# Simulations of Mpemba Effect in WATER, Lennard-Jones and Ising Models:
Metastability vs Critical Fluctuations

Soumik Ghosh  Purnendu Pathak  Sohini Chatterjee  Subir K. Das
<EMAIL_ADDRESS>Theoretical Sciences Unit and School of Advanced Materials,
Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur P.O.,
Bangalore 560064, India

Via molecular dynamics simulations we study ICE formation in the TIP4P/Ice
model that is known to describe structure and dynamics in various phases of
WATER accurately. For this purpose well equilibrated configurations from
different initial temperatures, $T_{s}$, belonging to the fluid phase, are
quenched to a fixed subzero temperature. Our results on kinetics, for a wide
range of $T_{s}$, following such quenches, show quicker crystallization of
samples that are hotter at the beginning. This implies the presence of the
puzzling Mpemba effect (ME). Via a similar study, we also identify ME in fluid
to solid transitions in a Lennard-Jones (LJ) model. In the latter case, the ME
appears purely as an outcome of the influence of critical fluctuations on the
nonequilibrium growth process, for which we present interesting scaling
results. For the TIP4P/Ice case, on the other hand, we show that delay in
nucleation, due to metastability, can alone be a driving factor for the
exhibition of ME. To substantiate the difference between the two cases, we
also present LJ-like scaling results for ME in a magnetic transition. Our
simulations indicate that in each of the systems the effect can be observed
independent of the cooling rate that may vary when samples from different
$T_{s}$ are brought in contact with a heat reservoir working at a fixed lower
temperature.

##  I Introduction

A hotter sample of Water may freeze faster, than a colder one, when kept
inside a refrigerator working at a subzero temperature [1, 2, 3, 4]. This
counterintuitive fact is referred to as the Mpemba Effect (ME) [5] and is
discussed since the time of Aristotle [6]. In recent times, in efforts to
generalize the effect, several other experimental systems and theoretical
models were shown to exhibit similar phenomena. Examples include cooling
granular gases [7, 8], clathrate hydrates [9], anti-ferromagnets [10] and spin
glasses[11], as well as, perhaps even more surprisingly, pure ferromagnetic
systems [3, 12, 13]. Despite the continuously growing list of such systems,
the ME remains still a puzzle. There exists no clear hint whether a common
fact is responsible for the observations in different systems. In fact, for
the original system, i.e., Water, even a demonstration via computer
simulations is nonexistent, though there are works by providing possible
explanations if the effect indeed exists [14, 15, 16, 17, 2, 1]. Such a status
is perhaps due to the difficulty owing to the complex natures of Water
molecules and related interactions [18] that do not allow simulations of
adequately large systems for long enough times. The long simulations are
needed to counter the metastable features that may severely delay nucleation
for a transition from a fluid phase to Ice. An important question here is to
ask: How the ME may be connected to the metastability – Should the longevity
of the latter be a function of the initial temperature? Or, growth, following
the nucleation, is the only contributor to the initial temperature dependence
of the transformation? Questions have also been raised if the ME is a result
of differences in times for reaching the final temperature, $T_{f}$, from
different starting temperatures, $T_{s}$ [14, 19]. To address these issues,
and obtain a more general picture, in addition to theoretical and
computational studies of Water, it should also be of interest to undertake
studies of simpler systems exhibiting fluid to solid transitions. Here, we
study such transitions in a model Water [20, 21] and a Lennard-Jones (LJ)
system [22], without impurities.

In each of the considered cases, we prepare equilibrium fluid configurations
at different $T_{s}$. These configurations are quenched to a fixed $T_{f}$, at
which the thermodynamic phase is a solid one. In the case of Water, achieving
Ice nucleation under homogeneous condition is a difficult task, for reason
already mentioned above. We choose a model [21] for which this was
demonstrated [20] to occur with reasonable ease. Note that higher possibility
of fluctuations should enhance the scope of nucleation. Consideration of small
systems, may severely restrict such chances. On the other hand, handling large
systems is a difficult task, when the requirement is to simulate for long
times that may be a necessity for the present problem for a certain range of
$T_{s}$. Thus, it is important to adopt a “trade-off” between system size and
run length [3]. Following this strategy, we have been able to realize
homogeneous Ice nucleations for a large range of $T_{s}$. From the analysis of
the corresponding data sets we find clear evidence of ME in this as well as
the LJ system. In the case of LJ, metastability is not a matter of concern;
expected thermodynamic structure for the chosen $T_{f}$ is always obtained
without encountering any barrier. This and the results for Water provide
interesting classification of ME based on the role of metastability. We also
present results for para-to-ferromagnetic transitions in a model system [13,
12, 23], Hamiltonian of which does not contain any element of frustration.
This is to further substantiate the conclusion from the LJ system that ME can
be observed in absence of metastability as well [12, 3, 24], unlike the case
of Water, driven, for example, by the differences in critical fluctuations at
various starting points [12, 3, 13].

##  II models and techniques

For the LJ system, we perform molecular dynamics (MD) simulations [22, 25]
using a truncated, shifted, and force-corrected potential, within which two
particles at a distance $r$ apart interact via [22, 26]
$u(r)=U(r)-U(r_{c})-(r-r_{c})(dU/dr)_{r=r_{c}}$. Here
$U(r)=4\varepsilon\left[(\frac{\sigma}{r})^{12}-(\frac{\sigma}{r})^{6}\right]$
is the standard LJ potential, $\sigma$ being the particle diameter and
$\epsilon$ deciding the strength of the interaction. The cut-off distance is
taken to be [26] $r_{c}=2.5\sigma$. The modified form captures the same basic
facts of transition (e.g., critical universality remains unchanged) as the
original one, while speeding up the simulations. In $d=2$, the space dimension
of our interest for this model, corresponding coexistence curve in the number
density ($\rho$)-temperature ($T$) plane was estimated using Monte Carlo (MC)
simulations [27]. The critical values of $\rho$ and $T$ were noted to be
$\rho_{c}=0.37$, and $T_{c}=0.41\varepsilon/k_{B}$, respectively, $k_{B}$
being the Boltzmann constant. While one observes practically a vapor-liquid
coexistence for $T$ close to $T_{c}$, far below $T_{c}$ the high density phase
is a solid one, in $d=2$ this being a (quasi long-range) hexatic one. For this
model, MD simulations are carried out in constant NVT ensemble, N and V being,
respectively, the number of particles and volume of the system, using a
hydrodynamics preserving Nosé-Hoover (NH) thermostat [28, 25], while quenching
homogeneous configurations from different $T_{s}$ $(>T_{c})$ to a $T_{f}$
belonging to the coexistence regime. We have used square boxes of linear
dimension $L=256\sigma$. The integration time step was fixed at $\Delta
t=0.005\tau$, $\tau$ $(=\sqrt{m\sigma^{2}/\varepsilon})$ being the LJ unit of
time, and $m$ the mass of each particle. We set $\varepsilon$, $\sigma$,
$k_{B}$ and $m$ to unity.

For the simulations of Water, we have chosen a rather realistic model [29,
20]: TIP4P/Ice. This is a four-point rigid Water model with transferable
intermolecular potential. There, in addition to the points related to the
positions of one oxygen (O) atom and two hydrogen (H) atoms, an additional
point exists. This point, having a (negative) charge $1.1794$, in electronic
unit, is placed at a fixed distance ($0.1577\AA$) from O along the bisector of
the H-O-H angle, having experimental value [21] $104.52\degree$. The O atoms
interact with each other via the full LJ potential, with $\varepsilon=0.21084$
kcal/mole. Each H-atom carries a positive charge of magnitude $0.5897$. All
charge points, expectedly, interact via the Coulombic potential [21]. The
masses of O and H atoms are set to be $15.9994$ amu and $1.008$ amu,
respectively. With this model we carry out NPT MD simulations, with $96$
molecules [30, 31, 32], using LAMMPS [33], having NH thermo- [28, 25] and
barostats [34], at pressure $P=4.5$ atm, for which the thermodynamic limit
boiling and freezing points stand [35] at $\simeq 420K$ and $\simeq 273K$,
respectively. Each simulation we run upto $500$ ns, to observe freezing, with
the integration time step $\Delta t=1$ fs. Results on average freezing time
are obtained by using the runs that exhibit freezing [36, 20].

For the magnetic case, we carry out MC simulations with the Ising model,
having the Hamiltonian [23] $H=-J\sum_{<i,j>}S_{i}S_{j}$, $S_{i}=\pm 1$ being
the possible spin states and $J$ $(=1)$ the interaction strength, in $d=2$,
for which $T_{c}\simeq 2.269J/k_{B}$. For this model, we have implemented the
Glauber spin-flip kinetics [23] on a square lattice having same box dimensions
as the LJ case. An MC trial move there is to randomly choose a lattice site
and change its spin state. Such a move is accepted following a standard
Metropolis criterion [23]. The time in these simulations are measured in units
of MC steps (MCS), a unit consisting of $L^{2}$ trial moves. For all the
models, we have applied periodic boundary conditions.

Figure 1: (a) An equilibrium configuration of Water molecules is shown for
$T_{s}=400K$. (b) Potential energy profile corresponding to a simulation run,
starting with the snapshot at (a), is shown, following a quench to
$T_{f}=230$K. In the main frame we have used a log scale for time, whereas in
the inset the scale is linear. The jump corresponds to metastable Water to Ice
transition. The horizontal lines capture the fluctuations of energy within the
metastable window. (c) A typical configuration is shown from the Ice regime of
the last simulation run. (d) Temperature is plotted versus time for the
simulation in (b). The assigned value of temperature is shown by the dashed
line. Figure 2:  (a) PE is plotted as a function of time for the TIP4P/Ice
simulations. Results are shown for quenches to $T_{f}=230$K, from a few
$T_{s}$. Data in each of the sets are thinned down for the sake of clarity.
The symbols show the locations of jumps for simulation runs with certain other
starting configurations. A unique colour is used for a given $T_{s}$. The
early parts of the energy decay are shown again in (b) by using a log scale
for the abscissa. The horizontal lines stand for approximately the average
values for intermediate metastable liquid and final Ice phase energies. (c)
The freezing time, averaged over $8$ runs, for each $T_{s}$, is plotted versus
$T_{s}$. The dashed line is a guide to the eye.

##  III Results

Fig. 1 provides information on the freezing of a configuration of water
molecules, when quenched from $T_{s}=400K$ to $T_{f}=230K$, following the
protocol described below. A system of Water molecules were first arranged on a
periodic array, with bond lengths and orientations taken from GitHub [37],
before heating up to $500K$ to rapidly break the periodicity. A fluid
configuration from this heating run is further simulated at the desired
$T_{s}$, to prepare initial configurations, before finally quenching to
$T_{f}$. A typical configuration (see a fluid-like snapshot in (a)) from the
equilibrium part of the later run is quenched to the previously mentioned
$T_{f}$. The potential energy (PE) corresponding to this simulation is shown
in part (b), over $500$ ns. For a long period, the energy fluctuates within a
window corresponding to a metastable liquid phase, before jumping down to
another window that relates to an Ice phase, which we have confirmed via
calculations of appropriate structural quantities. Corresponding hexagonal
structure is shown in part (c). A temperature profile for this run can be seen
in part (d). Clearly, the temperature settles at the assigned $T_{f}$ within a
short time interval.

Figure 3:  (a) Schematic diagram describing the protocol for studying kinetics
in the LJ model. In (b) and (c) we show typical equilibrium snapshots from two
different $T_{s}$. (d) A late-time snapshot, captured during an evolution,
following a quench from $T_{s}=0.6$ to $T_{f}=0.2$, is shown. (e) An enlarged
portion of the snapshot in (d) is showing a hexatic arrangement of particles.
(f) Plots of PE, following quenches to $T_{f}=0.2$, from a few $T_{s}$, are
shown against time. The upper half displays the early-time behavior and the
lower one corresponds to the late-time decays.

Fig. 2 (a) shows the energy profiles for the earliest freezings for a set of
$T_{s}$ values. The transformation times, from the intermediate metastable to
the Ice phase, appear systematic, for the presented range of $T_{s}$. For a
clear visualization of the arrivals at the metastable energy levels, certain
early parts of the plots are shown in a semi-log scale in Fig. 2 (b). These
results are indicative of faster decay of PE for higher $T_{s}$. This
resembles ME in other systems that we will describe below. In Fig. 2 (a), we
have marked the freezing times corresponding to simulation runs starting with
some other configurations. For each $T_{s}$, the spread of the times is
restricted to a rather narrow window. We present the key results, on $t_{f}$,
the average freezing time, in Fig. 2 (c). This plot strongly points towards
the existence of the ME, while nicely resembling the nonmonotonic character
seen in the original work [5], even without impurity, i.e., for homogeneous
nucleation. Since in experiments heterogeneous nucleation cannot be avoided,
our results discard a possible belief that ME in Water may be an outcome
related to impurities. Noting that the temperature settles at the final value
rather early, see Fig. 1 (d), compared to the freezing times, differences in
heat transport or cooling rate for different $T_{s}$ are also unlikely to have
roles to play in the effect. Before further discussing the origin of this
fascinating behavior in Water, we take a look at the results from other
systems.

Fig. 3 corresponds to the LJ model. In part (a) we show the protocol. Systems
from various $T_{s}$ are quenched to a fixed $T_{f}$ inside the coexistence
curve. Typical initial configurations are shown in (b) and (c). Clearly, these
snapshots differ from each other in terms of spatial fluctuations that can be
quantified via $\xi$, the equilibrium correlation length [38]. Fig. 3 (d)
shows a typical nonequilibrium configuration obtained during an evolution
following a quench to $T_{f}=0.2$, from $T_{s}=0.6$. An enlarged portion of
this snapshot is displayed in (e) which shows a regular arrangement of
particles [39]. In part (f) we show the decay of potential energy for quenches
from a few $T_{s}$. The upper and lower parts of this frame contain,
respectively, the early and late time data. Opposite sequences of appearances
of the curves for different $T_{s}$, in the two time regimes, indicate
crossings that imply a faster rate of equilibration for a hotter system, the
basic requirement of ME [11, 12]. Tendency of such crossings can be
appreciated from early data in the case of Water as well – see the plots in
Fig. 2 (b) – faster rate of fall for higher $T_{s}$ is clear. In this case,
the crossings, however, got delayed due to the appearances of metastability.
Before discussing further the case of Water, we present more results on the LJ
case, as well as similar ones from the Ising model.

We have noted the times, $t_{c,T_{s}^{\rm{ref}}}$, corresponding to the
crossings of the PE curves for various $T_{s}$ with that for a reference case
$T_{s}^{\rm{ref}}=0.6$. These are plotted in Fig. 4 (a), as a function of
$T_{s}$. In Fig. 4 (b) we show a similar plot from the Ising model with
$T_{s}^{\rm{ref}}=2.32$ as the reference case. Very systematic trend is
visible, implying that the systems from each higher $T_{s}$ are approaching
the new equilibrium quicker than those from any lower $T_{s}$. Flattening of
the plots with the increase of $T_{s}$, for both the models, however, implies
that for very high $T_{s}$ values, even if the difference is large, between
two starting $T_{s}$, ME will be weak. Thus, the effect in these two models
may be related to differences in critical fluctuations [3] at different
$T_{s}$, that die out when $T_{s}>>T_{c}$. To facilitate a quantitative
understanding, we calculate $d{\rm(PE)}/dt$, the rate of decay of $\rm{PE}$,
for different $T_{s}$. For the ME to exist, $|d{\rm(PE)}/dt|$ should decrease
with the decrease of $T_{s}$, when estimated at a fixed value of $t$. If the
ME is truly due to the differences in critical fluctuations at the initial
states, we may expect a scaling behavior of this quantity, when seen against
$\epsilon=T_{s}-T_{c}$. Given that these two models belong to the same static
universality class [23, 26, 27, 38], this critical behavior can possibly be
reasonably close, barring deviation due to some differences in the values of
the domain growth exponent [23, 40] and finite-size effects between the two
cases. In Fig. 4 (c) and (d), we demonstrate it, for specific choices of $t$,
for the LJ and the Ising cases, respectively. Indeed, asymptotically,
reasonably similar power-law behavior emerge, while indicating very slow
equilibration as $T_{s}\rightarrow T_{c}$. On the other hand, the rate is
nearly constant for $T_{s}>>T_{c}$. This overall picture supports our
conjecture on the role of critical fluctuation. To appreciate the statement on
such fluctuations, it is worth looking at the snapshots in Figs. 3 (b) and
(c).

Figure 4:  (a) Plot of the crossing times, $t_{c,T_{s}^{\rm{ref}}}$, as a
function of $T_{s}$, for the LJ model. (b) Same as (a), but here it is for the
Ising model with $T_{f}=0$. The results were obtained after averaging over
runs with nearly $20000$ independent initial configurations. (c) Negative of
the decay rates of $\rm{PE}$, for different $T_{s}$, are plotted versus
$\epsilon$, for the LJ model, with $t=7$. (d) Same as (c), but here it is for
the Ising model, with $t=50$. The dashed lines in (c) and (d) represent power-
laws.

In the case of Water, on the other hand, such differences in spatial
correlation are practically nonexistent, for the considered initial state
points. It should be noted that the vapour-liquid critical point in this case
has the coordinates $(T_{c},P_{c})\equiv(647.096\rm K,22.064\rm MPa)$. This is
far from the trajectory of our initial states. We have, nevertheless, verified
this quantitatively by calculating the density field structure factors for
large enough systems and comparing with the Ornstein-Zernike picture [41].
Thus, for Water we have a different problem in hand.

##  IV Conclusion

We have presented results from the studies of kinetics of transitions in pure
Water, from fluid phases to that of Ice. The relaxation, quantified via the
transition time, starting from various initial temperatures, follows an
interesting nonmonotonic behavior. For a large range of starting temperatures,
surprizingly, a hotter sample freezes earlier than a colder one. This nicely
resembles the Mpemba effect [5]. For comparison, we have also presented
results from simulations of simpler systems for which differences in critical
fluctuations at the initial states appear to be the driving factor for the ME.
It is evident that the origin is different in Water. In this case, the
behavior in Fig. 2 (c) is related to the jumps from metastable intermediates
to the Ice phase. The longevity at a metastable level should be connected to
the properties, associated with molecular orientations, hydrogen bonding,
etc., at the corresponding initial state. In addition, it will also be
interesting to undertake studies by combining the distribution of nuclei in
space, and of their occurrences in time, with the growth process, for large
enough systems.

##  V author contribution

SKD proposed and designed the overall project, supervised the work and wrote
the manuscript. SG carried out all the works on Water. PP and SC contributed,
respectively, the results on the LJ and the Ising models.

##  VI acknowledgement

The authors acknowledge computation times in the clusters of National
Supercomputer Mission located in JNCASR.

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# Zhou valuations and jumping numbers

Qi’an Guan  Qi’an Guan: School of Mathematical Sciences, Peking University,
Beijing, 100871, China<EMAIL_ADDRESS>and  Zheng Yuan  Zheng Yuan:
Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese
Academy of Sciences, Beijing 100190, China<EMAIL_ADDRESS>
###### Abstract.

In this article, we prove that for any Zhou valuation $\nu$, there exists a
graded sequence of ideals $\mathfrak{a}_{\bullet}$ and a nonzero ideal
$\mathfrak{q}$ such that $\nu$ $\mathscr{A}-$computes the jumping number
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})$, and that for the
subadditive sequence $\mathfrak{b}^{\varphi}_{\bullet}$ related to a
plurisubharmonic function $\varphi$, there exists a Zhou valuation which
$\mathscr{A}-$computes
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}^{\varphi}_{\bullet})$, where the
“$\mathscr{A}-$compute” coincides with the “compute” in Jonsson-Mustaţă’s
Conjecture when the Zhou valuation $\nu$ is quasimonomial. We also give a
characterization for a valuation being a Zhou valuation.

###### Key words and phrases:

Zhou valuation, jumping number, compute, Jonsson-Mustaţă’s Conjecture

###### 2020 Mathematics Subject Classification:

32U25, 13A18, 14B05, 32U35

##  1\. Introduction

Let $u$ be a plurisubharmonic function near the origin $o$ of
$\mathbb{C}^{n}$. The Lelong number of $u$ at $o$ was defined as

$\nu(u,o):=\sup\\{c\geq 0:u(z)\leq c\log|z|+O(1)\text{ near }o\\},$

which gives some information on the asymptotic behaviour of $u$ near $o$ (see
[6, 5, 17, 3]). In [17], Rashkovskii introduced a generalization of the Lelong
number:

_Let $\varphi$ be a maximal weight with isolated singularity $o$ (see [17,
3]), and let $u$ be a plurisubharmonic function near $o$. The relative type of
$u$ relative to $\varphi$ was defined as: $\sigma(u,\varphi):=\sup\\{c\geq
0:u\leq c\varphi+O(1)\text{ near }o\\}.$_

Let $\psi$ be a plurisubharmonic function on a complex manifold. The
multiplier ideal sheaf $\mathcal{I}(\psi)$ (see [16, 7, 8, 4]) was defined as
the sheaf of germs of holomorphic functions $f$ such that $|f|^{2}e^{-2\psi}$
is local integrable. In [3], Boucksom-Favre-Jonsson showed that the relative
types to all tame maximal weights characterize the multiplier ideal sheaves of
plurisubharmonic functions.

Based on the solution of the strong openness conjecture:
$\mathcal{I}(\psi)=\mathcal{I}_{+}(\psi):=\cup_{p>1}\mathcal{I}(p\psi)$ (see
[4, 12]), Bao-Guan-Mi-Yuan [1] obtained a class of tame maximal weights
$\Phi_{o,\max}$ (called Zhou weights), whose relative types characterize the
multiplier ideal sheaves of plurisubharmonic functions and are well suited for
the tropical structure of the cone of plurisubharmonic functions:

$\sigma(u+v,\Phi_{o,\max})=\sigma(u,\Phi_{o,\max})+\sigma(v,\Phi_{o,\max})$

and

$\sigma(\max\\{u,v\\},\Phi_{o,\max})=\min\\{\sigma(u,\Phi_{o,\max}),\sigma(v,\Phi_{o,\max})\\}.$

Thus, the relative types to Zhou weights are valuations on the local ring
$\mathcal{O}_{o}$, denoted by:

$\nu(f):=\sigma(\log|f|,\Phi_{o,\max})\text{\,\,\,\,\,\,\,\, for any
$(f,o)\in\mathcal{O}_{o}$,}$

which are called by Zhou valuations [1]. Zhou valuations characterize the
division relations of $\mathcal{O}_{o}$ [1].

Let $\mathfrak{q}$ be a nonzero ideal of $\mathcal{O}_{o}$, and let $\varphi$
be a plurisubharmonic function near $o$. Recall that the jumping number (see
[14, 15])

$c_{o}^{\mathfrak{q}}(\varphi):=\sup\\{c\geq
0:|\mathfrak{q}|^{2}e^{-2c\varphi}\text{ is integrable near }o\\},$

where $|\mathfrak{q}|^{2}:=\sum_{1\leq j\leq k}|{h}_{j}|^{2}$ and
$\mathfrak{q}$ is generated by $({h}_{1},o),\ldots,({h}_{k},o)$. For any local
Zhou weight $\Phi_{o,\max}$ and its corresponding Zhou valuation $\nu$,
$c_{o}^{f}(\Phi_{o,\max})$ and $\nu(f)$ for all $(f,o)\in\mathcal{O}_{o}$ are
linearly controlled by each other [1] (see also Lemma 2.2).

Let $\mathfrak{a}_{\bullet}$ be a graded sequence of ideals of
$\mathcal{O}_{o}$: it is a sequence of ideals $(\mathfrak{a}_{m})_{m\geq 1}$
of $\mathcal{O}_{o}$ such that
$\mathfrak{a}_{p}\cdot\mathfrak{a}_{q}\subset\mathfrak{a}_{p+q}$ for any
positive integers $p$ and $q$. Let $\nu$ on $\mathcal{O}_{o}$ be a valuation,
and recall that (see [14, 15])

$\nu(\mathfrak{a}_{\bullet}):=\inf\frac{\nu(\mathfrak{a}_{m})}{m}=\lim_{m\rightarrow+\infty}\frac{\nu(\mathfrak{a}_{m})}{m},$

where $\nu(\mathfrak{c}):=\inf\\{\nu(f):(f,o)\in\mathfrak{c}\\}$ for any ideal
$\mathfrak{c}$ of $\mathcal{O}_{o}$. Recall that (see [14, 15])

$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet}):=\lim_{m\rightarrow+\infty}m\cdot\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{m})=\sup_{m}m\cdot\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{m})\in\mathbb{R}_{\geq
0}\cup\\{+\infty\\},$

where the jumping number of $\mathfrak{a}_{m}$ with respect to $\mathfrak{q}$

$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{m}):=c_{o}^{\mathfrak{q}}(\log|\mathfrak{a}_{m}|).$

Note that (see [14, 15])

$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})=\inf_{\nu}\frac{A(\nu)+\nu(\mathfrak{q})}{\nu(\mathfrak{a}_{\bullet})},$

where the infimum is over all valuations on $\mathcal{O}_{o}$ (it is enough to
only consider quasimonomial valuations (see [14, 15, 19])) and $A(\nu)$ is the
log discrepancy of $\nu$ (see [14, 15, 19]).

In [15], Jonsson-Mustaţă reduced the strong openness conjecture to a purely
algebraic statement:

######  Conjecture 1.1.

For any nonzero ideal $\mathfrak{q}$ and any graded sequence
$\mathfrak{a}_{\bullet}$ of ideals of $\mathcal{O}_{o}$ such that
$\mathfrak{m}^{p}\subset\mathfrak{a}_{1}$ for the maximal ideal $\mathfrak{m}$
of $\mathcal{O}_{o}$ and some $p\geq 1$, there is a quasimonomial valuation
$\nu$ on $\mathcal{O}_{o}$ that computes
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})$, i.e.,

$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})=\frac{A(\nu)+\nu(\mathfrak{q})}{\nu(\mathfrak{a}_{\bullet})}.$

A subadditive sequence of ideals $\mathfrak{b}_{\bullet}$ is a sequence
$(\mathfrak{b}_{t})_{t\in\mathbb{R}_{>0}}$ of nonzero ideals of
$\mathcal{O}_{o}$ satisfying
$\mathfrak{b}_{s+t}\subset\mathfrak{b}_{s}\cdot\mathfrak{b}_{t}$ for any
$s,t\in\mathbb{R}_{>0}$. For any valuation $\nu$ on $\mathcal{O}_{o}$ and
nonzero ideal $\mathfrak{q}$ of $\mathcal{O}_{o}$, recall that (see [14, 15])

$\nu(\mathfrak{b}_{\bullet}):=\sup_{t}\frac{\nu(\mathfrak{b}_{t})}{t}=\lim_{t\rightarrow+\infty}\frac{\nu(\mathfrak{b}_{t})}{t}\in\mathbb{R}_{\geq
0}\cup\\{+\infty\\}$

and

$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}):=\inf_{t}t\cdot\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{t})=\lim_{t\rightarrow+\infty}t\cdot\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{t}).$

We have (see [14, 15])

$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})=\inf_{\nu}\frac{A(\nu)+\nu(\mathfrak{q})}{\nu(\mathfrak{b}_{\bullet})},$

where the infimum is over all valuations on $\mathcal{O}_{o}$ (it is enough to
only consider quasimonomial valuations).

The following remark shows that the asymptotic multiplier ideals of
$\mathfrak{a}_{\bullet}$ is a subadditive sequence of ideals.

######  Remark 1.2.

Let $\mathfrak{a}_{\bullet}$ be a graded sequence of ideals. The asymptotic
multiplier ideals of $\mathfrak{a}_{\bullet}$ was defined by

$\mathfrak{b}_{t}:=\mathcal{I}(\frac{t}{m}\log|\mathfrak{a}_{m}|)_{o},$

where $m$ is divisible enough (depending on $t>0$). It is clear that
$\mathfrak{a}_{m}\subset\mathfrak{b}_{m}$ for any $m$, and it follows from
Lemma 3.7 (the subadditive Theorem, see [7]) that $(\mathfrak{b}_{t})_{t>0}$
is a subadditive system of ideals. We have
$\nu(\mathfrak{a}_{\bullet})=\nu(\mathfrak{b}_{\bullet})$ whenever
$A(\nu)<+\infty$ and

$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})=\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})$

for any nonzero ideal $\mathfrak{q}$ (see [14]).

Jonsson-Mustaţă also presented an equivalent conjecture [15] of Conjecture
1.1:

######  Conjecture 1.3.

Let $\mathfrak{q}$ be a nonzero ideal of $\mathcal{O}_{o}$, and let
$\mathfrak{b}_{\bullet}$ be a subadditive sequence of ideals of
$\mathcal{O}_{o}$. Assume that $\mathfrak{b}_{\bullet}$ is of controlled
growth (i.e.,
$\nu(\mathfrak{b}_{\bullet})\leq\frac{\nu(\mathfrak{b}_{t})}{t}+\frac{A(\nu)}{t}$
for any $t>0$ and any quasimonomial valuations $\nu$), and there is a positive
integer $p$ such that $\mathfrak{m}^{pj}\subset\mathfrak{b}_{j}$ for any $j$
and the maximal ideal $\mathfrak{m}$ of $\mathcal{O}_{o}$, then there is a
quasimonomial valuation $\nu$ on $\mathcal{O}_{o}$ that computes
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})$, i.e.,

$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})=\frac{A(\nu)+\nu(\mathfrak{q})}{\nu(\mathfrak{b}_{\bullet})}.$

The 2-dimensional Jonsson-Mustaţă’s Conjecture has been proved in [14] (some
related results seen in [9, 15]). Recently, Xu [19] proved the Jonsson-
Mustaţă’s Conjecture for the case $\mathfrak{q}=1$, which completed the
algebraic approach of solving the openness conjecture (see [8, 9, 2]).

In this article, we prove that for any Zhou valuation $\nu$, there exists a
graded sequence of ideals $\mathfrak{a}_{\bullet}$ and a nonzero ideal
$\mathfrak{q}$ that $\nu$ $\mathscr{A}-$computes the jumping number
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})$. For subadditive
sequence $\mathfrak{b}^{\varphi}_{\bullet}$ related to a plurisubharmonic
function $\varphi$, we show that there exists a Zhou valuation that
$\mathscr{A}-$computes
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}^{\varphi}_{\bullet})$. When the Zhou
valuation $\nu$ is quasimonomial, “$\mathscr{A}-$compute” coincides with
“compute”. Hence these results provide some support for Jonsson-Mustaţă’s
Conjecture. In the second part of this article, we give a characterization for
a valuation being a Zhou valuation.

###  1.1. Zhou valuation $\mathscr{A}-$computing the jumping number

Let us recall the definitions of Zhou weights and Zhou numbers (see [1]). Let
$f_{0}=(f_{0,1},\cdots,f_{0,m})$ be a vector, where $f_{0,1},\cdots,f_{0,m}$
are holomorphic functions near $o$. Let $\varphi_{0}$ be a plurisubharmonic
function near $o$, such that $|f_{0}|^{2}e^{-2\varphi_{0}}$ is integrable near
$o$.

######  Definition 1.4 ([1]).

We call that $\Phi^{f_{0},\varphi_{0}}_{o,\max}$ ($\Phi_{o,\max}$ for short)
is a local Zhou weight related to $|f_{0}|^{2}e^{-2\varphi_{0}}$ near $o$, if
the following three statements hold

(1) $|f_{0}|^{2}e^{-2\varphi_{0}}|z|^{2N_{0}}e^{-2\Phi_{o,\max}}$ is
integrable near $o$ for large enough $N_{0}\gg 0$;

(2) $|f_{0}|^{2}e^{-2\varphi_{0}}e^{-2\Phi_{o,\max}}$ is not integrable near
$o$;

(3) for any plurisubharmonic function $\varphi^{\prime}\geq\Phi_{o,\max}+O(1)$
near $o$ such that $|f_{0}|^{2}e^{-2\varphi_{0}}e^{-2\varphi^{\prime}}$ is not
integrable near $o$, $\varphi^{\prime}=\Phi_{o,\max}+O(1)$ holds.

For any local Zhou weight $\Phi_{o,\max}$, we have $\Phi_{o,\max}\geq
N\log|z|$ near $o$ for large enough $N$ (see [1]). Let $\Phi_{o,\max}$ be a
local Zhou weight related to $|f_{0}|^{2}e^{-2\varphi_{0}}$ near $o$, then
$\big{(}1+\sigma(\varphi_{0},\Phi_{o,\max})\big{)}\Phi_{o,\max}$ is a local
Zhou weight related to $|f_{0}|^{2}$ (see [1]). Thus, we only consider the
case $\varphi_{0}\equiv 0$ in this article.

We call a valuation $\nu$ on $\mathcal{O}_{o}$ is a Zhou valuation related to
$|f_{0}|^{2}$ if there exists a local Zhou weight $\Phi_{o,\max}$ related
$|f_{0}|^{2}$ near $o$ such that

$\sigma(\log|f|,\Phi_{o,\max})=\nu(f)$

for any $(f,o)\in\mathcal{O}_{o}$.

Let $\mathfrak{a}_{\bullet}$ be a graded sequence of ideals of
$\mathcal{O}_{o}$, and let $\mathfrak{q}$ be a nonzero ideal of
$\mathcal{O}_{o}$. Let $\nu$ on $\mathcal{O}_{o}$ be a Zhou valuation related
to $|\mathfrak{q}_{0}|^{2}$, where $\mathfrak{q}_{0}$ is a nonzero ideal of
$\mathcal{O}_{o}$. We define

$\mathscr{A}(\nu):=\sup_{\mathfrak{q},\mathfrak{a}_{\bullet}}\left(\nu(\mathfrak{a}_{\bullet})\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})-\nu(\mathfrak{q})\right),$

where the supremum is over all nonzero ideals $\mathfrak{q}$ and graded
sequences $\mathfrak{a}_{\bullet}$ satisfying
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})<+\infty$.

As
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})=\inf_{\nu}\frac{A(\nu)+\nu(\mathfrak{q})}{\nu(\mathfrak{a}_{\bullet})}$
(see [14]), we have the log discrepancy $A(\nu)\geq\mathscr{A}(\nu)$ for Zhou
valuation $\nu$. When the Zhou valuation $\nu$ is quasimonomial (see [14,
15]), $\mathscr{A}(\nu)$ actually coincides with $A(\nu)$ (by Proposition 2.5
in [15]).

######  Definition 1.5.

A Zhou valuation $\nu$ $\mathscr{A}-$computes
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})$, for a nonzero ideal
$\mathfrak{q}$ and a graded sequence of ideals $\mathfrak{a}_{\bullet}$ if
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})=\frac{\mathscr{A}(\nu)+\nu(\mathfrak{q})}{\nu(\mathfrak{a}_{\bullet})}.$

By definitions, a quasimonomial Zhou valuation $\nu$ $\mathscr{A}-$computes
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})$ if and only if $\nu$
computes $\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})$.

Let $\mathfrak{a}^{\nu}_{\bullet}$ be a graded sequence of ideals of
$\mathcal{O}_{o}$ such that
$\mathfrak{a}^{\nu}_{m}=\\{(f,o)\in\mathcal{O}_{o}:v(f)\geq m\\}$ for any $m$.
The following theorem shows that for any Zhou valuation $\nu$, there exists a
graded sequence of ideals $\mathfrak{a}_{\bullet}$ and a nonzero ideal
$\mathfrak{q}$ that $\nu$ $\mathscr{A}-$computes the jumping number
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})$.

######  Theorem 1.6.

Let $\mathfrak{q}_{0}$ be a nonzero ideal of $\mathcal{O}_{o}$, and let $\nu$
is a Zhou valuation related to $|\mathfrak{q}_{0}|$. Then $\nu$
$\mathscr{A}-$computes
$\mathrm{lct}^{\mathfrak{q}_{0}}(\mathfrak{a}^{\nu}_{\bullet})$, i.e.,

$\mathrm{lct}^{\mathfrak{q}_{0}}(\mathfrak{a}^{\nu}_{\bullet})=\frac{\mathscr{A}(\nu)+\nu(\mathfrak{q}_{0})}{\nu(\mathfrak{a}^{\nu}_{\bullet})}.$

The following remark gives a relation between $\mathscr{A}(\nu)$ with Tian
functions.

######  Remark 1.7.

$\mathscr{A}(\nu)=1-\nu(\mathfrak{q}_{0})=\sup_{(h,o)\in\mathcal{O}^{*}_{o}}(c_{o}^{h}(\Phi_{o,\max})-\nu(h))$,
where $\Phi_{o,\max}$ is the corresponding local Zhou weight of $\nu$ and
$\mathcal{O}^{*}_{o}:=\\{(f,o)\in\mathcal{O}_{o}:f\not=0\\}$. Denote that the
Tian function (see [1])

$\mathrm{Tn}(t):=\sup\\{c\geq
0:|\mathfrak{q}_{0}|^{2t}e^{-2c\Phi_{o,\max}}\text{ is integrable near }o\\}$

for $t\in\mathbb{R}$. Then
$\mathrm{Tn}(t)=\nu(\mathfrak{q}_{0})t+\mathscr{A}(\nu)$ for any $t\geq 1$.

Let $\mathfrak{b}_{\bullet}$ be a subadditive sequence of ideals. Proposition
3.18 shows that for any subadditive sequence $\mathfrak{b}_{\bullet}$ and any
nonzero ideal $\mathfrak{q}$,

$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})=\inf_{\nu}\frac{\mathscr{A}(\nu)+\nu(\mathfrak{q})}{\nu(\mathfrak{b}_{\bullet})},$

where the infimum is over all Zhou valuation $\nu$. We call a Zhou valuation
$\nu$ $\mathscr{A}-$computes
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})$ if
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})=\frac{\mathscr{A}(\nu)+\nu(\mathfrak{q})}{\nu(\mathfrak{b}_{\bullet})}$.

In the following, we consider a class of subadditive sequences
$\mathfrak{b}^{\varphi}_{\bullet}$ related to plurisubharmonic functions
$\varphi$ and show that there exist Zhou valuations, which
$\mathscr{A}-$compute
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}^{\varphi}_{\bullet})$.

Let $\varphi$ be a plurisubharmonic function near $o$. Denote that

$\mathfrak{b}_{t}^{\varphi}:=\mathcal{I}(t\varphi)_{o},$

then $\mathfrak{b}^{\varphi}_{\bullet}=(\mathfrak{b}^{\varphi}_{t})_{t>0}$ is
a subadditive sequence of ideals satisfying that
$\mathfrak{b}^{\varphi}_{s+t}\subset\mathfrak{b}^{\varphi}_{s}\cdot\mathfrak{b}^{\varphi}_{t}$
for any $s,t\in\mathbb{R}_{>0}$ by Lemma 3.7 (the subadditive Theorem, see
[7]). Assume that $\varphi\geq N\log|z|+O(1)$ near $o$ for large enough $N$.

######  Theorem 1.8.

For any nonzero ideal $\mathfrak{q}$ of $\mathcal{O}_{o}$, there exists a Zhou
valuation $\nu$ related to $|\mathfrak{q}|^{2}$, which $\mathscr{A}-$computes
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}^{\varphi}_{\bullet})$, i.e.,

$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}^{\varphi}_{\bullet})=\frac{\mathscr{A}(\nu)+\nu(\mathfrak{q})}{\nu(\mathfrak{b}^{\varphi}_{\bullet})}.$

###  1.2. A characterization for a valuation being a Zhou valuation

A valuation on $\mathcal{O}_{o}$ is a non-constant map
$\nu:\mathcal{O}_{o}^{*}\rightarrow\mathbb{R}_{\geq 0}$ satisfying the
following

$(1)$ $\nu(fg)=\nu(f)+\nu(g);$

$(2)$ $\nu(f+g)\geq\min\\{\nu(f),\nu(g)\\};$

$(3)$ $\nu(c)=0$, where $c\not=0$ is a constant function,

where $\mathcal{O}^{*}_{o}:=\\{(f,o)\in\mathcal{O}_{o}:f\not=0\\}$. We set
$\nu(0)=+\infty$ for any valuation $\nu$.

In a private conversation with the authors, Jonsson posed the following
question:

######  Question 1.9.

Can one characterize the valuations on $\mathcal{O}_{o}$ that appear as Zhou
valuations?

Let $\mathfrak{q}_{0}$ be a nonzero ideal of $\mathcal{O}_{o}$. For any
valuation $\nu$ on $\mathcal{O}_{o}$, denote that

$\mathfrak{a}_{m}^{\nu}:=\\{(f,o)\in\mathcal{O}_{o}:\nu(f)\geq m\\}$

is an ideal of $\mathcal{O}_{o}$.

We give a characterization for a valuation being a Zhou valuation, which gives
an answer to Question 1.9.

######  Theorem 1.10.

A valuation $\nu$ on $\mathcal{O}_{o}$ is a Zhou valuation related to
$|\mathfrak{q}_{0}|^{2}$ if and only if the following three statements hold:

$(1)$ $\\{(f,o)\in\mathcal{O}_{o}:v(f)>0\\}$ is the maximal ideal of
$\mathcal{O}_{o}$;

$(2)$ for any positive integer $m$,

$\mathfrak{q}_{0}\not\subset\mathcal{I}(\frac{1}{m}\log|\mathfrak{a}_{m}^{\nu}|)_{o};$

$(3)$ if there exists a valuation $\tilde{\nu}$ on $\mathcal{O}_{o}$
satisfying the above two statements and $\tilde{\nu}(f)\geq\nu(f)$ for any
$(f,o)\in\mathcal{O}_{o}$, then $\tilde{\nu}=\nu$.

If a valuation $\nu$ on $\mathcal{O}_{o}$ is a Zhou valuation related to
$|\mathfrak{q}_{0}|^{2}$, its corresponding Zhou weight is given by ([1], see
also Lemma 3.11)

$\Phi_{o,\max}(w)=\sup\left\\{\frac{\log|f(w)|}{\nu(f)}:f\in\mathcal{O}(D),\
\sup_{D}|f|\leq 1,\ f(o)=0,\ f\not\equiv 0\right\\},$

where $D$ is the unit ball in $\mathbb{C}^{n}$.

The following proposition shows the existence of Zhou valuations.

######  Proposition 1.11.

Let $\nu$ be a valuation on $\mathcal{O}_{o}$. Assume that $\nu$ satisfies the
statements $(1)$ and $(2)$ in Theorem 1.10. Then there exists a Zhou valuation
$\nu^{\mathrm{Zh}}$ related to $|\mathfrak{q}_{0}|^{2}$ such that
$\nu^{\mathrm{Zh}}\geq\nu$.

##  2\. Proofs of Theorem 1.6 and Remark 1.7

In this section, we prove Theorem 1.6 and Remark 1.7.

Firstly, we recall some results.

######  Lemma 2.1 (see [5]).

Let $D\subset\mathbb{C}^{n}$ be a bounded pseudoconvex domain, and let
$\varphi$ be a plurisubharmonic function $D$. For any positive integer $m$,
let $\\{\sigma_{m,k}\\}_{k=1}^{\infty}$ be an orthonormal basis of
$A^{2}(D,2m\varphi):=\big{\\{}f\in\mathcal{O}(D):\int_{D}|f|^{2}e^{-2m\varphi}d\lambda<+\infty\big{\\}}$.
Denote that

$\varphi_{m}:=\frac{1}{2m}\log\sum_{k=1}^{\infty}|\sigma_{m,k}|^{2}$

on $D$. Then there exist two positive constants $c_{1}$ (depending only on $n$
and diameter of $D$) and $c_{2}$ such that

(2.1)
$\varphi(z)-\frac{c_{1}}{m}\leq\varphi_{m}(z)\leq\sup_{|\tilde{z}-z|<r}\varphi(\tilde{z})+\frac{1}{m}\log\frac{c_{2}}{r^{n}}$

for any $z\in D$ satisfying
$\\{\tilde{z}\in\mathbb{C}^{n}:|\tilde{z}-z|<r\\}\subset\subset D$.
Especially, $\varphi_{m}$ converges to $\varphi$ pointwisely and in
$L^{1}_{\mathrm{loc}}$ on $D$.

Given a local Zhou weight $\Phi_{o,\max}$ near $o$ related to $|f_{0}|^{2}$,
the following lemma shows the jumping number $c_{o}^{G}(\Phi_{o,\max})$ and
the Zhou valuation $\nu(G,\Phi_{o,\max}):=\sigma(\log|G|,\Phi_{o,\max})$ for
all $(G,o)\in\mathcal{O}_{o}$ are linearly controlled by each other.

######  Lemma 2.2 ([1]).

For any holomorphic function $G$ near $o$, we have the following relation
between the jumping number $c^{G}_{o}(\Phi_{o,\max})$ and the Zhou valuation
$\nu(\cdot,\Phi_{o,\max})$,

$\begin{split}\nu(G,\Phi_{o,\max})+c_{o}(\Phi_{o,\max})\leq&c^{G}_{o}(\Phi_{o,\max})\\\
\leq&\nu(G,\Phi_{o,\max})-\sigma(\log|f_{0}|,\Phi_{o,\max})+1.\end{split}$

Especially, if $|f_{0}|^{2}\equiv 1$, we have

$\nu(G,\Phi_{o,\max})+1=c^{G}_{o}(\Phi_{o,\max}).$

Let $\psi$ be a plurisubharmonic function near $o$. Recall that the Tian
function (see [1]) was defined as

$\mathrm{Tn}(t;\psi):=\sup\\{c:|\mathfrak{q}_{0}|^{2}e^{2t\psi-2c\Phi_{o,\max}}\text{
is integrable near }o\\}$

for any $t\in\mathbb{R}$. Then we have

######  Lemma 2.3 ([1]).

$\mathrm{Tn}(t;\psi)=\mathrm{Tn}(0;\psi)+\sigma(\psi,\Phi_{o,\max})t$ holds
for any $t\geq 0$.

Let $\nu$ be a Zhou valuation related to $|\mathfrak{q}_{0}|^{2}$, where
$\mathfrak{q}_{0}$ is a nonzero ideal. Let
$\mathfrak{a}^{\nu}=(\mathfrak{a}^{\nu}_{m})_{m\geq 1}$ be a graded sequence
of ideals such that
$\mathfrak{a}_{m}^{\nu}=\\{(f,o)\in\mathcal{O}_{o}:\nu(f)\geq m\\}$.

######  Lemma 2.4.

$\nu(\mathfrak{a}^{\nu}_{\bullet})=1$ and
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}^{\nu}_{\bullet})=c_{o}^{\mathfrak{q}}(\Phi_{o,\max})$,
where $\mathfrak{q}$ is any nonzero ideal of $\mathcal{O}_{o}$ and
$\Phi_{o,\max}$ is the corresponding local Zhou weight of $\nu$.

###### Proof.

Note that $\nu(\mathfrak{a}^{\nu}_{m})\geq m$, which deduces
$\nu(\mathfrak{a}^{\nu}_{\bullet})\geq 1$. Let $(f_{1},o)\in\mathcal{O}_{o}$
such that $\nu(f_{1})\in(0,+\infty)$. Then $\nu(f_{1}^{k})=k\nu(f_{1})$ for
any positive integer $k$. As
$\mathfrak{a}^{\nu}_{m}=\\{(f,o)\in\mathcal{O}_{o}:v(f)\geq m\\}$, we have
$\nu(\mathfrak{a}_{m})\leq m+\nu(f_{1})$ for any $m$. Thus we get
$\nu(\mathfrak{a}^{\nu}_{\bullet})=1.$

Note that $\nu(f)=\sigma(\log|f|,\Phi_{o,\max}):=\sup\\{c\geq 0:\log|f|\leq
c\Phi_{o,\max}+O(1)$ near $o\\}$. It follows from Lemma 2.2 that there exists
a constant $C$ such that
$\mathcal{I}(m\Phi_{o,\max})_{o}\subset\mathfrak{a}_{m+C}^{\nu}$ for any $m$,
which shows

$\Phi_{o,\max}\leq\frac{1}{m}\log|\mathcal{I}(m\Phi_{o,\max})_{o}|+O(1)\leq\frac{1}{m}\log|\mathfrak{a}_{m+C}^{\nu}|+O(1)$

near $o$ by Lemma 2.1. Then we have

$c_{o}^{\mathfrak{q}}(\Phi_{o,\max})\leq\liminf_{m\rightarrow+\infty}c_{o}^{\mathfrak{q}}(\frac{1}{m}\log|\mathfrak{a}_{m+C}^{\nu}|)=\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}^{\nu}_{\bullet}).$

On the other hand, note that
$\frac{1}{m}\log|\mathfrak{a}_{m}^{\nu}|\leq\Phi_{o,\max}+O(1)$ near $o$,
which implies $\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}^{\nu}_{\bullet})\leq
c_{o}^{\mathfrak{q}}(\Phi_{o,\max}).$ Thus, we obtain

$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}^{\nu}_{\bullet})=c_{o}^{\mathfrak{q}}(\Phi_{o,\max}).$

∎

Now, we prove Theorem 1.6.

###### Proof of Theorem 1.6.

By definition of $\mathscr{A}(\nu)$, it suffices to prove

$\nu(\mathfrak{a}_{\bullet})\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})-\nu(\mathfrak{q})\leq\nu(\mathfrak{a}^{\nu}_{\bullet})\mathrm{lct}^{\mathfrak{q}_{0}}(\mathfrak{a}^{\nu}_{\bullet})-\nu(\mathfrak{q}_{0})$

for any nonzero ideal $\mathfrak{q}$ and graded sequence
$\mathfrak{a}_{\bullet}$ such that
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a})<+\infty$.
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})<+\infty$ shows that
$\lim_{m\rightarrow+\infty}\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{m})=0$.
Note that

(2.2)
$\begin{split}\nu(\mathfrak{a}_{\bullet})\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})&=\lim_{m\rightarrow+\infty}\nu(\mathfrak{a}_{m})\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{m})\\\
&=\lim_{m\rightarrow+\infty}[\nu(\mathfrak{a}_{m})]\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{m})\\\
&\leq\lim_{m\rightarrow+\infty}[\nu(\mathfrak{a}_{m})]\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}^{\nu}_{[\nu(\mathfrak{a}_{m})]}),\end{split}$

where $[x]:=\sup\\{y\in\mathbb{Z}:y\leq x\\}$ for any $x\in\mathbb{R}$. There
exists a positive number $N$ such that $\Phi_{o,\max}\geq N\log|z|+O(1)$ near
$o$. As
$\lim_{m\rightarrow+\infty}\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{m})=0$,
for any $N_{1}>0$ there exists $N_{2}>0$ such that $\log|\mathfrak{a}_{m}|\leq
N_{1}\log|z|+O(1)\leq\frac{N_{1}}{N}\Phi_{o,\max}+O(1)$ near $o$ for any
$m>N_{2}$, which implies that

$\lim_{m\rightarrow+\infty}\nu(\mathfrak{a}_{m})=+\infty.$

Using inequality (2.2), it suffices to consider the case
$\mathfrak{a}_{\bullet}=\mathfrak{a}_{\bullet}^{\nu}$. Note that
$c_{o}^{\mathfrak{q}_{0}}(\Phi_{o,\max})=1$, then following from Lemma 2.4, we
only need to prove that the equality

$c_{o}^{\mathfrak{q}}(\Phi_{o,\max})-\nu(\mathfrak{q})\leq
1-\nu(\mathfrak{q}_{0})$

holds for any nonzero ideal $\mathfrak{q}$. Assume that $\mathfrak{q}$ is
generated by $f_{1},\ldots,f_{l}$. Then Lemma 2.2 shows that

$c_{o}^{\mathfrak{q}}(\Phi_{o,\max})=\min_{j}c_{o}^{{f_{j}}}(\Phi_{o,\max})\leq\min_{j}\\{\nu(f_{j})+1-\nu(\mathfrak{q}_{0})\\}=\nu(\mathfrak{q})+1-\nu(\mathfrak{q}_{0}).$

Thus, Theorem 1.6 holds. ∎

Following the notations in Theorem 1.6, we prove Remark 1.7.

###### Proof of Remark 1.7.

Theorem 1.6 shows that

$\mathscr{A}(\nu)=\nu(\mathfrak{a}^{\nu}_{\bullet})\mathrm{lct}^{\mathfrak{q}_{0}}(\mathfrak{a}^{\nu}_{\bullet})-\nu(\mathfrak{q}_{0}).$

Following from Lemma 2.4, we have $\nu(\mathfrak{a}^{\nu}_{\bullet})=1$ and
$\mathrm{lct}^{\mathfrak{q}_{0}}(\mathfrak{a}^{\nu}_{\bullet})=c_{o}^{\mathfrak{q}_{0}}(\Phi_{o,\max})$,
where $\Phi_{o,\max}$ is the corresponding local Zhou weight related to
$|\mathfrak{q}_{0}|^{2}$ of $\nu$. Thus
$c_{o}^{\mathfrak{q}_{0}}(\Phi_{o,\max})=1$ and

$\mathscr{A}(\nu)=1-\nu(\mathfrak{q}_{0}).$

For any $(h,o)\in\mathcal{O}^{*}_{o}$, we have

$c_{o}^{h}(\Phi_{o,\max})-\nu(h)=\nu(\mathfrak{a}^{\nu}_{\bullet})\mathrm{lct}^{(h)_{o}}(\mathfrak{a}^{\nu}_{\bullet})-\nu((h)_{o})\leq\mathscr{A}(\nu),$

where $(h)_{o}$ is the ideal of $\mathcal{O}_{o}$ generated by $(h,o)$. There
exists $(h_{1},o)\in\mathfrak{q}_{0}$ such that
$\nu(h_{1})=\nu(\mathfrak{q}_{0})$. It follows from Lemma 2.2 that

$c_{o}^{h_{1}}(\Phi_{o,\max})\leq\nu(h_{1})-\nu(\mathfrak{q}_{0})+1=1.$

As $c_{o}^{\mathfrak{q}_{0}}(\Phi_{o,\max})=1$, we have
$c_{o}^{h_{1}}(\Phi_{o,\max})=1$. Thus,

$\mathscr{A}(\nu)=\sup_{(h,o)\in\mathcal{O}^{*}_{o}}(c_{o}^{h}(\Phi_{o,\max})-\nu(h)).$

Note that $\sigma(\log|\mathfrak{q}_{0}|,\Phi_{o,\max})=\nu(\mathfrak{q}_{0})$
and $c_{o}^{\mathfrak{q}_{0}}(\Phi_{o,\max})=1$. Taking
$\psi=\log|\mathfrak{q}_{0}|$, Lemma 2.3 implies that

$\mathrm{Tn}(t)=1+\nu(\mathfrak{q}_{0})(t-1)=\nu(\mathfrak{q}_{0})t+\mathscr{A}(\nu).$

∎

##  3\. Preparations

In this section, we do some preparations for the proofs of Theorem 1.8 and
Theorem 1.10.

###  3.1. Some results about plurisubharmonic functions

In this section, we recall some results about plurisubharmonic functions,
which will be used in future discussions.

Let $f$ be a holomorphic function near $o$, and let $\phi$ be a
plurisubharmonic function near $o$. Recall that (see [13, 11])

$C_{f,\phi}(U):=\inf\left\\{\int_{U}|\tilde{f}|^{2}:(\tilde{f}-f,o)\in\mathcal{I}(\phi)_{o}\
\&\ \tilde{f}\in\mathcal{O}(U)\right\\},$

where $U\subseteq\Delta^{n}$ is a domain with $o\in U$.

Let $\\{\phi_{m}\\}_{m\in\mathbb{N}^{+}}$ be a sequence of negative
plurisubharmonic functions on $\Delta^{n}$, which is convergent to a negative
Lebesgue measurable function $\phi$ on $\Delta^{n}$ in Lebesgue measure.

In [13], Guan-Zhou presented the following lower semicontinuity property of
plurisubharmonic functions with a multiplier.

######  Proposition 3.1 ([13]).

Let $f$ be a holomorphic function near $o$. Assume that for any small enough
neighborhood $U$ of $o$, the pairs $(f,\phi_{m})$ $(m\in\mathbb{N}^{+})$
satisfies

(3.1) $\inf_{m}C_{f,\varphi_{0}+\phi_{m}}(U)>0.$

Then $|f_{0}|^{2}e^{-2\varphi_{0}}e^{-2\phi}$ is not integrable near $o$.

The Noetherian property of multiplier ideal sheaves (see [6]) shows that

######  Remark 3.2.

Assume that

$(1)$ $\phi_{m+1}\geq\phi_{m}$ holds for any $m$;

$(2)$ $|f_{0}|^{2}e^{-2\varphi_{0}}e^{-2\phi_{m}}$ is not integrable near $o$
for any $m$.

Then inequality (3.1) holds.

Proposition 3.1 and Remark 3.2 imply the strong openness property
$\mathcal{I}(\varphi)=\mathcal{I}_{+}(\varphi)$ of plurisubharmonic functions
$\varphi$.

Recall that for any plurisubharmonic function $\varphi$ near $o$, the Lelong
number $\nu(\varphi,o):=\sup\\{c\geq 0:\varphi\leq c\log|z|$ near $o\\}$ (see
[6]). We recall the following famous result.

######  Lemma 3.3 ([18]).

For any plurisubharmonic function $\varphi$ near $o$, if the Lelong number
$\nu(\varphi,o)<1$, then $e^{-2\varphi}$ is integrable near $o$.

The following lemma gives a convergence property of jumping numbers.

######  Lemma 3.4 (see [1]).

Let $\varphi$ be any plurisubharmonic function near $o$, and let $f\not\equiv
0$ be any holomorphic function near $o$. Then
$\lim_{t\rightarrow+\infty}c_{o}^{f}(\max\\{\varphi,N\log|z|\\})=c_{o}^{f}(\varphi)$.

Let us recall two results about upper envelope of plurisubharmonic functions.

######  Lemma 3.5 (_Choquet’s lemma_ , see [6]).

Every family $(u_{\alpha})$ of uppersemicontinuous functions has a countable
subfamily $(v_{j})=(u_{\alpha(j)})$, such that its upper envelope
$v=\sup_{j}v_{j}$ satisfies $v\leq u\leq u^{*}=v^{*}$, where
$u=\sup_{\alpha}u_{\alpha}$, $u^{*}(z):=\lim_{\varepsilon\to
0}\sup_{\mathbb{B}^{n}(z,\varepsilon)}u$ and $v^{*}(z):=\lim_{\varepsilon\to
0}\sup_{\mathbb{B}^{n}(z,\varepsilon)}v$ are the regularizations of $u$ and
$v$.

######  Proposition 3.6 (see Proposition (4.24) in [6]).

If all $(u_{\alpha})$ are subharmonic, the upper regularization $u^{*}$ is
subharmonic and equals almost everywhere to $u$.

Let $\varphi$ be a plurisubharmonic function near $o$. Let us recall the
subadditive Theorem of the multiplier ideal sheaves.

######  Lemma 3.7 (see [7]).

For any positive numbers $t$ and $s$, we have
$\mathcal{I}((s+t)\varphi)_{o}\subset\mathcal{I}(t\varphi)_{o}\cdot\mathcal{I}(s\varphi)_{o}$.

###  3.2. Zhou weights

In this section, we recall some results about Zhou weights (see [1]).

Let $f_{0}=(f_{0,1},\cdots,f_{0,m})$ be a vector, where
$f_{0,1},\cdots,f_{0,m}$ are holomorphic functions near $o$. Let $\varphi_{0}$
be a plurisubharmonic function near $o$, such that
$|f_{0}|^{2}e^{-2\varphi_{0}}$ is integrable near $o$.

Let $D$ be a bounded domain in $\mathbb{C}^{n}$. Let us recall the definition
of the global Zhou weights.

######  Definition 3.8 ([1]).

We call a negative plurisubharmonic function
$\Phi^{f_{0},\varphi_{0},D}_{o,\max}$ ($\Phi^{D}_{o,\max}$ for short) on $D$ a
global Zhou weight related to $|f_{0}|^{2}e^{-2\varphi_{0}}$ if the following
statements hold:

$(1)$ $|f_{0}|^{2}e^{-2\varphi_{0}}|z|^{2N_{0}}e^{-2\Phi^{D}_{o,\max}}$ is
integrable near $o$ for large enough $N_{0}$;

$(2)$ $|f_{0}|^{2}e^{-2\varphi_{0}-2\Phi^{D}_{o,\max}}$ is not integrable near
$o$;

$(3)$ for any negative plurisubharmonic function $\tilde{\varphi}$ on $D$
satisfying that $\tilde{\varphi}\geq\Phi^{D}_{o,\max}$ on $D$ and
$|f_{0}|^{2}e^{-2\varphi_{0}-2\tilde{\varphi}}$ is not integrable near $o$,
$\tilde{\varphi}=\Phi^{D}_{o,\max}$ holds on $D$.

As $D$ is a bounded domain, any global Zhou weight is also a local Zhou
weight, and for any local Zhou weight $\Phi_{o,\max}$ related to
$|f_{0}|^{2}e^{-2\varphi_{0}}$ there exists a global Zhou weight
$\Phi^{D}_{o,\max}$ related to $|f_{0}|^{2}e^{-2\varphi_{0}}$ such that
$\Phi^{D}_{o,\max}=\Phi_{o,\max}+O(1)$ near $o$ (see [1]).

######  Lemma 3.9 ([1]).

Let $\Phi_{o,\max}$ be a local Zhou weight related to
$|f_{0}|^{2}e^{-2\varphi_{0}}$ near $o$, and denote that

$\begin{split}L(\Phi_{o,\max}):=&\big{\\{}\tilde{\varphi}(z)\in\mathrm{PSH}(D):\tilde{\varphi}<0\\\
&\ \&\ |f_{0}|^{2}e^{-2\varphi_{0}-2\tilde{\varphi}}\text{ is not integrable
near $o$}\ \&\ \tilde{\varphi}\geq\Phi_{o,\max}+O(1)\text{ near
$o$}\big{\\}}.\end{split}$

Then

$\Phi^{D}_{o,\max}(z):=\sup\big{\\{}\tilde{\varphi}(z):\tilde{\varphi}\in
L(\Phi_{o,\max})\big{\\}},\ \forall z\in D$

is a global Zhou weight related to $|f_{0}|^{2}e^{-2\varphi_{0}}$ on $D$
satisfying that

$\Phi^{D}_{o,\max}=\Phi_{o,\max}+O(1)$

near $o$.

The following lemma shows the existence of Zhou weight.

######  Lemma 3.10 ([1]).

Assume that there exists a negative plurisubharmonic function $\varphi$ on $D$
such that $|f_{0}|^{2}e^{-2\varphi_{0}-2\varphi}|z|^{2N_{0}}$ is integrable
near $o$ for large enough $N_{0}$ and
$(f_{0},o)\not\in\mathcal{I}(\varphi+\varphi_{0})_{o}.$

Then there exists a global Zhou weight $\Phi^{D}_{o,\max}$ on $D$ related to
$|f_{0}|^{2}e^{-2\varphi_{0}}$ such that $\Phi^{D}_{o,\max}\geq\varphi$ on
$D$.

We recall an approximation result of global Zhou weights.

######  Lemma 3.11 ([1]).

If $D$ is a bounded strictly hyperconvex domain, and $\Phi^{D}_{o,\max}$ is a
global Zhou weight related to some $|f_{0}|^{2}e^{-2\varphi_{0}}$ on $D$ near
$o$, then for any $w\in D$, we have

$\Phi^{D}_{o,\max}(w)=\sup\left\\{\frac{\log|f(w)|}{\sigma(\log|f|,\Phi^{D}_{o,\max})}:f\in\mathcal{O}(D),\
\sup_{D}|f|\leq 1,\ f(o)=0,\ f\not\equiv 0\right\\}.$

###  3.3. Some results about valuations

In this section, we give some lemmas about valuations on $\mathcal{O}_{o}$.

Let $\mathfrak{q}_{0}$ be a nonzero ideal of $\mathcal{O}_{o}$, and let $\nu$
be a valuation on $\mathcal{O}_{o}$. Assume that $\nu$ satisfies the
statements $(1)$ and $(2)$ in Theorem 1.10, i.e.,
$\\{(f,o)\in\mathcal{O}_{o}:v(f)>0\\}$ is the maximal ideal of
$\mathcal{O}_{o}$ and

$\mathfrak{q}_{0}\not\subset\mathcal{I}(\frac{1}{m}\log|\mathfrak{a}_{m}^{\nu}|)_{o}$

for any positive integer $m$, where

$\mathfrak{a}_{m}^{\nu}:=\\{(f,o)\in\mathcal{O}_{o}:\nu(f)\geq m\\}$

is an ideal of $\mathcal{O}_{o}$. Then we can use $\nu$ to construct a
plurisubharmonic function $\varphi_{\nu}$.

######  Lemma 3.12.

There exists a negative plurisubharmonic function $\varphi_{\nu}$ on $D$ such
that for any $f\in\mathcal{O}(D)$ satisfying $|f|\leq 1$ on $D$,

$\log|f|\leq\nu(f)\varphi_{\nu}$

on $D$, $|\mathfrak{q}_{0}|^{2}e^{-2\varphi_{\nu}}$ is not integrable near $o$
and $|\mathfrak{q}_{0}|^{2}e^{-2\varphi_{\nu}}|z|^{N}$ is integrable near $o$
for large enough $N$, where $D$ is the unit ball in $\mathbb{C}^{n}$.

###### Proof.

Denote that

$\begin{split}\varphi(w)=\sup\Bigg{\\{}\frac{\log|f(w)|}{a_{f}}:f\in\mathcal{O}(D),&\
\sup_{D}|f|\leq 1,\ f(o)=0,\ f\not\equiv 0\\\
&\&\,a_{f}\in\mathbb{Q}\cap(0,\nu(f)]\text{ is a
constant}\Bigg{\\}}\end{split}$

on $D$. Let $\varphi_{\nu}=\varphi^{*}$ be the upper semicontinuous
regularization of $\varphi$, which is a plurisubharmonic function on $D$.

By Lemma 3.5, there is a sequence
$\left\\{\frac{\log|f_{j}(w)|}{a_{f_{j}}}\right\\}_{j\in\mathbb{Z}_{>0}}$ such
that

$\varphi_{\nu}(w)=\left(\sup_{j}\frac{\log|f_{j}(w)|}{a_{f_{j}}}\right)^{*}$

on $D$. Denote that

$\tilde{\varphi}_{k}(w):=\max_{1\leq j\leq
k}\frac{\log|f_{j}(w)|}{a_{f_{j}}}=\log\max_{1\leq j\leq
k}|f_{j}(w)|^{\frac{1}{a_{f_{j}}}}$

for any positive integer $k$. There exists a positive number $m_{k}$ such that
$\frac{m_{k}}{a_{f_{j}}}\in\mathbb{Z}_{>0}$ for any $1\leq j\leq k$. Thus,

$\tilde{\varphi}_{k}(w)=\frac{1}{m_{k}}\log\max_{1\leq j\leq
k}|f_{j}(w)^{\frac{m_{k}}{a_{f_{j}}}}|$

and

$\nu(f_{j}^{\frac{m_{k}}{a_{f_{j}}}})={\frac{m_{k}}{a_{f_{j}}}}\nu(f_{j})\geq
m_{k}.$

The Statement $(2)$ shows that
$|\mathfrak{q}_{0}|^{2}e^{-2\tilde{\varphi}_{k}}$ is not integrable near $o$
for any $k$. Note that $\tilde{\varphi}_{k}$ is increasingly convergent to
$\varphi_{\nu}$ a.e. by Proposition 3.6. It follows from Proposition 3.1 and
Remark 3.2 that $|\mathfrak{q}_{0}|^{2}e^{-2\varphi_{\nu}}$ is not integrable
near $o$. By definition, we have

$\log|f|\leq\nu(f)\varphi_{\nu}$

on $D$, which shows that

$\max_{1\leq j\leq n}\frac{1}{\nu(z_{j})}\log|z_{j}|\leq\varphi_{\nu},$

where $(z_{1},\ldots,z_{m})$ is the coordinate system near $o$ and
$\nu(z_{j})>0$ for any $1\leq j\leq n$. Thus,
$|\mathfrak{q}_{0}|^{2}e^{-2\varphi_{\nu}}|z|^{N}$ is integrable near $o$ for
large enough $N$. ∎

We recall a closedness property of the ideals of $\mathcal{O}_{o}$.

######  Lemma 3.13 (see [10]).

Let $\mathfrak{a}$ be an ideal of $\mathcal{O}_{o}$, and let $\\{f_{j}\\}_{j}$
be a sequence of holomorphic functions on an open neighborhood $U$ of $o$.
Assume that all germs $(f_{j},o)\in\mathfrak{a}$ and the $f_{j}$ converge
uniformly on $U$ towards a holomorphic function $f$. Then
$(f,o)\in\mathfrak{a}$.

The following lemma will be used in the proof of Theorem 1.10.

######  Lemma 3.14.

Let $\nu$ be a valuation on $\mathcal{O}_{o}$, and let $\\{f_{j}\\}_{j}$ be a
sequence of holomorphic functions on an open neighborhood $U$ of $o$. Assume
that $f_{j}$ converge uniformly on $U$ towards a holomorphic function $f$.
Then $\nu(f)\geq\limsup_{j\rightarrow+\infty}\nu(f_{j})$.

Especially, assume that $\\{(h,o)\in\mathcal{O}_{o}:\nu(h)>0\\}$ is the
maximal ideal of $\mathcal{O}_{o}$, and let
$g(w)=\sum_{\alpha\in\mathbb{Z}_{\geq 0}^{n}}a_{\alpha}w^{\alpha}$ (Taylor
expansion) be a holomorphic function near $o$, then we have

$\nu(g)=\lim_{j\rightarrow+\infty}\nu(g_{j}),$

where $g_{j}(w):=\sum_{|\alpha|\leq j}a_{\alpha}w^{\alpha}$ and
$|\alpha|:=\sum_{1\leq k\leq n}\alpha_{k}$ for any $\alpha\in\mathbb{Z}_{\geq
0}^{n}.$

###### Proof.

For any

$x<\limsup_{j\rightarrow+\infty}\nu(f_{j}),$

there exists a subsequence of $\\{f_{j}\\}_{j}$ denoted also by
$\\{f_{j}\\}_{j}$ such that $\nu(f_{j})>x$ for any $j$. Note that
$\\{(h,o)\in\mathcal{O}_{o}:\nu(h)>x\\}$ is an ideal of $\mathcal{O}_{o}$ and
$f_{j}$ converge uniformly towards $f$, then it follows from Lemma 3.14 that
$\nu(f)>x$. Thus, we have

$\nu(f)\geq\limsup_{j\rightarrow+\infty}\nu(f_{j}).$

Note that $g_{j}$ converge uniformly towards $g$ on a neighborhood of $o$,
thus we have

(3.2) $\nu(g)\geq\limsup_{j\rightarrow+\infty}\nu(g_{j}).$

As $\\{(h,o)\in\mathcal{O}_{o}:\nu(h)>0\\}$ is the maximal ideal of
$\mathcal{O}_{o}$, we have

$c:=\min_{1\leq l\leq n}\\{\nu(w_{l})\\}>0,$

which implies that

$\begin{split}\nu(g-g_{j})&\geq\limsup_{k\rightarrow+\infty}\nu(g_{k}-g_{j})\\\
&\geq\limsup_{k\rightarrow+\infty}\min_{j<|\alpha|\leq
k}\\{\nu(a_{\alpha}w^{\alpha})\\}\\\ &\geq jc.\end{split}$

Take $j_{0}$ such that $j_{0}c>\nu(g)$. For any $j>j_{0}$, we have

$\nu(g_{j})\geq\min\\{\nu(g),\nu(g-g_{j})\\}=\nu(g).$

Combining equality (3.2), we have

$\nu(g)=\lim_{j\rightarrow+\infty}\nu(g_{j}).$

Thus, Lemma 3.14 holds. ∎

Let $\mathfrak{a}_{\bullet}$ be a graded sequence of ideals, and let
$\mathfrak{b}_{\bullet}$ be the corresponding subadditive sequence of
asymptotic multiplier ideals of $\mathfrak{a}_{\bullet}$. The following lemma
gives a relation between $\nu(\mathfrak{a}_{m})$ and $\nu(\mathfrak{b}_{m})$.

######  Lemma 3.15.

For any $m$ such that $\mathfrak{a}_{m}$ is nonzero, we have

$\nu(\mathfrak{a}_{\bullet})-\frac{\mathscr{A}(\nu)}{m}<\frac{\nu(\mathfrak{b}_{m})}{m}\leq\frac{\nu(\mathfrak{a}_{m})}{m}$

for any Zhou valuation $\nu$. In particular,
$\nu(\mathfrak{a}_{\bullet})=\nu(\mathfrak{b}_{\bullet})$.

###### Proof.

Let $\nu$ be a Zhou valuation related to $|\mathfrak{q}|^{2}$, whose
corresponding local Zhou weight is denoted by $\Phi_{o,\max}$. It follows from
$\mathfrak{a}_{m}\subset\mathfrak{b}_{m}$ that

$\nu(\mathfrak{a}_{m})\geq\nu(\mathfrak{b}_{m})$

for any $m$. For any $m$ such that $\mathfrak{a}_{m}$ is nonzero, there exists
$m_{1}$ large enough such that
$\mathfrak{b}_{m}=\mathcal{I}(\frac{m}{m_{1}}\log|\mathfrak{a}_{m_{1}}|)_{o}$,
then it follows from Lemma 2.2 and $\mathscr{A}(\nu)=1-\nu(\mathfrak{q})$ that

$\begin{split}\frac{\nu(\mathfrak{b}_{m})}{m}&=\frac{\nu(\mathcal{I}(\frac{m}{m_{1}}\log|\mathfrak{a}_{m_{1}}|)_{o})}{m}\\\
&\geq\frac{c_{o}^{\mathcal{I}(\frac{m}{m_{1}}\log|\mathfrak{a}_{m_{1}}|)_{o}}(\Phi_{o,\max})-\mathscr{A}(\nu)}{m}\\\
&\geq\frac{c_{o}^{\mathcal{I}(\frac{m}{m_{1}}\log|\mathfrak{a}_{m_{1}}|)_{o}}(\frac{\log|\mathfrak{a}_{m_{1}}|}{\nu(\mathfrak{a}_{m_{1}})})-\mathscr{A}(\nu)}{m}\\\
&>\frac{\frac{m\nu(\mathfrak{a}_{m_{1}})}{m_{1}}-\mathscr{A}(\nu)}{m}\\\
&\geq\nu(\mathfrak{a}_{\bullet})-\frac{\mathscr{A}(\nu)}{m},\end{split}$

where the above “$>$” follows from the strong openness property of the
multiplier ideal sheaves(see [12], see also Proposition 3.1 and Remark 3.2).
Thus, Lemma 3.15 holds. ∎

Recall that

(3.3)
$\mathscr{A}(\nu):=\sup\\{\nu(\mathfrak{a}_{\bullet})\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})-\nu(\mathfrak{q}):\text{for
any $\mathfrak{q}$ and $\mathfrak{a}_{\bullet}$ such that
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})<+\infty$}\\},$

where $\mathfrak{a}_{\bullet}$ is any graded sequence of ideals. The following
remark shows that equality (3.3) also holds when replacing
$\mathfrak{a}_{\bullet}$ by subadditive sequences of ideals
$\mathfrak{b}_{\bullet}$.

######  Remark 3.16.

Let $\nu$ be a Zhou valuation related to $|\mathfrak{q}_{0}|^{2}$, where
$\mathfrak{q}_{0}$ is a nonzero ideal. Then

$\mathscr{A}(\nu)=\sup\\{\nu(\mathfrak{b}_{\bullet})\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})-\nu(\mathfrak{q}):\text{for
any $\mathfrak{q}$ and $\mathfrak{b}_{\bullet}$ such that
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})<+\infty$}\\}.$

###### Proof.

For any nonzero ideal $\mathfrak{q}$ and any subadditive sequence
$\mathfrak{b}_{\bullet}$ satisfying
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})<+\infty$, if
$\nu(\mathfrak{b}_{\bullet})>0$, it is clear that

$\begin{split}\nu(\mathfrak{b}_{\bullet})\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})=&\lim_{t\rightarrow+\infty}\nu(\mathfrak{b}_{t})\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{t})\\\
\leq&\lim_{t\rightarrow+\infty}[\nu(\mathfrak{b}_{t})]\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}^{\nu}_{[\nu(\mathfrak{b}_{t})]})\\\
=&\nu(\mathfrak{a}^{\nu})\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}^{\nu}_{\bullet}),\end{split}$

where $[s]:=\sup\\{x\in\mathbb{Z}:x\leq s\\}$ for any $s\in\mathbb{R}$. Thus,
by the definition of $\mathscr{A}(\nu)$ we have

$\nu(\mathfrak{b}_{\bullet})\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})-\nu(\mathfrak{q})\leq\mathscr{A}(\nu)$

for any $\mathfrak{q}$ and $\mathfrak{b}_{\bullet}$ such that
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})<+\infty$. Note that for
any graded sequence of ideals $\mathfrak{a}_{\bullet}$, the corresponding
subadditive sequence $\mathfrak{b}_{\bullet}$ of asymptotic multiplier ideals
of $\mathfrak{a}_{\bullet}$ satisfies that
$\nu(\mathfrak{a}_{\bullet})=\nu(\mathfrak{b}_{\bullet})$ and
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})=\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})$.
Thus, using the definition of $\mathscr{A}(\nu)$, we have

$\mathscr{A}(\nu)=\sup\\{\nu(\mathfrak{b}_{\bullet})\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})-\nu(\mathfrak{q}):\text{for
any $\mathfrak{q}$ and $\mathfrak{b}_{\bullet}$ such that
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})<+\infty$}\\}.$

∎

Let $\varphi$ be a plurisubharmonic function near $o$. Denote that

$\mathfrak{b}_{t}^{\varphi}:=\mathcal{I}(t\varphi)_{o}$

for any $t>0$. Lemma 3.7 (the subadditive Theorem, see [7]) shows that
$\mathfrak{b}^{\varphi}_{\bullet}=(\mathfrak{b}^{\varphi}_{t})_{t>0}$ is a
subadditive system of ideals satisfying that
$\mathfrak{b}^{\varphi}_{s+t}\subset\mathfrak{b}^{\varphi}_{s}\cdot\mathfrak{b}^{\varphi}_{t}$
for any $s,t\in\mathbb{R}_{>0}$. Then we have

######  Lemma 3.17.

Let $\nu$ be a Zhou valuation, whose corresponding local Zhou weight is
denoted by $\Phi_{o,\max}$. Then

$\nu(\mathfrak{b}_{\bullet}^{\varphi})=\sigma(\varphi,\Phi_{o,\max})$

and

$\frac{\nu(\mathfrak{b}^{\varphi}_{t})}{t}>\nu(\mathfrak{b}^{\varphi}_{\bullet})-\frac{\mathscr{A}(\nu)}{t}.$

###### Proof.

It follows from Lemma 2.1 that
$\varphi\leq\frac{1}{t}\log|\mathcal{I}(t\varphi)_{o}|+O(1)$ near $o$ for any
$t>0$, which deduces that

$\begin{split}\sigma(\varphi,\Phi_{o,\max})&\geq\limsup_{t\rightarrow+\infty}\sigma(\frac{1}{t}\log|\mathcal{I}(t\varphi)_{o}|,\Phi_{o,\max})\\\
&=\limsup_{t\rightarrow+\infty}\frac{\nu(\mathfrak{b}^{\varphi}_{t})}{t}\\\
&=\nu(\mathfrak{b}_{\bullet}^{\varphi}).\end{split}$

Note that $\varphi\leq\sigma(\varphi,\Phi_{o,\max})\Phi_{o,\max}+O(1)$ near
$o$, then it follows from Lemma 2.2 that

$\begin{split}\nu(\mathfrak{b}_{\bullet}^{\varphi})&=\lim_{t\rightarrow+\infty}\frac{\nu(\mathfrak{b}^{\varphi}_{t})}{t}\\\
&=\lim_{t\rightarrow+\infty}\frac{c_{o}^{\mathcal{I}(t\varphi)_{o}}(\Phi_{o,\max})}{t}\\\
&\geq\lim_{t\rightarrow+\infty}\frac{c_{o}^{\mathcal{I}(t\varphi)_{o}}(\frac{\varphi}{\sigma(\varphi,\Phi_{o,\max})})}{t}\\\
&\geq\sigma(\varphi,\Phi_{o,\max}).\end{split}$

Thus, we have
$\nu(\mathfrak{b}_{\bullet}^{\varphi})=\sigma(\varphi,\Phi_{o,\max}).$
Following from Proposition 3.1 and Remark 3.2,

$c_{o}^{\mathfrak{b}^{\varphi}_{t}}(\varphi)>t$

for any $t>0$. Using Lemma 2.2, we have

$\begin{split}\frac{\nu(\mathfrak{b}^{\varphi}_{t})}{t}&\geq\frac{c_{o}^{\mathfrak{b}^{\varphi}_{t}}(\Phi_{o,\max})-\mathscr{A}(\nu)}{t}\\\
&\geq\frac{c_{o}^{\mathfrak{b}^{\varphi}_{t}}(\frac{\varphi}{\sigma(\varphi,\Phi_{o,\max})})-\mathscr{A}(\nu)}{t}\\\
&>\sigma(\varphi,\Phi_{o,\max})-\frac{\mathscr{A}(\nu)}{t}\\\
&=\nu(\mathfrak{b}^{\varphi}_{\bullet})-\frac{\mathscr{A}(\nu)}{t}.\end{split}$

Thus, Lemma 3.17 holds. ∎

Remark 3.16 shows that

$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})\leq\inf_{\nu}\frac{\mathscr{A}(\nu)+\nu(\mathfrak{q})}{\nu(\mathfrak{b}_{\bullet})},$

where the infimum is over all Zhou valuations $\nu$. The following Proposition
tells us that it is actually an equality.

######  Proposition 3.18.

Let $\mathfrak{b}_{\bullet}$ be a subadditive system of ideals, and let
$\mathfrak{q}$ be a nonzero ideal. Then

(3.4)
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})=\inf_{\nu}\frac{\mathscr{A}(\nu)+\nu(\mathfrak{q})}{\nu(\mathfrak{b}_{\bullet})},$

where the infimum is over all Zhou valuations $\nu$.

###### Proof.

Firstly, we prove that
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})=+\infty$ holds if and
only if $\nu(\mathfrak{b}_{\bullet})=0$ for any Zhou valuation $\nu$. If
$\nu(\mathfrak{b}_{\bullet})>0$ for some Zhou valuation $\nu$ (its
corresponding local Zhou weight is denoted by $\Phi_{o,\max}$), there exists
$c>0$ such that $\nu(\mathfrak{b}_{t})>tc$ for large enough $t$, i.e.,
$\log|\mathfrak{b}_{t}|<tc\Phi_{o,\max}+O(1)$ near $o$, which implies that

$\begin{split}\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})&=\lim_{t\rightarrow+\infty}t\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{t})\\\
&\leq\lim_{t\rightarrow+\infty}tc_{o}^{\mathfrak{q}}(tc\Phi_{o,\max})\\\
&=\frac{c_{o}^{\mathfrak{q}}(\Phi_{o,\max})}{c}\\\ &<+\infty.\end{split}$

Thus, $\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})=+\infty$ implies
$\nu(\mathfrak{b}_{\bullet})=0$ for any Zhou valuation $\nu$. Note that the
Lelong number $\nu_{L}:=\nu(\log|\cdot|,o)$ is also a Zhou valuation. If
$\nu_{L}(\mathfrak{b}_{\bullet})=0$, i.e.,

$\lim_{t\rightarrow+\infty}\nu(\frac{1}{t}\log|\mathfrak{b}_{t}|,o)=0,$

it follows from Lemma 3.3 that

$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})=\lim_{t\rightarrow+\infty}c_{o}^{\mathfrak{q}}(\frac{1}{t}\log|\mathfrak{b}_{t}|)=+\infty.$

In the following, we prove equality (3.4) with assumption
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})<+\infty$. Remark 3.16
shows that

(3.5)
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})\leq\frac{\mathscr{A}(\nu)+\nu(\mathfrak{q})}{\nu(\mathfrak{b}_{\bullet})}$

for any Zhou valuation $\nu$.

Let $x>\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})$ be a constant.
Following from the definition of
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})$ and Lemma 3.4, there
exist two constants $t$ and $N$ large enough satisfying

$tc_{o}^{\mathfrak{q}}(\max\\{\log|\mathfrak{b}_{t}|,N\log|z|\\})<x.$

For simplicity, denote $\max\\{\log|\mathfrak{b}_{t}|,N\log|z|\\}$ by
$\varphi_{t,N}$. The strong openness property of multiplier ideal sheaves (see
Proposition 3.1 and Remark 3.2) shows that

$|\mathfrak{q}|^{2}e^{-2c_{o}^{\mathfrak{q}}(\varphi_{t,N})\varphi_{t,N}}$

is not integrable near $o.$ By remark 3.10, there exists a local Zhou weight
$\Phi_{o,\max}$ (whose corresponding Zhou valuation is denoted by $\nu_{t,N}$)
related to $|\mathfrak{q}|^{2}$ satisfying that

$\Phi_{o,\max}\geq c_{o}^{\mathfrak{q}}(\varphi_{t,N})\varphi_{t,N}$

near $o$ and $|\mathfrak{q}|^{2}e^{-2\Phi_{o,\max}}$ is not integrable near
$o.$ It is clear that $\sigma(\varphi_{t,N},\Phi_{o,\max}):=\sup\\{c\geq
0:\varphi_{t,N}\leq c\Phi_{o,\max}+O(1)$ near
$o\\}=\frac{1}{c_{o}^{\mathfrak{q}}(\varphi_{t,N})}$. Note that
$\mathscr{A}(\nu_{t,N})+\nu_{t,N}(\mathfrak{q})=1$. Thus, we have

$x>tc_{o}^{\mathfrak{q}}(\varphi_{t,N})=\frac{t}{\sigma(\varphi_{t,N},\Phi_{o,\max})}>\frac{\mathscr{A}(\nu_{t,N})+\nu_{t,N}(\mathfrak{q})}{\frac{\sigma(\log|\mathfrak{b}_{t}|,\Phi_{o,\max})}{t}}\geq\frac{\mathscr{A}(\nu_{t,N})+\nu_{t,N}(\mathfrak{q})}{\nu_{t,N}(\mathfrak{b}_{\bullet})}.$

Combining with inequality (3.5), we have

$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})=\inf_{\nu}\frac{\mathscr{A}(\nu)+\nu(\mathfrak{q})}{\nu(\mathfrak{b}_{\bullet})},$

where the infimum is over all Zhou valuations $\nu$. ∎

Using Proposition 3.18, we have the following remark.

######  Remark 3.19.

Note that for any local Zhou weight $\Phi_{o,\max}$, there exists $N\gg 0$
such that $N\log|z|+O(1)\leq\Phi_{o,\max}\leq\frac{1}{N}\log|z|+O(1)$ near
$o$. Thus Proposition 3.18 implies the following three statements are
equivalent:

$(1)$ $\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})=+\infty$;

$(2)$ $\nu(\mathfrak{b}_{\bullet})=0$ for any Zhou valuation $\nu$;

$(3)$ there exists a Zhou valuation $\nu$ such that
$\nu(\mathfrak{b}_{\bullet})=0$.

Let $\mathfrak{a}_{\bullet}$ be any graded sequence of ideals, and let
$\mathfrak{q}$ be any nonzero ideal. let $\mathfrak{b}_{\bullet}$ be the
corresponding subadditive sequence of asymptotic multiplier ideals of
$\mathfrak{a}_{\bullet}$, then we have
$\nu(\mathfrak{a}_{\bullet})=\nu(\mathfrak{b}_{\bullet})$ and
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})=\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}_{\bullet})$.
Thus, Proposition 3.18 implies the following

######  Proposition 3.20.

$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})=\inf_{\nu}\frac{\mathscr{A}(\nu)+\nu(\mathfrak{q})}{\nu(\mathfrak{a}_{\bullet})},$

where the infimum is over all Zhou valuations $\nu$.

##  4\. Proofs of Theorem 1.8, Theorem 1.10 and Proposition 1.11

In this section, we prove Theorem 1.8, Theorem 1.10 and Proposition 1.11.

Finally, we prove Theorem 1.8.

###### Proof of Theorem 1.8.

Following from Lemma 3.10, there is a local Zhou weight $\Phi_{o,\max}$
related to $|\mathfrak{q}|^{2}$ satisfying that

$\Phi_{o,\max}\geq c_{o}^{\mathfrak{q}}(\varphi)\varphi$

near $o$, and denote its corresponding Zhou valuation by $\nu$. Then we have
$\mathscr{A}(\nu)+\nu(\mathfrak{q})=1$ and
$\sigma(\varphi,\Phi_{o,\max})=\frac{1}{c_{o}^{\mathfrak{q}}(\varphi)}$. Note
that
$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}^{\varphi}_{\bullet})=c_{o}^{\mathfrak{q}}(\varphi)$
(see [15]). Using Lemma 3.17, we get

$\mathrm{lct}^{\mathfrak{q}}(\mathfrak{b}^{\varphi}_{\bullet})\cdot\nu(\mathfrak{b}^{\varphi}_{\bullet})=1=\mathscr{A}(\nu)+\nu(\mathfrak{q}).$

Theorem 1.8 holds. ∎

Now, we prove Theorem 1.10.

###### Proof of Theorem 1.10.

Firstly, we prove the necessity. Assume that there exists a local Zhou weight
$\Phi_{o,\max}$ related $|\mathfrak{q}_{0}|^{2}$ near $o$ such that

$\nu(f,\Phi_{o,\max}):=\sigma(\log|f|,\Phi_{o,\max})=\nu(f)$

for any $(f,o)\in\mathcal{O}_{o}$. Note that there exist two positive
constants $N_{1}$ and $N_{2}$ such that

$N_{1}\log|z|+O(1)\leq\Phi_{o,\max}\leq N_{2}\log|z|+O(1)$

near $o$, then we obtain that $\nu(f,\Phi_{o,\max})=\sup\\{c\geq 0:\log|f|\leq
c\Phi_{o,\max}+O(1)$ near $o\\}=0$ if and only if $f(o)\not=0$, which shows
that the statement $(1)$ holds. By the definition of $\mathfrak{a}_{m}^{\nu}$,
we have

$\log|\mathfrak{a}_{m}^{\nu}|\leq m\Phi_{o,\max}+O(1)$

near $o$. Since $|\mathfrak{q}_{0}|^{2}e^{-2\Phi_{o,\max}}$ is not integrable
near $o$, $|\mathfrak{q}_{0}|^{2}e^{-\frac{2}{m}\log|\mathfrak{a}_{m}^{\nu}|}$
is not integrable near $o$ for any $m$.

If there exists a valuation $\tilde{\nu}$ on $\mathcal{O}_{o}$ satisfying the
statements $(1)$ and $(2)$, and

(4.1) $\tilde{\nu}(f)\geq\nu(f)=\nu(f,\Phi_{o,\max})$

for any $(f,o)\in\mathcal{O}_{o}$. It follows from Lemma 3.12 that there
exists a negative plurisubharmonic function $\varphi_{\tilde{\nu}}$ on $D$
such that for any $f\in\mathcal{O}(D)$ satisfying $|f|\leq 1$ on $D$,

(4.2) $\log|f|\leq\tilde{\nu}(f)\varphi_{\nu}$

on $D$, $|\mathfrak{q}_{0}|^{2}e^{-2\varphi_{\tilde{\nu}}}$ is not integrable
near $o$ and $|\mathfrak{q}_{0}|^{2}e^{-2\varphi_{\tilde{\nu}}}|z|^{N}$ is
integrable near $o$ for large enough $N$, where $D$ is the unit ball in
$\mathbb{C}^{n}$. By the construction of $\varphi_{\tilde{\nu}}$ in the proof
of Lemma 3.12, we have

(4.3)
$\varphi_{\tilde{\nu}}=\left(\sup\left\\{\frac{\log|f(w)|}{\tilde{\nu}(f)}:f\in\mathcal{O}(D),\
\sup_{D}|f|\leq 1,\ f(o)=0,\ f\not\equiv 0\right\\}\right)^{*}.$

Lemma 3.9 shows that there is a global Zhou weight $\Phi_{o,\max}^{D}$ on $D$
related to $|\mathfrak{q}_{0}|^{2}$ satisfying

$\Phi_{o,\max}^{D}=\Phi_{o,\max}+O(1)$

near $o$. It follows from inequality (4.1), equality (4.3) and Lemma 3.11 that

$\begin{split}\varphi_{\tilde{\nu}}&=\left(\sup\left\\{\frac{\log|f(w)|}{\tilde{\nu}(f)}:f\in\mathcal{O}(D),\
\sup_{D}|f|\leq 1,\ f(o)=0,\ f\not\equiv 0\right\\}\right)^{*}\\\
\geq&\left(\sup\left\\{\frac{\log|f(w)|}{\nu(f)}:f\in\mathcal{O}(D),\
\sup_{D}|f|\leq 1,\ f(o)=0,\ f\not\equiv 0\right\\}\right)^{*}\\\
=&\Phi_{o,\max}^{D}.\end{split}$

By the definition of the global Zhou weight $\Phi_{o,\max}^{D}$, we have
$\varphi_{\tilde{\nu}}=\Phi_{o,\max}^{D}$ on $D$. Using inequality (4.2), we
have

$\log|f|\leq\tilde{\nu}(f)\varphi_{\tilde{\nu}}=\tilde{\nu}(f)\Phi_{o,\max}^{D},$

for any $f\in\mathcal{O}(D)$ satisfying $|f|\leq 1$ on $D$, which shows that
$\nu(f,\Phi_{o,\max})\geq\tilde{\nu}(f)$ for any polynomial $f$ on
$\mathbb{C}^{n}$. Using Lemma 3.14, we have

$\nu(f,\Phi_{o,\max})\geq\tilde{\nu}(f)$

for any $(f,o)\in\mathcal{O}_{o}$, which shows that

$\nu(\cdot,\Phi_{o,\max})=\tilde{\nu}$

by the assumption $\tilde{\nu}\geq\nu(\cdot,\Phi_{o,\max})$. Thus, the
statement $(3)$ holds.

Now, we prove the sufficiency. Assume that the three statements in Theorem
1.10 hold. Lemma 3.12 shows that there exists a plurisubharmonic function
$\varphi_{\nu}$ on $D$ such that for any $f\in\mathcal{O}(D)$ satisfying
$|f|\leq 1$ on $D$,

(4.4) $\log|f|\leq\nu(f)\varphi_{\nu}$

on $D$, $|\mathfrak{q}_{0}|^{2}e^{-2\varphi_{\nu}}$ is not integrable near $o$
and $|\mathfrak{q}_{0}|^{2}e^{-2\varphi_{\nu}}|z|^{N}$ is integrable near $o$
for large enough $N$, where $D$ is the unit ball in $\mathbb{C}^{n}$.

It follows from Lemma 3.10 that there exists a global Zhou weight
$\Phi_{o,\max}^{D}$ related to $|\mathfrak{q}_{0}|^{2}$ such that

(4.5) $\Phi_{o,\max}^{D}\geq\varphi_{\nu}$

on $D$. Following from inequalities (4.4) and (4.5), we obtain that the Zhou
valuation

$\nu(f,\Phi_{o,\max}^{D})\geq\nu(f)$

for any polynomial $f$ on $\mathbb{C}^{n}$. Using Lemma 3.14, we obtain that

$\nu(f,\Phi_{o,\max}^{D})\geq\nu(f)$

for any $(f,o)\in\mathcal{O}_{o}$. Note that the Zhou valuation
$\nu(\cdot,\Phi_{o,\max}^{D})$ satisfies the statements $(1)$ and $(2)$, then
it follows from the statement $(3)$ that

$\nu(\cdot,\Phi_{o,\max}^{D})=\nu,$

i.e., $\nu$ is a Zhou valuation related to $|\mathfrak{q}_{0}|^{2}$.

Thus, Theorem 1.10 holds. ∎

Finally, we prove Proposition 1.11.

###### Proof of Proposition 1.11.

Lemma 3.12 shows that there exists a negative plurisubharmonic function
$\varphi_{\nu}$ on $D$ such that for any $f\in\mathcal{O}(D)$ satisfying
$|f|\leq 1$ on $D$,

(4.6) $\log|f|\leq\nu(f)\varphi_{\nu}$

on $D$, $|\mathfrak{q}_{0}|^{2}e^{-2\varphi_{\nu}}$ is not integrable near $o$
and $|\mathfrak{q}_{0}|^{2}e^{-2\varphi_{\nu}}|z|^{N}$ is integrable near $o$
for large enough $N$, where $D$ is the unit ball in $\mathbb{C}^{n}$. It
follows from Lemma 3.10 there exists a global Zhou weight $\Phi_{o,\max}^{D}$
on $D$ related to $|\mathfrak{q}_{0}|^{2}$ such that

$\Phi_{o,\max}^{D}\geq\varphi_{\nu}$

on $D$. For any $f\in\mathcal{O}(D)$ satisfying $|f|\leq 1$ on $D$,

$\log|f|\leq\nu(f)\varphi_{\nu}\leq\nu(f)\Phi_{o,\max}^{D}.$

Combining Lemma 3.14, we obtain that for any $(f,o)\in\mathcal{O}_{o}$, the
Zhou valuation

$\nu(f,\Phi_{o,\max}^{D}):=\sigma(\log|f|,\Phi_{o,\max})\geq\nu(f).$

Thus, Proposition 1.11 holds. ∎

Acknowledgements. The authors would like to thank Professor Mattias Jonsson
for helpful discussions. The authors would also like to thank Xun Sun, Dr.
Shijie Bao and Dr. Zhitong Mi for checking the manuscript. The first author
was supported by National Key R&D Program of China 2021YFA1003100 and
NSFC-11825101. The second author was supported by China Postdoctoral Science
Foundation BX20230402.

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