}(f(P))$

for every $f,g$ rational functions and $P\in K\cup\\{\infty\\}$.

The result is as follows.

######  Lemma 2.6.

Let $\psi\in K(z)$ be a nontrivial rational function, let $S\subset M_{K}$ be
a finite set of absolute values on $K$, each extended in some way to
$\bar{K}$, and let $A,P\in\mathbb{P}^{1}(K)$. Then

$\displaystyle\sum_{v\in
S}\max\limits_{A^{\prime}\in\psi^{-1}(A)}e_{A^{\prime}}(\psi)$
$\displaystyle\lambda_{v}(P,A^{\prime})\geq$ $\displaystyle\sum_{v\in
S}\lambda_{v}(\psi(P),A)-O(h(A)+h(\psi)+1),$

where the implied constant depends only on the degree of the map $\psi$.

###### Proof.

This is stated in [8, Proposition 7] over number fields. Its proof uses a
higher dimensional version of Lemma 2.4(b) applied for maps in dimension 1,
thus Lemma 2.4 is enough and works over function fields. More importantly, the
proof uses strong distribution value theorems related with inverse function
theorem due to Silverman. It was found an error in these proofs, which were
accordingly corrected in [11, Sections 4 and 5], that work over global fields.
∎

######  Lemma 2.7.

[8, Lemma 9] Fix an integer $d\geq 2$. Then there exist two positive constants
$\kappa_{1}>0$ and $0<\kappa_{2}<1$ depending only on $d$ such that for all
rational functions $\phi:\mathbb{P}^{1}(K)\rightarrow\mathbb{P}^{1}(K)$, all
points $Q$ that are not exceptional for $\phi$, all integers $m\geq 1$, and
all $P\in\phi^{-m}(Q)$, we have that

$e_{P}(\phi^{m})\leq\kappa_{1}(\kappa_{2}d)^{m}$ for any $m\geq 0$.

###### Proof.

The proof of ([14, Lemma 3.52]) works here, since the Riemann-Hurwitz formula
works for this context as well.

∎

The third result is the following effective version of Roth’s theorem over
function fields due to Wang.

######  Lemma 2.8.

[16] Let $S$ be a finite subset of $M_{K}$. We assume that each place in $S$
is extended to $\bar{K}$ in some fashion. Assume that for each $v\in S$, we
have an element $\beta_{v}\in\bar{K}$. Then, for any $\mu>2$, the elements
$x\in K$ satisfying

$\sum_{v\in S}\log^{+}|x-\beta_{v}|_{v}^{-1}\geq\mu h(x)$

have their heights bounded by an effective constant depending on $\mu,|S|,$
the genus of $K$, and the elements $\beta_{v}$.

###  2.3. A bound for the number of quasiintegral points in an orbit

In this section, we show explicit bounds for the number of $S$-integral points
in a given orbit of a wandering point for a dynamical system of rational
functions extending previous work by Hsia and Silverman [8].

The next quantitative theorem generalizes Theorem 11 of Hsia and Silverman [8]
to function fields of zero characteristic. The definitions and strategy of the
proof are inspired by their ideas with diophantine approximation.

######  Theorem 2.9.

Let $\phi\in K(z)$ be a non-isotrivial rational function of respective degree
$d\geq 2$, and $P\in\mathbb{P}^{1}(K)$ not preperiodic for $\phi$. Fix
$A\in\mathbb{P}^{1}(K)$ which is not an exceptional point of $\phi$. For any
finite set of places $S\subset M_{K}$ and any constant $1\geq\epsilon>0$,
define a set of nonnegative integers by

$\Gamma_{\phi,S}(A,P,\epsilon):=\\{\phi^{(n)}(P):\sum_{v\in
S}\lambda_{v}(\phi^{n}(P),A)\geq\epsilon\hat{h}_{\phi}(\phi^{n}(P))\\}$.

(a) There exist effective constants

$\gamma_{1}=\gamma_{1}(\phi,\epsilon,|S|,K,A)$ and
$\gamma_{2}=\gamma_{2}(\phi,\epsilon,|S|,K,A)$

such that

$\left\\{\phi^{(n)}(P)\in\Gamma_{\phi,S}(A,P,\epsilon):n>\gamma_{1}+\log_{d}^{+}\left(\dfrac{\hat{h}_{\phi}(A)+h(\phi)}{\hat{h}_{\phi}(P)}\right)\right\\}$

has bounded height from above by $\gamma_{2}$
(b) If $P$ is not $\phi$-preperiodic, there is an effective constant
$\gamma_{3}(\phi,\epsilon,|S|,K,A)$ that is independent of $P$ such that

$\displaystyle\max_{P}\\{n\geq
0:\phi^{(n)}(P)\in\Gamma_{\phi,S}(A,P,\epsilon)\\}\leq\gamma_{3}+\log_{d}^{+}\left(\dfrac{h(\phi)}{\inf_{\hat{h}_{\phi}(P)>0}\hat{h}_{\phi}(P)}\right)$.

###### Proof.

For simplicity, we write $\Gamma_{S}(\epsilon)$ instead of
$\Gamma_{\Phi,S}(A,P,\epsilon)$. Taking $\kappa_{1}$ and $\kappa_{2}<1$ the
constants from Lemma 2.7, we choose $m\geq 1$ minimal such that
$\kappa^{m}_{2}\leq\epsilon/5\kappa_{1}$. Then $\kappa_{1},\kappa_{2}$ and $m$
depend only on $d$ and on $\epsilon$.

If $n\leq m$ for all $n$ such that $\phi^{n}(P)\in\Gamma_{S}(\epsilon)$, then

$\\#\Gamma_{S}(\epsilon)\leq
m\leq\dfrac{\log(5\kappa_{1})+\log(\epsilon^{-1})}{\log(\kappa_{2}^{-1})}+1$,

which is in the desired form. If there is an $n$ with
$\phi^{n}(P)\in\Gamma_{S}(\epsilon)$ such that $n>m$, we fix $n$ for instance.
Then by definition of $\Gamma_{S}(\epsilon)$ we have

(2.1) $\epsilon\hat{h}_{\phi}(\phi^{n}(P))\leq\sum_{v\in
S}\lambda_{v}(\phi^{n}(P),A).$

We can write $\phi^{n}=\phi^{m}\circ\phi^{n-m}$ and $\psi=\phi^{m}$.

For our chosen $m$, we denote

$\textbf{e}_{m}:=\max\limits_{A^{\prime}\in\psi^{-1}(A)}e_{A^{\prime}}(\psi)$.

By Lemma 2.7 and our choice of $m$, we notice that

$\textbf{e}_{m}\leq\kappa_{1}(\kappa_{2})^{m}\deg\psi\leq\epsilon\deg\psi/5$

Therefore, Lemma 2.6 yields, for $Q\in\mathbb{P}^{1}(K)$, that

(2.2) $\sum_{v\in
S}\lambda_{v}(\psi(Q),A)-O(h(A)+h(\psi)+1)\leq\textbf{e}_{m}\sum_{v\in
S}\max\limits_{A^{\prime}\in\psi^{-1}(A)}\lambda_{v}(Q,A^{\prime}).$

Gathering 2.1 and 2.2 with $Q:=\phi^{n-m}(P)$, we obtain that

$\epsilon\hat{h}_{\phi}(\phi^{n}(P))\leq\textbf{e}_{m}\sum_{v\in
S}\max\limits_{A^{\prime}\in\psi^{-1}(A)}\lambda_{v}(\phi^{n-m}(P),A^{\prime})+O(h(A)+h(\psi)+1)$,

where the involved constants depend only on the degree $d^{m}$, $d$ and on
$\epsilon$.

For each $v\in S$, we choose $A_{v}^{\prime}\in\psi^{-1}(A)$ such that

$\lambda_{v}(\phi^{n-m}(P),A_{v}^{\prime})=\max\limits_{A^{\prime}\in\psi^{-1}(A)}\lambda_{v}(\phi^{n-m}(P),A^{\prime})$,

so that

$\epsilon\hat{h}_{\phi}(\phi^{n}(P))\leq\textbf{e}_{m}\sum_{v\in
S}\lambda_{v}(\phi^{n-m}(P),A_{v}^{\prime})+O(h(A)+h(\psi)+1)$.

For instance, we can assume that $z(A^{\prime})\neq\infty$ for all
$A^{\prime}\in\psi^{-1}(A)$. If this is not the case, we use $z$ for some of
the $A^{\prime}$ and $z^{-1}$ for the others.

Let $S^{\prime}\subset S$ be the set of places in $S$ defined by

$S^{\prime}=\\{v\in
S;\lambda_{v}(\phi^{n-m}(P),A_{v}^{\prime})>\lambda_{v}(A_{v}^{\prime},\infty)\\}$.

Set $S^{\prime\prime}:=S-S^{\prime}$. Applying Lemma 2.2 to the places in
$S^{\prime}$ and using the definition of $S^{\prime\prime}$ we find that

$\displaystyle\epsilon\hat{h}_{\phi}(\phi^{n}(P))$
$\displaystyle\leq\textbf{e}_{m}\displaystyle\sum\limits_{v\in
S}\lambda_{v}(\phi^{n-m}(P),A_{v}^{\prime})+O(h(A)+h(\psi)+1)$
$\displaystyle\leq\textbf{e}_{m}\displaystyle\sum\limits_{v\in
S^{\prime}}(2\lambda_{v}(A_{v}^{\prime},\infty)-\log|z(\phi^{n-m}(P))-z(A_{v}^{\prime})|+\log
l_{v})$
$\displaystyle\qquad\qquad+\textbf{e}_{m}\displaystyle\sum\limits_{v\in
S^{\prime\prime}}(\lambda_{v}(A_{v}^{\prime},\infty)+\log
l_{v})+O(h(A)+h(\psi)+1)$
$\displaystyle\leq\textbf{e}_{m}\displaystyle\sum\limits_{v\in
S^{\prime}}\log|z(\phi^{n-m}(P))-z(A_{v}^{\prime})|^{-1}$
$\displaystyle\qquad\qquad+\textbf{e}_{m}\displaystyle\sum\limits_{v\in
S}(2\lambda_{v}(A_{v}^{\prime},\infty)+\log l_{v})+O(h(A)+h(\psi)+1).$

Now using Lemma 2.4 it can be checked that

$\displaystyle\displaystyle\sum\limits_{v\in
S}\lambda_{v}(A_{v}^{\prime},\infty)$
$\displaystyle\leq\displaystyle\sum\limits_{A^{\prime}\in\psi^{-1}(A)}\displaystyle\sum\limits_{v\in
S}\lambda_{v}(A^{\prime},\infty)$
$\displaystyle\leq\displaystyle\sum\limits_{A^{\prime}\in\psi^{-1}(A)}h(A^{\prime})$
$\displaystyle\leq\displaystyle\sum\limits_{A^{\prime}\in\psi^{-1}(A)}\hat{h}_{\phi}(A^{\prime})+O(h(\phi)+1)$
$\displaystyle=\displaystyle\sum\limits_{A^{\prime}\in\psi^{-1}(A)}(\deg\psi)^{-1}\hat{h}_{\phi}(\psi(A^{\prime}))+O(h(\phi)+1)$
$\displaystyle\leq\displaystyle\sum\limits_{A^{\prime}\in\psi^{-1}(A)}(\deg\psi)^{-1}\hat{h}_{\phi}(A)+O(h(\phi)+1)$
$\displaystyle\leq\hat{h}_{\phi}(A)+O(h(\phi)+1).$

The constants depend only on $m$ and $d$. Also, from Proposition 2.3 it
follows that $h(\psi)=O(h(\phi)+1)$.

All the inequalities above together imply that

$\epsilon(\hat{h}_{\phi}(\phi^{n}(P))\leq\textbf{e}_{m}(\displaystyle\sum\limits_{v\in
S^{\prime}}\log|z(\phi^{n-m}(P))-z(A_{v}^{\prime})|^{-1})+O(\hat{h}_{\phi}(A)+h(\phi)+1)$.

Let us set some definitions in order to apply Wang’s Theorem. We define
$\beta_{v}:=A_{v}^{\prime}$ and analyze the points $x=\phi^{n-m}(P)$ for
$\phi^{n}(P)\in\Gamma_{S}(\epsilon)$. Applying Lemma 2.8 for the set of places
$S^{\prime}$ and $\mu=5/2$, yields that there exist a constant $r_{1}$
depending only on $K,\phi,|S|$ and $\epsilon$ such that the set of
$\phi^{n}(P)\in\Gamma_{S}(\epsilon)$ with $n>m$ can be written as a union

$\\{\phi^{n}(P)\in\Gamma_{S}(\epsilon):n>m\\}=T_{1}\cup T_{2}\cup T_{3}$

such that

$T_{1}$ has all its elements with height bounded from above by $r_{1}$,

$T_{2}=\\{\phi^{n}(P)\in\Gamma_{S}(\epsilon):n>m,\displaystyle\sum\limits_{v\in
S^{\prime}}d_{v}\log|z(\phi^{n-m}(P))-z(A_{v}^{\prime})|^{-1}\leq\frac{5}{2}h(\phi^{n-m}(P))\\},$

We already have a bound for the height of $T_{1}$. We consider the set
$T_{2}$. Again using Lemmas 2.4 we derive

$h(\phi^{n-m}(P))\leq\hat{h}_{\phi}(\phi^{n-m}(P))+c_{1}h(\phi)+c_{2}\newline
\newline =d^{n-m}\hat{h}_{\phi}(P)+c_{1}h(\phi)+c_{2}$,

and then, for $n$ with $\phi^{n}(P)\in T_{2}$, using that
$\textbf{e}_{m}\leq\epsilon\deg\psi/5$

$\displaystyle\epsilon\hat{h}_{\phi}(\phi^{n}(P))$ $\displaystyle=\epsilon
d^{n}\hat{h}_{\Phi}(P)$
$\displaystyle\leq\textbf{e}_{m}(\displaystyle\sum\limits_{v\in
S^{\prime}}\log|z(\phi^{n-m}(P))-z(A_{v}^{\prime})|^{-1})+c_{13}(\hat{h}_{\phi}(A)+h(\phi)+1)$
$\displaystyle\leq(\epsilon\frac{\deg\psi}{5})\frac{5}{2}(d^{n-m})\hat{h}_{\phi}(P)+c_{10}(\hat{h}_{\phi}(A)+h(\phi)+1)$
$\displaystyle=\frac{\epsilon}{2}d^{n}\hat{h}_{\phi}(P)+c_{14}(\hat{h}_{\phi}(A)+h(\phi)+1).$

Thus

$\frac{\epsilon}{2}d^{n}\hat{h}_{\phi}(P)\leq
c_{14}(\hat{h}_{\phi}(A)+h(\phi)+1)$, which implies that

$\frac{\epsilon}{2}d_{1}^{n}\hat{h}_{\phi}(P)\leq
c_{14}(\hat{h}_{\phi}(A)+h(\phi)+1)$,

equivalent to

$n\leq
c_{15}+\log_{d}^{+}\left(\dfrac{\hat{h}_{\phi}(A)+h(\phi)}{\hat{h}_{\phi}(P)}\right)$.

We observe that the set $r_{1}$ does not depend on the point, so the elements
in $T_{1}$ have height bounded independently of $P$ by the constant $r_{1}$.
We also note that the quantity

$\hat{h}^{\text{min}}_{\phi,K}:=\inf\\{\hat{h}_{\phi}(P):P\in\mathbb{P}^{1}(K)$
is not preperiodic for $\Phi\\}$

is strictly positive. This is a consequence of Lemma 2.5. For $\phi^{n}(P)\in
T_{1}$, we can see from this and Lemma 2.4 that
$d^{n}\hat{h}_{\phi}(P)=\hat{h}_{\phi}(\phi^{n}(P))\leq r_{1}+O(h(\phi)+1)$,
and thus

$n\leq\log_{d}\left(\dfrac{r_{1}+O(h(\phi)+1)}{\hat{h}^{\text{min}}_{\phi,K}}\right)$

in this case.

Therefore, $\max\\{n:\phi^{n}(P)\in(T_{1}\cup T_{2})\\}$ can be bounded
independently of $P$. ∎

######  Corollary 2.10.

Let $S\subset M_{K}$ be a finite set of places, let $R_{S}$ be the ring os
$S$-integers of $K$, and let $2\leq d$. Then, there is an effective constant
$\gamma=\gamma(\phi,|S|,K)$ such that for all $\phi\in K(z)$ non-isotrivial
rational maps of degrees $d\geq 2$ with $\phi^{2}\notin\bar{K}[z]$, all
$P\in\mathbb{P}^{1}(K)$ that are not preperiodic for $\phi$, the number of
$S$-integers in the $\phi-$orbit of $P$ is bounded by

$\\#\\{n\geq 1;z(\phi^{n}(P))\in
R_{S}\\}\leq\gamma+\log^{+}_{d}\left(\dfrac{h(\phi)}{\hat{h}_{\phi}(P)}\right)$.

###### Proof.

An element $\alpha\in K$ is in $R_{S}$ if and only if $|\alpha|_{v}\leq 1$ for
all $v\not\in S$, or equivalently, if and only if

$h(\alpha)=\sum_{v\in S}\log\max\\{|\alpha|_{v},1\\}$.

Another fact is that

$\log\max\\{|\alpha|_{v},1\\}\leq\lambda_{v}(\alpha,\infty)$.

This implies for $\alpha\in R_{S}$ that $h(\alpha)\leq\sum_{v\in
S}\lambda_{v}(\alpha,\infty)$.

Let $n\geq 1$ satisfy $z(\phi^{n}(P))\in R_{S}$. Then

$h(\phi^{n}(P))\leq\sum_{v\in S}\lambda_{v}(\phi^{n}(P),\infty)$.

Lemmas 2.4 tell us that

$h(\phi^{n}(P))\geq\hat{h}_{\phi}(\phi^{n}(P))-c_{3}h(\phi)-c_{4}=\deg(\phi^{n})\hat{h}_{\phi}(P)-c_{3}h(\phi)-c_{4}$,

which implies that

$d^{n}\hat{h}_{\phi}(P)-c_{3}h(\phi)-c_{4}\leq\sum_{v\in
S}d_{v}\lambda_{v}(\phi^{n}(P),\infty)$.

The rest of the proof is divided in two cases: First one, when

$d^{n}\hat{h}_{\phi}(P)\leq 2c_{3}h(\phi)+2c_{4}$.

In this case,

$n\leq\log_{d}^{+}\left(\dfrac{2c_{3}h(\phi)+2c_{4}}{\hat{h}_{\phi}(P)}\right)$.

In the second case , $d^{n}\hat{h}_{\phi}(P)\geq 2c_{3}h(\phi)+2c_{4}$.
Therefore

$\sum_{v\in
S}\lambda_{v}(\phi^{n}(P),\infty)\geq\frac{1}{2}d^{n}\hat{h}_{\phi}(P)=\frac{1}{2}\hat{h}_{\phi}(\phi^{n}(P))$.

Now Theorem 2.9 with $\epsilon=1/2,A=\infty$ ($\infty$ is not exceptional for
$\phi$) tells us that $n$ is at most

$\gamma+\log^{+}_{d}\left(\dfrac{h(\phi)+\hat{h}_{\phi}(\infty)}{\hat{h}_{\phi}(P)}\right)$,

for an effective constant $\gamma$ depending only on $K,\phi$ and $|S|$. Both
bounds are on the desired form since $\hat{h}_{\phi}(\infty)\leq
h(\infty)+O(1)=0+O(1)$. ∎

######  Remark 2.11.

Theorem 2.9 delivers, in particular, under its conditions, an explicit upper
bound for

$\\#\\{n\geq 1;\dfrac{1}{\phi^{n}(P)-A}$ is quasi-$(S,\epsilon)$-integral
$\\}$.

######  Corollary 2.12.

Under the hypothesis of Theorem 2.9,

$\displaystyle\lim_{n\rightarrow\infty}\dfrac{\lambda_{v}(\phi^{n}(P),A)}{d^{n}}=\displaystyle\lim_{n\rightarrow\infty}\dfrac{\lambda_{v}(\phi^{n}(P),A)}{\hat{h}_{\phi}(\phi^{n}(P))}=0$
for every $v\in M_{K}$.

###### Proof.

Applying Theorem 2.9 for the set of places that contains just the place $v$,
we conclude that for every natural $n$ big enough, it will be true that

$\dfrac{\lambda_{v}(\phi^{n}(P),A)}{d^{n}}\leq\epsilon\hat{h}_{\phi}(P)$.

Choosing $\epsilon$ sufficiently small, the result is proven.

∎

##  3\. Multiplicative dependence in orbits over function fields

###  3.1.  $S$-units, algebraic dynamics, and multiplicative dependence.

######  Lemma 3.1.

Let $\varphi(z)\in K(z)$ suppose that $|\varphi^{-1}(\infty)|\geq 3$. Then

$\\{f\in K:\varphi(f)\in R_{S}\\}$

is finite, and each $f$ in this set has height bounded by a constant
$C(\varphi,K,|S|)$

###### Proof.

This is [9, Theorem 12(i)]. ∎

The following is a version of [4, Theorem 1.2] for function fields.

######  Theorem 3.2.

Let $\varphi(z)\in K(z)$ of degree at least two, and assume that $\varphi$ is
not isotrivial. Then
(a) If $|\varphi^{-1}(\\{0,\infty\\})|\geq 3$, then

$\\{f\in K:\varphi(f)\in R^{*}_{S}\\}$

is finite, and its elements have height bounded from above effectively in
terms of $\varphi,|S|$ and $K$. In other words, if the referred set is
infinite, then $\varphi$ has one of the following forms:

$\varphi(X)=f(X-g)^{\pm d},f\neq 0,\text{ or
}\varphi(X)=f(X-g)^{d}/(X-h)^{d},f(g-h)\neq 0.$

(b) Let

$\mathcal{F}_{2}(K,\varphi,R_{S}^{*})=\\{(n,\alpha)\in\mathbb{Z}_{\geq
2}\times\text{Wander}_{K}(\varphi):\varphi^{(n)}(\alpha)\in R_{S}^{*}\\}.$

If $(n,\alpha)\in\mathcal{F}_{2}(K,\varphi,R_{S}^{*})$, then $n$ belongs to a
finite set bounded by an explicit constant depending on $\varphi,K,|S|$ and
$\inf_{\hat{h}_{\varphi}(P)>0}\hat{h}_{\varphi}(P)$, and $\alpha$ to a set of
height bounded effectively from above effectively in terms of $\varphi,|S|$
and $K$, unless $\varphi$ has the form $\varphi(X)=fX^{\pm d}$.

###### Proof.

For (a), such fact follows from the ideas in the proof of [10, Proposition
1.5(a)], namely, by our hypothesis, the function
$\psi(z):=\varphi(z)+1/\varphi(z)$ satisfies the hypothesis of the previous
Lemma, thus $\\{f\in K:\psi(f)\in R_{S}\\}$ is finite. But $\psi(\beta)\in
R_{S}$ whenever $\varphi(\beta)\in R^{*}_{S}$, hence it follows that $\\{f\in
K:\varphi(f)\in R^{*}_{S}\\}$ is finite, and each $f$ in it has height bounded
by $C(\varphi+1/\varphi,K,|S|)$.

For (b), we consider the well-defined map

$\mathcal{F}_{2}(K,\varphi,R_{S}^{*})\rightarrow\\{f\in K:\varphi^{(2)}(f)\in
R^{*}_{S}\\}$

sending $(n,\alpha)$ to $\varphi^{(n-2)}(\alpha)$. By (a), if $\varphi^{(2)}$
is in the same conditions of $\varphi$ in (a), then there are only finitely
many possibilities for such $\varphi^{(n-2)}(\alpha)$, whose heights are
bounded by $C(\varphi^{(2)}+1/\varphi^{(2)},K,|S|)$. Hence we also obtain an
effective bound $C_{1}(\varphi,K,|S|)$ for the canonical heights of such
points, and thus also for the heights of the referred $\alpha$’s. Taking a
certain $\varphi^{(n-2)}(\alpha)$ among such possibilities, and using by Lemma
2.5 that $\inf_{\hat{h}_{\varphi}(P)>0}\hat{h}_{\varphi}(P)$ exists, we have
by the same calculations in the proof of [4, Lemma 2.3] that $n$ is bounded by
$2+\log\left(\frac{C_{1}(\varphi,K,|S|)}{inf_{\hat{h}_{\varphi}(P)>0}\hat{h}_{\varphi}(P)}\right)$,
unless $\varphi^{(2)}$ has one of the special forms from (a). Using the
Riemann-Hurwitz formula, which is valid in characteristic zero, the same
method carried out in the end of the the proof of [4, Theorem 1.2] can be
performed, concluding the desired results.

∎

The next result generalizes [4, Theorem 1.3] to function fields.

######  Theorem 3.3.

Let $r,s\in\mathbb{Z}$ with $rs\neq 0$, and set

$\rho=\dfrac{\log(|s|/|r|)}{\log d}+1$

Let $\phi\in K(z)$ not isotrivial with degree $d\geq 2$. Assume that $0$ is
not a periodic point for $\phi$ and that $|\phi^{-1}(\\{0,\infty\\})|\geq 3$.
We let $\mathcal{E}_{\rho}(K,\phi,S,r,s)$ to be

$\left\\{(n,k,f,u)\in\mathbb{Z}_{\geq\rho}\times\mathbb{Z}_{\geq
0}\times\textit{Wander}_{K}(\phi)\times
R_{S}^{*}:\phi^{(n+k)}(f)^{r}=u\phi^{(k)}(f)^{s}\right\\}.$

If $(n,k,f,u)\in\mathcal{E}_{\rho}(K,\phi,S,r,s)$, then $n$ and $k$ are
bounded effectively in terms of $\phi,K,\rho,|S|$ and
$\inf_{\hat{h}_{\varphi}(P)>0}\hat{h}_{\varphi}(P)$, and $f$ and $u$ have
height bounded from above effectively in terms of $\phi,\rho,|S|$ and $K$.

###### Proof.

Arguing in the same way as in the proof of [4, Theorem 1.3], we can assume
that $0$ is not an exceptional point for $\phi$, which allows us to apply
Theorem 2.9(b) with $A=0$, and $\epsilon$ to be specified later, yielding an
effective constant $\gamma_{3}(\phi,\epsilon,|S|,K)$ such that

$\max\left\\{n\geq 0:\displaystyle\sum_{v\in
S}\log^{+}(|\phi^{n}(\alpha)|_{v}^{-1})\geq\epsilon\hat{h}_{\phi}(\phi^{n}(\alpha))\right\\}\leq\gamma_{3}$

We study triples $(n,k,\alpha)\in\mathbb{Z}_{\geq 1}\times\mathbb{Z}_{\geq
0}\times\text{Wander}_{K}(\phi)$ such that

$|\phi^{n+k}(\alpha)|_{v}^{r}=|\phi^{k}(\alpha)|_{v}^{s}\quad\text{ for all
}v\in M_{K}\setminus S.$

We then divide the proof in two cases.
Case 1: $n+k\geq\gamma_{3}$.
In this case Theorem 2.9 tells us that $(n,k,\alpha)$ satisfies

(3.1) $\displaystyle\sum_{v\in
S}\log^{+}(|\phi^{n+k}(\alpha)|_{v}^{-1})\leq\epsilon\hat{h}_{\phi}(\phi^{n+k}(\alpha)).$

we compute

$\displaystyle h(\phi^{n+k}(\alpha))$
$\displaystyle=h(\phi^{n+k}(\alpha)^{-1})=\sum_{v\in
S}\log^{+}|\phi^{n+k}(\alpha)|^{-1}_{v}+\sum_{v\notin
S}\log^{+}|\phi^{n+k}(\alpha)|^{-1}_{v}$
$\displaystyle\leq\epsilon\hat{h}_{\phi}(\phi^{n+k}(\alpha))\ +\sum_{v\notin
S}\log^{+}|\phi^{k}(\alpha)|^{-s/r}_{v}$
$\displaystyle\leq\epsilon\hat{h}_{\phi}(\phi^{n+k}(\alpha))\
+\left|\frac{s}{r}\right|\sum_{v\notin S}h(\phi^{k}(\alpha)^{-1})$
$\displaystyle\leq\epsilon\hat{h}_{\phi}(\phi^{n+k}(\alpha))\
+d^{\rho-1}h(\phi^{k}(\alpha))$
$\displaystyle\leq\epsilon\hat{h}_{\phi}(\phi^{n+k}(\alpha))+d^{\rho-1}(\hat{h}_{\phi}(\phi^{k}(\alpha))+O(h(\phi)+1))$

implying that

$(1-\epsilon)d^{n+k}\hat{h}_{\phi}(\alpha)\leq
d^{\rho-1+k}\hat{h}_{\phi}(\alpha)+O(d^{\rho-1}(h(\phi))+1),$

which with $\epsilon=1/3$ yields that

$d^{n+k-1}\hat{h}_{\phi}(\alpha)\leq O(d^{\rho-1}(h(\phi))+1),$

and hence

$n,k\leq\gamma_{4}+\log^{+}_{d}\left(\dfrac{h(\phi)}{\inf_{\hat{h}_{\phi}(P)>0}\hat{h}_{\phi}(P)}\right)$

for an effective constant $\gamma_{4}(K,|S|,\phi,\rho)$. Given that there are
at most finitely many possibilities for $n$ and $k$, this also yields a
desired effective height bound for the referred $\alpha$’s.
Case 2: $n+k\leq\gamma_{3}$.
Since$n$ and $k$ are bounded, we may assume that they are fixed. In this case,
we let

$g(z)=\phi^{n}(z)^{r}/z^{s}$

so that we have

$g(\phi^{k}(\alpha))=\phi^{n+k}(\alpha)^{r}/\phi^{k}(\alpha)^{s}\in
R_{S}^{*}.$

This says that

$\phi^{k}(\alpha)\in\\{f\in K:g(f)\in R^{*}_{S}\\}$

By 3.1, the set above has cardinality effectively bounded, and its elements as
well as the referred $\alpha$’s have height effectively bounded from above as
desired, except if $g$ has at most two zeros and poles. But the assumption
that $0$ is not periodic implies that $0$ is a pole of $g$, and the assumption
that $\phi(X)\neq cX^{\pm d}$ implies that $\phi^{n}$ has at least two poles
or zeros distinct from $0$. Hence $g$ has at least three poles and zeros. ∎

###  3.2. Polynomial dynamics and multiplicative dependence over function
fields in one variable.

In this subsection, we consider $K$ to be an algebraic function field in one
variable of characteristic zero. When we deal with polynomials in this
setting, one can obtain independence on the exponents $r,s$ from Theorem 3.3.

The result stated below is analogous to a Schinzel-Tijdeman result, now over
the fields treated here.

######  Lemma 3.4.

[6] Let $f\in K[X]$ be a polynomial of degree $n$ with at least two distinct
zeros in some algebraic closure of $K$. Then the equation

$f(x)=y^{m}\text{ in }x,y\in K,m\in\mathbb{Z}_{\geq 0},y\notin k$

implies

$m\leq B(f,K),$

where $B=B(f,K)$ is an effective constant.

The next result generalizes [4, Theorem 1.7] to function fields in one
variable.

######  Theorem 3.5.

Let $\phi\in K[z]$ be a non-isotrivial polynomial of degree $d\geq 2$ such
that $\phi$ and $\phi^{2}$ have no multiple roots, and $0$ is not a periodic
point of $\phi$. Let $S\in M_{K}$ be a finite set of places. Then there exist
a set $T_{1}$ with cardinality effectively bounded in terms of $K,\phi$ and
$S$, and a set $T_{2}$ whose elements have height effectively bounded from
above in terms of $K,\phi$ and $S$, such that

$\\{f\in K:(\phi^{m}(f))^{r}=u(\phi^{n}(f))^{s},u\in R^{*}_{S},m>n\geq
0,r,s\in\mathbb{Z}\\}=T_{1}\cup T_{2}.$

###### Proof.

Let $\alpha\in\text{Wander}_{\phi}(K)$ such that there exist non-negative
integers $m>n>0$, integers $r$ and $s$, and $u\in R^{*}_{S}$ such that

(3.2) $(\phi^{m}(\alpha))^{r}=u(\phi^{n}(\alpha))^{s}.$

If $r=0$ or $s=0$, the result follows from Theorem 3.2, so we may assume that
$rs\neq 0.$ Also, due to the saturation $R^{*}_{S}$, we may assume that
gcd$(r,s)=1$. We enlarge $S$ by the places for which $\phi$ has bad reduction,
namely, those $v\in M_{K}$ such that if $f(z)=c_{0}+c_{1}z+...+c_{d}z^{d}$
then $v(c_{i})<0$ for some $i$ or $v(c_{d})>0$, obtaining a new finite set of
places $S_{\phi}$. In fact, proving the results for the larger $S_{f}$ gives
results that are stronger than the original statements. In this case, one can
see that

$S_{\phi^{m}}\subset S_{\phi}\text{ for all }m\geq 1.$

Since $rs\neq 0,$ replacing $r,s$ by $-r,-s$ if necessary, we may assume that
$r>0$. We now divide the proof in cases.
Case A: $\alpha\in R_{S_{\phi}}$.
By definition one can check that $\phi^{k}(\alpha)\in R_{S_{\phi}}$ for every
$k\geq 0$, so that

$v(\phi^{k}(\alpha))\geq 0\text{ for all }k\geq 0,v\notin S_{\phi}.$

We now divide this step in some subcases.
Case A.1: $r>0,s<0$.
Here, equation 3.2 becomes

$(\phi^{m}(\alpha))^{r}(\phi^{n}(\alpha))^{t}=u\text{ with }t=-s>0.$

As $u\in R^{*}_{S_{\phi}},$ it follows that

$rv(\phi^{m}(\alpha))+tv((\phi^{n}(\alpha))=0\text{ for all }v\notin
S_{\phi}.$

Since $r,t>0$, this implies that

$v(\phi^{m}(\alpha))=v((\phi^{n}(\alpha))=0\text{ for all }v\notin S_{\phi},$

and hence that $\phi^{m}(\alpha)\in R^{*}_{S}$. The conclusions come then from
Theorem 3.2.
Case A.2: $r>0,s\geq 2$.
Since gcd$(r,s)$, we can choose $a$ and $b$ with $ar+bs=1$ so that 3.2 becomes

$\phi^{m}(\alpha)=u^{a}\left((\phi^{n}(\alpha))^{a}(\phi^{m}(\alpha))^{b}\right)^{s}.$

Since $\phi^{m}(\alpha)\in R_{S_{\phi}}$ and $u\in R^{*}_{S_{\phi}}$, we have
that $(\phi^{n}(\alpha))^{a}(\phi^{m}(\alpha))^{b}\in R_{S_{\phi}}$. If
$(\phi^{n}(\alpha))^{a}(\phi^{m}(\alpha))^{b}\in R_{S_{\phi}}^{*}$, then
$\phi^{m}(\alpha)\in R_{S_{\phi}}^{*}$ and we have the desired results by
Theorem 3.2. If not, then we also have that
$(\phi^{n}(\alpha))^{a}(\phi^{m}(\alpha))^{b}\notin k$. Writing

$\phi^{m}(\alpha)=\phi(\phi^{m-1}(\alpha)),$

we use Lemma 3.4 to conclude that the exponent $s\geq 2$ is effectively
bounded in terms of $\phi$ and $K$.

If we first assume that $\deg\phi\geq 3$, then we can apply [2, Proposition
4.6] to effectively bound the referred $\phi^{m-1}(\alpha)$’s, and then $m$
and $h(\alpha)$ as desired after using the similar method to the one used in
the proof of Theorem 3.2(b).

If otherwise $\deg\phi=2$, we can apply [2, Proposition 4.7], and conclude
similarly.
Case A.3: $r\geq 2,s=1$.
If $n\geq 2$, then the same discussion as above holds again (replacing $m$ by
$n$ and $r$ by $s$). It is enough then to consider the case $n=1$, so that 3.2
becomes

$\phi(\alpha)=u^{-1}(\phi^{m}(\alpha))^{r}.$

If $r\geq 3$, then we can apply [2, Proposition 4.7] again to conclude. If
otherwise $r=2$, we can apply Theorem 3.3 to conclude as desired.
Case A.4: $r=1,s=1$.
This case is covered by Theorem 3.3.
Case B: $\alpha\notin R_{S_{\phi}}$.
Choosing $v\in M_{K}\setminus S_{\phi}$ such that $v(\alpha)<0$, we have also
that $v\notin S_{\phi^{k}}$ for all $k\geq 1$. In this case, we have that
$v(\phi^{k}(\alpha))=d^{k}v(\alpha)$ for all $k\geq 0$ and hence 3.2 implies

$rd^{m}v(\alpha)=sd^{n}v(\alpha).$

Therefore $r=1$ and $s=d^{m-n}$, and 3.2 becomes

$\phi^{n+k}(\alpha)=u(\phi^{k}(\alpha))^{d^{n}}.$

If $n=1$, $\phi^{1+k}(\alpha)=u(\phi^{k}(\alpha))^{d^{n}},$ and we can use
Theorem 3.2(a) applied to the rational function $f(X)/X^{d}$ to conclude the
desired.

Otherwise, we first enlarge $S_{\phi}$ in a controlled way such that
$\text{disc}(\phi),\text{disc}(\phi^{2})\in R^{*}_{S_{\phi}}$. Moreover, we
note that the group $R^{*}_{S_{\phi}}/(R^{*}_{S_{\phi}})^{d^{2}}$ is finite
and effectively bounded due to [12, III, Prop.2.3.2], allowing us to replace
$K$ by the finite extension $L$ generated by the set of values

$\\{\beta\in\overline{K}:\beta^{d^{2}}\in R^{*}_{S_{\phi}}\\},$

which depends only on $K,d$ and $S$.

If $d\geq 3$ or $d=2$, we consider respectively the curves

$\phi(X)=Y^{d^{2}}\text{ or }\phi^{2}(X)=Y^{d^{2}}.$

In both cases, we are able to apply [7, Theorem 2] to obtain a set $T_{1}$ as
in the statement with the desired finiteness properties.∎

###  3.3. Iterates as zeros of split polynomials

Here we would like to point that Theorem 1.10 of [4] also is true over
function fields of characteristic $0$, making use of Lemma 2.5. We start
recalling the following definition

######  Definition 3.6.

We define a multilinear polynomial with split variables to be a vector of
polynomials

$F(\textbf{T}_{1},...,\textbf{T}_{k})=\sum_{i=1}^{r}c_{i}\prod_{j\in
J_{i}}\textbf{T}_{j}\in K[\textbf{T}_{1},...,\textbf{T}_{k}]$

for some disjoint partition $J_{1}\cup...\cup J_{r}=\\{1,...,s\\}$ and
$c_{i}\in K^{*},i=1,...,r$.

######  Theorem 3.7.

Let $K$ be a function field of a smooth projective curve over an algebraically
closed field of characteristic $0$. Let
$F(\textbf{T}_{1},...,\textbf{T}_{k})\in K[\textbf{T}_{1},...,\textbf{T}_{k}]$
be a multilinear polynomial with split variables and let $\phi\in K(z)$ be a
non-isotrivial rational function degree $d\geq 2$.

The set of $\alpha\in\overline{K}$ not preperiodic for $\phi$, for which there
exists a $k$-tuple of distinct non-negative integers $(n_{1},...,n_{k})$
satisfying

$F(\phi^{(n_{1})}(\alpha),...,\phi^{(n_{k})}(\alpha))=0$

is a set of bounded height. If $d\geq 3$, such heights are bounded effectively
in terms of $\phi,F$ and $K$.

Moreover, there are only finitely many $k$-tuples of integers
$n_{1}>n_{2}>...>n_{k}$ satisfying
$F(\phi^{(n_{1})}(\alpha),...,\phi^{(n_{k})}(\alpha))=0$, and there is a bound
for such integers that depend only on $\phi,F,K$ and
$\inf_{\hat{h}_{\phi}(P)>0}\hat{h}_{\phi}(P)$, independent on $\alpha$.

###### Proof.

The proof follows the number field situation [4, Theorem 1.10] almost ipsis
literis, except that we have to point that the quanitity $C_{2}(K,f)$ in that
proof is replaced here by $\inf_{\hat{h}_{\phi}(P)>0}\hat{h}_{\phi}(P)$, which
we know to exist due to Lemma 2.5. ∎

Acknowledgements: I am thankful to Australian Research Council Discovery Grant
DP180100201 and UNSW for supporting me in this research.

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