# On Delay Performance in Mega Satellite Networks with Inter-Satellite Links

Kosta Dakic†, Chiu Chun Chan‡, Bassel Al Homssi⋆,
Kandeepan Sithamparanathan†, and Akram Al-Hourani†
† School of Engineering, RMIT University, Melbourne, Australia
‡ School of Engineering, Australian National University, Canberra, Australia
⋆ School of Engineering & Information Technology, UNSW Canberra, Canberra,
Australia
Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Utilizing Low Earth Orbit (LEO) satellite networks equipped with Inter-
Satellite Links (ISL) is envisioned to provide lower delay compared to
traditional optical networks. However, LEO satellites have constrained energy
resources as they rely on solar energy in their operations. Thus requiring
special consideration when designing network topologies that do not only have
low-delay link paths but also low-power consumption. In this paper, we study
different satellite constellation types and network typologies and propose a
novel power-efficient topology. As such, we compare three common satellite
architectures, namely; (i) the theoretical random constellation, the widely
deployed (ii) Walker-Delta, and (iii) Walker-Star constellations. The
comparison is performed based on both the power efficiency and end-to-end
delay. The results show that the proposed algorithm outperforms long-haul ISL
paths in terms of energy efficiency with only a slight hit to delay
performance relative to the conventional ISL topology.

###### Index Terms:

Low Earth orbit constellations, inter-satellite links, mega satellite
constellations, dense satellite constellations, delay.

Figure 1: Different constellations evaluated in the study: random
constellation (left), the Walker-Star with 16 orbital planes (middle), and the
Walker-Delta with 16 orbital planes (right). The orange lines represent the
satellite orbits, and satellites are illustrated as blue dots. The
constellation size is 200 in this illustration. Figure 2: An illustration of
the satellite distance and the great circle distance between two ground
stations.

##  I Introduction

The deployment of Low Earth Orbit (LEO) constellation networks is at an
accelerating pace aiming to provide global connectivity. Many satellite
projects, such as OneWeb, SpaceX’s Starlink, and Amazon’s Kupier [1] are
currently been deployed with thousands of satellites into LEO orbit to provide
seamless Internet coverage around the globe. These constellations will
complement the existing terrestrial communications including both 5G and
future 6G networks.

One of the key enablers of such constellations envisioned inter-satellite
connectivity. A connection between two satellites is referred to as Inter-
Satellite Link (ISL) and is being widely anticipated (and demonstrated) in the
upcoming LEO constellations. ISLs relay data directly between satellites,
unlike current methods which depend on the large network of ground stations
[2]. ISLs could liberate constellations from the burden of establishing
costly, and sometimes infeasible, ground station networks. An additional
advantage is that the data carried by ISLs travel in free space and thus at
the exact speed of light as opposed to conventional optical fiber networks.
The average propagation speed in a typical single-mode optical fiber cables
network is around $65-70\%$ the speed of light [3]. Hence, ISLs can
potentially enable new low-delay applications such as remote control industry
operations, cloud-controlled autonomous vehicles and farming, and telesurgery,
in addition to enabling faster financial transactions. However, despite these
advantages, satellite networks with ISLs face their own set of challenges,
such as power constraints due to reliance on solar energy and the need to
maintain reliable inter-satellite connections in a dynamic orbital
environment. Consequently, the development of power-efficient and low-delay
satellite constellation topologies that effectively leverage ISLs remains an
active area of research and innovation.

Nevertheless, using satellites rather than optical fiber submarine cable
possesses the potential to decrease the propagation delay by a few tens of
milliseconds depending on the distance. Apart from delay reduction, the LEO
satellite constellation can provide an access network for remote and rural
communities, and to locations with extreme terrain such as mountains [2]. This
is also particularly important for industries that require real-time
monitoring and control, such as manufacturing and logistics. Recent studies
have also shown that LEO satellites with ISLs offer better resilience to
cyber-attacks, making them a more secure option for data transfer in Industry
4.0 applications [4, 5]. Furthermore, due to their low altitude, LEO
satellites offer lower latency and higher bandwidth capacity compared to
traditional geostationary satellites, enabling faster and more efficient
communication services for users in remote areas [6]. Recent studies have
demonstrated the potential benefits of LEO satellite networks for improving
connectivity in developing countries and bridging the digital divide [7].

Simulations presented in [8] and in [9] illustrate the comparative delay
advantages of ISL as a data relay technology versus optical cable.
Additionally, authors in [10], delve deeper into the idea of using optical
satellite links rather than terrestrial fiber by developing a crossover
function to optimize delay. The crossover function is a mathematical formula
that determines the optimal point at which to switch from using a terrestrial
fiber link to an optical satellite link. Authors in [11] revealed the
limitations of traditional network design approaches in the context of ISL-
enable satellite communications and suggested the use of repetitive
3-satellite link patterns to address the temporal dynamics, achieving a higher
efficiency than previous state-of-the-art methods. Nevertheless, more
investigation is needed to better evaluate the benefits of optical satellite
links for data relaying as well as to develop satellite-aware ISL topologies
rather than just applying the shortest path ISLs. For example, a power-
efficient ISL topology would take into consideration the power limitations of
satellites due to their reliance on intermittent solar energy.

In this paper, we further analyze the prospect of using LEO satellite ISLs to
relay data. The analysis is made through the simulation of different LEO
satellite constellations (shown in Fig. 1) where network topologies are
proposed; (i) Nearest hop topology and (ii) Cutoff distance topology. We
concentrate on the performance in terms of delay between the transmitting and
receiving device because ISL is expected to facilitate lower delays, however,
the performance of ISL-enabled networks needs to be carefully studied and
assessed. An illustration of data communications with ISLs is shown in 2. The
contributions of this work are summarized as follows:

  * •

We compare the performance of three common satellite constellations, namely
the theoretical random constellation, the widely deployed Walker-Delta, and
the Walker-Star constellations in terms of end-to-end delay.

  * •

We propose and evaluate two network topologies for LEO satellite
constellations with ISLs: Nearest hop topology and Cutoff distance topology,
focusing on their impact on delay performance comparing a theoretical great-
circle optical fiber connection.

  * •

We show that our proposed topologies achieve competitive delay performance
relative to the conventional ISL topology, highlighting their potential for
use in future LEO satellite networks.

##  II System Model

###  II-A Geomtric model

In this section, we introduce the three constellation models used for
benchmarking the performance.

####  II-A1 Random

The random distribution of satellites with random circular orbits. This is
distribution is used in the simulations, as outlined in [12], with the
assumption that satellite collisions are disregarded. The left-hand side of
Fig. 1 provides an illustration of the random constellation along with the
common Walker constellations.

####  II-A2 Walker-Star Constellation

Satellite providers, such as OneWeb and Iridium use the Walker-Star
constellation. The orbits in such constellation follow a near-polar
configuration which has an inclination angle close to 90∘, this ensures global
coverage, including the poles. However, this inherently results in an
increased density of satellites with higher latitudes. The right ascension of
the ascending node (RAAN) of the orbital planes in a Walker-Star configuration
is spread across $0$ to $\pi$, unlike the Walker-Delta constellation which
uses $0$ to $2\pi$. The middle of Fig. 1 depicts a typical Walker-Star
constellation with 200 satellites along with their orbital planes.

####  II-A3 Walker-Delta Constellation

The Walker-Delta constellation, employed in satellite networks such as Kupier
and Starlink [13, 14], reduces inter-satellite distance variations. These
networks are currently considering inter-satellite links (ISLs) to support
end-to-end communication without the need for a large network of terrestrial
gateways. In the Walker-Delta configuration, the orbital planes are equally
spaced and rotate around the Earth’s axis of rotation, with a RAAN of
$\Omega\in{0,\frac{2\pi}{P},2\frac{2\pi}{P},\dots,(P-1)\frac{2\pi}{P}}$. The
right-hand side of Fig. 1 displays the constellation and orbital planes for
the Walker-Delta constellation.

###  II-B Footprint model

In order to avoid terrestrial interference and heavy signal fading, the user-
terminal only connects to satellites that have an elevation angle larger than
a given threshold $\theta_{\mathrm{min}}$. One reasonable threshold is
$25^{\circ}$ as FCC filings by Starlink [13]. Thus, according to [15], the
effective footprint of a satellite is bounded by the minimum permissible
elevation angle. In a practical sense, the footprint might be even smaller
than this bound depending on the antenna beamwidth $\psi$. For the purpose of
this study the footprint projection is assumed to be an ideal spherical cap
bounded by an earth-centered zenith angle, denoted as $\varphi$, see Fig. 3
for details. In order to calculate the beamwidth, the maximum slant distance
between the satellite and the ground device needs to be calculated with the
cosine rule as follows,

$a=R_{\mathrm{e}}\cos\left(\frac{\pi}{2}+\theta_{\mathrm{min}}\right)+\sqrt{R^{2}-R_{\mathrm{e}}\sin\left(\frac{\pi}{2}+\theta_{\mathrm{min}}\right)^{2}},$
(1)

where $R_{\mathrm{e}}$ is Earth’s average radius, $R=R_{\mathrm{e}}+h$, and
$h$ is the satellite altitude above the Earth’s mean sea level. Thus, the
maximum effective beamwidth can be again calculated with the cosine rule as
follows [15],

$\psi=\mathrm{acos}\left(\frac{R_{\mathrm{e}}^{2}+R^{2}-a^{2}}{2Ra}\right).$
(2)

Then the earth-centered zenith angle is calculated using the law of sines as
follows [12],

$\varphi=\operatorname{asin}\left(\frac{1}{\alpha}\sin\frac{\psi}{2}\right)-\frac{\psi}{2},$
(3)

where $\alpha=R_{\mathrm{e}}/R$. Finally, the area of the spherical cap
(footprint) of the beam is calculated as follows,

$A_{\mathrm{fp}}=2\pi R_{\mathrm{e}}^{2}\left(1-\cos\varphi\right),$ (4)

The perimeter of a spherical cap can then be drawn on the earth’s surface to
define each satellite footprint, where if a device is located within the
footprint, it is able to connect to the satellite. For defining the perimeter
of the footprint, the latitude and longitude of the footprint boundary need to
be calculated with the heading formulae [16] as follows,

$\phi_{\mathrm{fp}}=\operatorname{asin}\left(\sin\phi_{\mathrm{sat}}\cos\varphi+\cos\phi_{\mathrm{sat}}\sin\varphi\cos\theta\right),$
(5)

and the longitude,

$\rho_{\mathrm{fp}}=\rho_{\mathrm{sat}}+\operatorname{atan2}(\sin\theta\sin\varphi\cos\phi_{\mathrm{sat}},\\\
\cos\varphi-\sin\phi_{\mathrm{sat}}\sin\phi_{\mathrm{fp}}),$ (6)

where $\theta$ is an array from 0 to $2\pi$ with 360 elements and
$\phi_{\mathrm{sat}}$ and $\rho_{\mathrm{sat}}$ is the latitude and longitude
of the satellite in radians. An illustration of the geometry of a LEO
satellite is shown in Fig. 3.

Figure 3: The concept of Earth-centered zenith angle $\varphi$ and the
satellite beamwidth $\psi$. Figure 4: An illustration of the lowest energy
next hop concept. Figure 5: Different constellations with the nearest hop
topology shown: random constellation (left), the Walker-Star (middle), and the
Walker-Delta (right).

###  II-C Communication Delay

In order to benchmark the satellite network, a direct point-to-point fiber
link is assumed between the communicating ground terminals. The link thus
follows the great-circle which is the shortest path on a spherical surface.
The delay is calculated as follows,

$\tau_{\mathrm{gc}}=d_{\mathrm{gc}}/(c/n)$ (7)

where $c$ is the speed of light, $n$ is the refractive index of the optical
fiber cable, and $d_{\mathrm{gc}}$ is the great circle distance between the
communicating ground terminals. When using the satellite network, the total
delay for related to the distance of the sum of all hops distances plus the
approximated processing delay and is formulated as follows,

$\tau_{\mathrm{gc}}=\left(d_{\mathrm{sat}}+d_{\mathrm{tx\rightarrow
sat}}+d_{\mathrm{sat\rightarrow
rx}}\right)/c+N_{\mathrm{hops}}\tau_{\mathrm{process}},$ (8)

where $d_{\mathrm{sat}}$ is the sum of all the ISL distances from the first
satellite point to the last satellite point, $d_{\mathrm{tx\rightarrow sat}}$
is the distance from the transmitter to the first satellite, and
$d_{\mathrm{sat\rightarrow rx}}$ is the distance from the last satellite in
the path to the receiver. The processing delay is denoted as
$\tau_{\mathrm{process}}$.

##  III Topologies

In this paper, we evaluate the end-to-end delay performance of different ISL-
enabled constellations with two different network topologies. The topologies
being utilized are (i) cutoff distance-based topology and (ii) nearest hop-
based topology. For both topologies, after the links are constructed,
Dijkstra’s algorithm is used to calculate the lowest delay path. For
Dijkstra’s algorithm, the distance of each link is used for the link weight
because it is directly proportional to the propagation delay.

###  III-A Nearest Hop-based Topology

In a practical LEO satellite constellation, the number of available optical
ISL ports (links) is limited due to various reasons including, cost, energy
consumption, and satellite size. Therefore, ISLs links to neighboring
satellites need to be optimized according to the given criteria. One way to
form such links is to connect with the next satellites (next hop) that
minimize the transmission energy. One method to establish the lowest energy
next hops is to evaluate every satellite within a given vicinity and then pick
up only the neighbors that if reached by a direct link would be more energy
efficient. In geometric terms, this topology can be realized using the
following steps:

  1. 1.

Draw a virtual sphere between the current satellite and the candidate
satellite residing on the opposing side of the diameter segment.

  2. 2.

Create a link from the test node to the candidate only if there is no other
candidate node in the sphere.

  3. 3.

Repeat for every candidate satellite in the vicinity

By referring to the illustration in Fig. 4, the figure shows an ISL path from
node A to node D. If we assume the transmission power is variable, the link
from A $\rightarrow$ C cannot be made as link A $\rightarrow$ B is more
efficient. The link between A and C cannot be made as $d_{\mathrm{AC}}^{2}\geq
d_{\mathrm{AB}}^{2}+d_{\mathrm{BC}}^{2}$. The inverse-square relationship
between distance and transmit power due to the free space path-loss (FSPL)
exponent means that less transmit power is required if the signal travels A
$\rightarrow$ B rather than directly from A $\rightarrow$ C. The FSPL is
calculated as follows,

$l=\left[{\frac{4\pi d}{\lambda}}\right]^{2},$ (9)

Then to calculate the total energy consumption of the system is formulated as
follows,

$E=\alpha\sum_{i=0}^{K}d_{i}^{2}+\left(E_{\text{processing}}\times K\right),$
(10)

where $K$ is the number of hops and $\alpha$ is the transmit power.

After all the ISLs are made, Dijkstra’s algorithm [17] is then used to
calculate the shortest path between the transmitter and receiver using the
satellite network. A figure showing the ISLs between each satellite using the
nearest hop algorithm is shown in Fig. 5. Additionally, the pseudocode for
constructing the nearest hop topology is shown in 1.

Algorithm 1 Nearest hop topology

$n>0$ $\triangleright$ Number of candidates

for $i=1~{}\text{to}~{}N$ do $\triangleright$ Loop through all satellites

$Sat_{\mathrm{candidate}}\leftarrow n$ number of closest satellites to
satellite $Sat(i)$

for $p=1~{}\text{to}~{}n$ do $\triangleright$ Check if a link can be made

Draw sphere $Sat(i)\leftrightarrow Sat_{\mathrm{candidate}}(p)$ with Eq.(6)
and (7)

if No other satellite in the sphere then

Connect $Sat(i)\leftrightarrow Sat_{\mathrm{candidate}}(p)$

end if

end for

end for

$\text{Tx}_{\mathrm{sat}}\leftarrow$ closest satellite to
$\text{Rx}_{\mathrm{coord}}(Lat,Lon)$

$\text{Rx}_{\mathrm{sat}}\leftarrow$ closest satellite to
$\text{Tx}_{\mathrm{coord}}(Lat,Lon)$

$path\leftarrow\text{DIJKSTRA}\left(\text{Tx}_{\mathrm{sat}},\text{Rx}_{\mathrm{sat}}\right)$

###  III-B Cutoff Distance -based Topology

This topology assumes that a satellite can link with any other satellite if it
is closer that a given distance threshold. The practical sense of such a
topology is that ISL links would have a maximum viable distance either limited
by the link budget or by the occlusion caused by the Earth’s curvature. For a
generic case, we take the assumption that the links are limited by the Earth’s
curvature which imposes the upper bound distance of feasible ISL links.
Accordingly, the connection between the neighbor pairs is removed when the
weight exceeds the maximum horizon range, denoted as $d_{\mathrm{max}}$. To be
more accurate, the practical visibility constraint of ISL is limited by the
troposphere which contains 99% of the atmospheric water vapor and aerosols
[18]. The troposphere has an average height of around 18 km above the Earth’s
surface so we use this as a threshold [18]. The maximum ISL links distance is
then calculated as follows,

$\centering
d_{\mathrm{max}}=2\sqrt{\left({R_{\oplus}}+h_{\mathrm{s}}\right)^{2}-\left({R_{\oplus}}+h_{\mathrm{t}}\right)^{2}},\@add@centering$
(11)

where $h_{\mathrm{s}}$ is the height of the satellite and $h_{\mathrm{t}}$ is
the average height of the troposphere. Dijkstra’s algorithm [17] is then
utilized to find the shortest path in the topology. Note, the cutoff algorithm
provides a lower bound for ISL delay, as this is the maximum distance at which
ISLs could be formed. An illustration of how the links for the Cutoff routing
concept are formed is shown in Fig. 6.

Figure 6: An illustration of the Cutoff concept. TABLE I: Simulation Parameters. Parameters | Symbol | Simulation value
---|---|---
Satellite constellation altitude | $h_{\mathrm{s}}$ | 550 [km]
Troposphere altitude | $h_{\mathrm{t}}$ | 18 [km]
Number of satellites | $N$ | [250 500 1000]
Orbital planes | $P$ | 12
Earth’s radius | $R_{\vbox{\hbox{\scalebox{0.5}{$\oplus$}}}}$ | 6371 [km]
Satellite processing delay | $\tau_{\mathrm{process}}$ | 5.6 [$\mu$s]

##  IV Results

In this section, we present the performance results compared to torrential
optical fiber connections between two arbitrary locations on Earth.
Additionally, we explore three different LEO satellite constellations, (i)
random, (ii) Walker-Star, and (iii) Walker-Delta. Note, we explore the
performance on the random constellation as a baseline because it has been
shown to be analytically tractable from coverage handover problems [19, 20,
21].

For calculating the processing delay, we assume a processing clock operating
at $533$ MHz, which is chosen as the Zynq UltraScale+ RFSoC [22] as an example
from a real-time single-chip radio platform. We also assume 3000 instructions
to decide the ISL for the next satellite, the low number of instructions is
due to the orbits being deterministic, thus all the topologies can be
calculated apriori and the satellites would just need to use a look-up table
to calculate the next link. The processing delay can then be calculated as
$D_{\mathrm{p}}=\text{Number of Instructions}/{CLK}$. As the processing delay
within satellites is likely to be fixed and similar to each other, we assumed
the same processing delay for all satellites. Therefore, the processing delay
is multiplied by the number of satellite hops $K$.

###  IV-A Distance vs. Delay

When comparing the delay of LEO satellite networks for data communication and
the great circle optical fiber path, we normalize the delay to the great
circle optical fiber path. The $\text{Improvement}={D_{\text{Optical
fiber}}}/{D_{\text{Satellite}}}-1$, where $D$ is the delay. As such, the
improvement relative to the great circle optical fiber path is plotted
relative to the great circle distance.

The great circle distance is plotted against the delay in Fig. 7, where we
normalize the delay to the delay achieved by using an optical fiber path. The
plot then shows a trend as the number of orbiting satellites increases, the
performance also improves. The improvement is due to the greater coverage of
satellites around the globe so the probability of a more efficient path
existing also increases. Additionally in the same plot, the performance of the
nearest hop topology is compared when processing is included in the
calculation as well as assumed negligible. The plot shows the performance
decreases very slightly due to the high clock rate of the hardware considered
in this work [22]. The performance of cutoff topology with and without
processing was also considered, however, due to the smaller hop count relative
to the nearest hop topology, the effect is negligible thus it is not shown on
the plot. Note, the random constellation is used as a baseline example.

In Fig. 8 the performance of the delay against distance for LEO walker
constellations, which are also normalized a great circle optical fiber path.
From the plot, it can be seen that like with the random constellation, the
performance in terms of delay increases as the number of satellites increases.
Also, the Walker-Star seems to outperform the Walker-Delta constellation when
the nearest hop topology is used. Furthermore, the performance contrast
between the Walker-Delta and Walker-Star is the greatest when the nearest hop
algorithm is used and also increases as the distance between the transmitting
device and the receiving device grows. The performance disparity between the
walker constellations is much small when the cutoff topology is used. The
Walker-Star constellation outperforms the Walker-Delta constellation in terms
of delay as the Walker-Star has better coverage over the globe compared to the
Walker-Delta and random locations for the ground-based transmitter and
receiver are used.

Figure 7: Delay plotted against the great circle distance between the
transmitter and receiver, where the ISL delay is benchmarked against
conventional optical fiber cable between the transmitter and receiver.  Figure
8: Delay plotted against the great circle distance between the transmitter and
receiver. The ISL delay is benchmarked against conventional optical fiber
cable between the transmitter and receiver with $N=250$.  Figure 9: Simulation
of a random constellation, where the transmitting device is in Perth,
Australia, and the receiving device is in Brest, France. The shortest hop path
is shown in orange while the cutoff topology ISL path is shown in yellow.

###  IV-B Alternate Paths

To determine the robustness of each LEO constellation to link failures, we
investigate how the link delay increases as we introduce alternate paths. We
construct the best possible path and calculate the delay, then remove the
links that correspond to the best path and form a new path. The delay of the
new path is then calculated. The process is repeated until we have 10 distinct
paths from the transmitting device to the receiving device. The results are
shown in Fig. 10 where the delay performance is shown for Walker-Delta as well
as Walker-Star satellite constellations. In addition, the delay for using
terrestrial optical fiber cable over the great circle distance between the
cities is shown. An illustration showing the different types of links is shown
in Fig. 9. The Walker-Delta constellation using the nearest hop topology has
the greatest deviation between cities, particularly for the path between New
York and London. Moreover, the deviation from using the cutoff algorithm is
very low, thereby showing greater robustness to failed links between
satellites. Finally, the performance of satellites for ISL paths compared to
traditional fiber has a greater pay-off when the distance between the two
locations increases such as in the delay between Perth and Brest.

Figure 10: The delay between major cities with $N=1000$.

##  V Conclusion

In this study, we analyzed ISL-enabled LEO satellite constellations as a low-
delay alternative to traditional terrestrial optical fiber networks. We
investigated three LEO constellations: Walker-Delta, Walker-Star, and random.
Additionally, we explored two topologies, presenting the power-efficient
nearest hop topology and comparing it to the delay-minimizing cutoff topology.
Our results demonstrated that satellite networks improve delay performance
compared to optical fiber connections as the transmitter-receiver distance
increases. The proposed nearest hop topology maintains a better delay
performance compared to the great-circle fiber path while also utilizing more
energy-efficient ISLs. Future work will involve using machine learning to
develop a topology for dynamically changing ISLs due to high traffic loads and
link failures.

The Walker-Star constellation had a lower delay, but the Walker-Delta
constellation was more power-efficient in terms of average ISL length with the
nearest hop algorithm.

##  VI Acknowledgment

The authors would like to acknowledge the discussions with Dr. Ben Allen and
Mr. Ben Moores, also the partial funding by SmartSat CRC under the UK-
Australia spacebridge program.

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# Frobenius test exponent for ideals generated by filter regular sequences

Duong Thi Huong  Department of Mathematics, Thang Long University, Hanoi,
Vietnam<EMAIL_ADDRESS>and  Pham Hung Quy  Department of
Mathematics, FPT University, Hanoi, Vietnam<EMAIL_ADDRESS>
###### Abstract.

Let $(R,\mathfrak{m})$ be a Noetherian local ring of prime characteristic
$p>0$, and $t$ an integer such that
$H_{\mathfrak{m}}^{j}(R)/0^{F}_{H^{j}_{\mathfrak{m}}(R)}$ has finite length
for all $j<t$. The aim of this paper is to show that there exists an uniform
bound for Frobenius test exponents of ideals generated by filter regular
sequences of length at most $t$.

###### Key words and phrases:

The Frobenius test exponent, The Hartshorne-Speiser-Lyubeznik number, Local
cohomology, Filter regular sequence.

2020 Mathematics Subject Classification: 13A35, 13D45.
The authors are partially supported by a fund of Vietnam National Foundation
for Science and Technology Development (NAFOSTED) under grant number
101.04-2020.10.

Dedicated to Professor Nguyen Tu Cuong on the occasion of his 70th birthday

##  1\. Introduction

Throughout this paper, let $R$ be a Noetherian commutative ring of prime
characteristic $p>0$, and $I$ an ideal of $R$. The key ingredient in study
ring of prime characteristic is using the Frobenius endomorphism $F:R\to
R;x\mapsto x^{p}$ and its $e$-th iterations $F^{e}$, $e\geq 1$. The $e$-th
Frobenius power of $I$ is the extension of $I$ via $F^{e}$,
$I^{[p^{e}]}=(x^{p^{e}}\mid x\in I)$. The Frobenius closure of $I$ is
$I^{F}=\\{x\mid x^{p^{e}}\in I^{[p^{e}]}\text{ for some }e\geq 0\\}$. Hence
$I^{F}$ is the set of all nilpotent elements under Frobenius endomorphism
modulo $I$. By the Noetherianess of $R$ there is an integer $e$, depending on
$I$, such that $(I^{F})^{[p^{e}]}=I^{[p^{e}]}$. The Frobenius test exponent of
$I$ is defined by
$\mathrm{Fte}(I)=\min\\{e\mid(I^{F})^{[p^{e}]}=I^{[p^{e}]}\\}$. Under mild
conditions, $R$ is $F$-pure if and only if $\mathrm{Fte}(I)=0$ for all $I$ by
Hochster [6]. In general, we can not expect an upper bound for all
$\mathrm{Fte}(I)$ by Brenner [1]. Restricting to the class of parameter
ideals, for any local ring $(R,\mathfrak{m})$ of dimension $d$ we define the
Frobenius test exponent for parameter ideals $\mathrm{Fte}(R)$ as follows

$\mathrm{Fte}(R)=\min\\{e\mid(\mathfrak{q}^{F})^{[p^{e}]}=\mathfrak{q}^{[p^{e}]}\,\text{for
all parameter ideals }\,\mathfrak{q}\\},$

and $\mathrm{Fte}(R)=\infty$ if we have no such integer. Katzman and Sharp
asked whether $\mathrm{Fte}(R)<\infty$ for any (equidimensional) local ring.
Moreover they [11] proved that it is the case when the ring is Cohen-Macaulay.
The main idea in [11] is connecting $\mathrm{Fte}(R)$ with an invariant
defined by the Frobenius actions on the local cohomology modules
$H^{i}_{\mathfrak{m}}(R)$, namely the Hartshorne-Speiser-Lyubeznik number of
$H^{i}_{\mathfrak{m}}(R)$. Recall that the Frobenius endomorphism induces
natural Frobenius actions on the local cohomology modules
$H^{i}_{\mathfrak{m}}(R)$ for all $i\geq 0$. The Frobenius closure of zero
submodule of $H^{i}_{\mathfrak{m}}(R)$, denoted by
$0^{F}_{H^{i}_{\mathfrak{m}}(R)}$, consists nilpotent elements of
$H^{i}_{\mathfrak{m}}(R)$ under the Frobenius action. The Hartshorne-Speiser-
Lyubeznik number of $H^{i}_{\mathfrak{m}}(R)$ is defined by

$\mathrm{HSL}(H^{i}_{\mathfrak{m}}(R))=\min\\{e\mid
0^{F}_{H^{i}_{\mathfrak{m}}(R)}=\mathrm{Ker}(H^{i}_{\mathfrak{m}}(R)\overset{F^{e}}{\longrightarrow}H^{i}_{\mathfrak{m}}(R))\\}.$

It is proved that $\mathrm{HSL}(H^{i}_{\mathfrak{m}}(R))$ always exists. The
Hartshorne-Speiser-Lyubeznik number of $R$ is
$\mathrm{HSL}(R)=\max\\{\mathrm{HSL}(H^{i}_{\mathfrak{m}}(R))\mid
i=0,\ldots,d\\}$. In fact Katzman and Sharp proved that
$\mathrm{Fte}(R)=\mathrm{HSL}(R)$ provided $R$ is Cohen-Macaulay. In general,
we have $\mathrm{Fte}(R)\geq\mathrm{HSL}(R)$ for any local ring by [9]. Beyond
the Cohen-Macaulay case, Huneke, Katzman, Sharp, and Yao [8] showed that
$\mathrm{Fte}(R)<\infty$ for generalized Cohen-Macaulay local rings by using
several concepts and techniques from commutative algebra, namely unconditioned
strong d-sequences, cohomological annihilators, and modules of generalized
fractions. In 2019, the second author [16] not only simplified the proof for
generalized Cohen-Macaulay rings but also proved $\mathrm{Fte}(R)<\infty$ for
weakly F-nilpotent rings, i.e.
$H^{i}_{\mathfrak{m}}(R)=0^{F}_{H^{i}_{\mathfrak{m}}(R)}$ for all $i<d$.
Recently, Maddox [13] extended this result for generalized weakly F-nilpotent
rings, i.e. $H^{i}_{\mathfrak{m}}(R)/0^{F}_{H^{i}_{\mathfrak{m}}(R)}$ has
finite length for all $i<d$.

Suppose $\mathrm{Fte}(R)<\infty$, and let $x_{1},\ldots,x_{d}$ be a system of
parameters of $R$. Then for any $t\leq d$ we have
$\mathrm{Fte}((x_{1},\ldots,x_{t}))\leq\mathrm{Fte}(R)$ by using Krull’s
intersection theorem. If we do not know about the finiteness of
$\mathrm{Fte}(R)$, in which conditions we have
$\mathrm{Fte}((x_{1},\ldots,x_{t}))<\infty$? For a regular sequence
$\underline{x}=x_{1},\ldots,x_{t}$, Katzman and Sharp [11, Corollary 4.3]
showed that there exists an integer $C_{\underline{x}}$ such that
$\mathrm{Fte}((x_{1}^{n_{1}},\ldots,x_{t}^{n_{t}}))\leq C_{\underline{x}}$ for
all $n_{1},\ldots,n_{t}\geq 1$. Since $C_{\underline{x}}$ depends on the
choice of $\underline{x}$, the previous result does not answer for the
following question.

######  Question 1.

Let $(R,\mathfrak{m})$ be a local ring of prime characteristic and of
dimension $d$, and $t\leq\mathrm{depth}(R)$ an integer. Does there exist a
positive integer $C$ such that for any regular sequence $x_{1},\ldots,x_{t}$
we have $\mathrm{Fte}((x_{1},\ldots,x_{t}))\leq C$?

Inspired by known results about the finiteness of $\mathrm{Fte}(R)$, we give
an affirmative answer for the above question in a more general form. It covers
all aforementioned results.

######  Theorem 1.1.

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d$ and of
prime characteristic $p>0$, and $t\leq d$ a positive integer such that
$H_{\mathfrak{m}}^{j}(R)/0^{F}_{H^{j}_{\mathfrak{m}}(R)}$ has finite length
for all $j<t$. Then there exists a positive integer $C$ such that for any
filter regular sequence $x_{1},\ldots,x_{t}$ we have
$\mathrm{Fte}((x_{1},\ldots,x_{t}))\leq C$.

In the next section we recall the basic notions and relevant materials. We
prove the main theorem in the last section.

##  2\. Preliminaries

###  2.1. Filter regular sequences

In this subsection, the assumption that $R$ is of prime characteristic is
unnecessary.

######  Definition 2.1.

Let $(R,\mathfrak{m})$ be a Noetherian local ring, and $x_{1},\ldots,x_{t}$ a
sequence of elements of $R$. Then we say that $x_{1},\ldots,x_{t}$ is a filter
regular sequence if the following conditions hold:

  1. (1)

$(x_{1},\ldots,x_{t})\subseteq\mathfrak{m}$,

  2. (2)

$x_{i}\notin\mathfrak{p}$ for all
$\mathfrak{p}\in\operatorname{Ass}\big{(}\frac{M}{(x_{1},\ldots,x_{i-1})M}\big{)}\setminus\\{\mathfrak{m}\\}$,
$i=1,\ldots,t$.

The notion of filter regular sequences was introduced by Cuong, Schenzel, and
Trung in [2]. The following is well-known.

######  Lemma 2.2.

Let $(R,\mathfrak{m})$ be a Noetherian local ring and
$x_{1},\ldots,x_{t}\subseteq\mathfrak{m}$. The following are equivalent

  1. (1)

$x_{1},\ldots,x_{t}$ is a filter regular sequence.

  2. (2)

For each $1\leq i\leq t$ the quotient

$\frac{(x_{1},\ldots,x_{i-1}):_{R}x_{i}}{(x_{1},\ldots,x_{i-1})}$

is an $R$-module of finite length.

  3. (3)

For each $1\leq i\leq t$ the sequence

$\frac{x_{1}}{1},\frac{x_{2}}{1},\ldots,\frac{x_{i}}{1}$

forms an $R_{\mathfrak{p}}$-regular sequence for every
$\mathfrak{p}\in\operatorname{Spec}(R/(x_{1},\ldots,x_{i}))\setminus\\{\mathfrak{m}\\}$.

  4. (4)

The sequence $x_{1}^{n_{1}},\ldots,x_{t}^{n_{t}}$ is a filter regular sequence
for all $n_{1},\ldots,n_{t}\geq 1$.

  5. (5)

The sequence $x_{1},\ldots,x_{t}$ is a filter regular sequence of
$\widehat{R}$, where $\widehat{R}$ is the $\mathfrak{m}$-adic completion of
$R$.

###  2.2. The Frobenius action on local cohomology

In this subsection, let $R$ be a Noetherian ring containing a field of
characteristic $p>0$. Let $F:R\to R,x\mapsto x^{p}$ denote the Frobenius
endomorphism. If we want to notationally distinguish the source and target of
the $e$-th Frobenius endomorphism $F^{e}:R\xrightarrow{x\mapsto x^{p^{e}}}R$,
we will use $F_{*}^{e}(R)$ to denote the target. $F_{*}^{e}(R)$ is an
$R$-bimodule, which is the same as $R$ as an abelian group and as a right
$R$-module, that acquires its left $R$-module structure via the $e$-th
Frobenius endomorphism $F^{e}$. By definition the $e$-th Frobenius
endomorphism $F^{e}:R\to F_{*}^{e}(R)$ sending $x$ to
$F_{*}^{e}(x^{p^{e}})=x\cdot F_{*}^{e}(1)$ is an $R$-homomorphism. We say $R$
is $F$-finite if $F_{*}(R)$ is a finite $R$-module.

######  Definition 2.3.

Let $I$ be an ideal of $R$, we define

  1. (1)

The $e$-th Frobenius power of $I$ is $I^{[p^{e}]}=(x^{p^{e}}\mid x\in I)$.

  2. (2)

The Frobenius closure of $I$, $I^{F}=\\{x\mid x^{p^{e}}\in I^{[p^{e}]}\text{
for some }e\geq 0\\}$.

######  Definition 2.4.

Let $I$ be an ideal of $R$. By the Noetherianess of $R$ there is an integer
$e$, depending on $I$ such that $(I^{F})^{[p^{e}]}=I^{[p^{e}]}$. The smallest
number $e$ satisfying the condition is called the Frobenius test exponent of
$I$, and denoted by $\mathrm{Fte}(I)$,

$\mathrm{Fte}(I)=\min\\{e\mid(I^{F})^{[p^{e}]}=I^{[p^{e}]}\\}.$

A problem of Katzman and Sharp [11, Introduction] asks in its strongest form:
does there exist an integer $e$, depending only on the ring $R$, such that for
every ideal $I$ we have $(I^{F})^{[p^{e}]}=I^{[p^{e}]}$. A positive answer to
this question, together with the actual knowledge of a bound for $e$, would
give an algorithm to compute the Frobenius closure $I^{F}$. Unfortunately,
Brenner [1] gave two-dimensional normal standard graded domains with no
Frobenius test exponent. In contrast, Katzman and Sharp showed the existence
of Frobenius test exponent if we restrict to class of parameter ideals in a
Cohen-Macaulay ring. Therefore it is natural to ask the following question.

######  Question 2.

Let $(R,\mathfrak{m})$ be an (equidimensional) local ring of prime
characteristic $p$. Then does there exist an integer $e$ such that for every
parameter ideal $\mathfrak{q}$ of $R$ we have
$(\mathfrak{q}^{F})^{[p^{e}]}=\mathfrak{q}^{[p^{e}]}$?

We define the Frobenius test exponent for parameter ideals of $R$,
$\mathrm{Fte}(R)$, the smallest integer $e$ satisfying the above condition and
$\mathrm{Fte}(R)=\infty$ if we have no such $e$. It should be noted that the
authors recently used the finiteness of $\mathrm{Fte}(R)$, if have, to find an
upper bound of the multiplicity of a local ring [10].
For any ideal $I=(x_{1},\ldots,x_{t})$, the Frobenius endomorphism $F:R\to R$
and its localizations induce a natural Frobenius action on local cohomology
$F:H^{i}_{I}(R)\to H^{i}_{I^{[p]}}(R)\cong H^{i}_{I}(R)$ for all $i\geq 0$. In
general, let $A$ be an Artinian $R$-module with a Frobenius action $F:A\to A$.
Then we define the Frobenius closure $0^{F}_{A}$ of the zero submodule of $A$
is the submodule of $A$ consisting all elements $z$ such that $F^{e}(z)=0$ for
some $e\geq 0$. Hence $0^{F}_{A}$ is the nilpotent part of $A$ under the
Frobenius action. By [5, Proposition 1.11], [12, Proposition 4.4] and [17] we
have the following.

######  Theorem 2.5.

Let $(R,\mathfrak{m})$ be a local ring of prime characteristic $p>0$, and $A$
an Artinian $R$-module with a Frobenius action $F:A\to A$. Then there exists a
non-negative integer $e$ such that
$0^{F}_{A}=\operatorname{Ker}(A\overset{F^{e}}{\longrightarrow}A)$.

######  Definition 2.6.

  1. (1)

Let $A$ be an Artinian $R$-module with a Frobenius action $F$. The Hartshorne-
Speiser-Lyubeznik number of $A$ is denoted by $\mathrm{HLS}(A)$ and is defined
to be

$\mathrm{HSL}(A)=\min\\{e\mid
0^{F}_{A}=\operatorname{Ker}(A\overset{F^{e}}{\longrightarrow}A)\\}.$

  2. (2)

Notice that $H^{i}_{\mathfrak{m}}(R)$ is always Artinian for all $i\geq 0$. We
define the Hartshorne-Speiser-Lyubeznik number of a local ring
$(R,\mathfrak{m})$ as follows

$\mathrm{HSL}(R):=\min\\{e\mid
0^{F}_{H^{i}_{\mathfrak{m}}(R)}=\operatorname{Ker}(H^{i}_{\mathfrak{m}}(R)\overset{F^{e}}{\longrightarrow}H^{i}_{\mathfrak{m}}(R))\text{
for all }i=0,\ldots,d\\}.$

As mentioned in the introduction, Question 2 was answered affirmatively in the
following cases:

  1. (1)

(Katzman-Sharp) $R$ is Cohen-Macaulay ring. Moreover,
$\mathrm{Fte}(R)=\mathrm{HSL}(R)$.

  2. (2)

(Huneke-Katzman-Sharp-Yao) $R$ is generalized Cohen-Macaulay ring, i.e.
$H^{i}_{\mathfrak{m}}(R)$ has finite length for all $i<d$.

  3. (3)

(Quy) $R$ is weakly $F$-nilpotent ring, i.e.
$H^{i}_{\mathfrak{m}}(R)=0^{F}_{H^{i}_{\mathfrak{m}}(R)}$ for all $i<d$.

  4. (4)

(Maddox) $R$ is generalized weakly $F$-nilpotent ring, i.e.
$H^{i}_{\mathfrak{m}}(R)/0^{F}_{H^{i}_{\mathfrak{m}}(R)}$ has finite length
for all $i<d$.

The main idea of the proofs of the affirmative cases is to make a connection
between the Frobenius test exponent for parameter ideals with Hartshorne-
Speiser-Lyubeznik numbers of the local cohomology modules. For example, if $R$
is weakly $F$-nilpotent, then the second author [16] used the notion of
relative Frobenius action on local cohomology to prove

$\mathrm{Fte}(R)\leq\sum_{i=0}^{d}\binom{d}{i}\mathrm{HSL}(H^{i}_{\mathfrak{m}}(R)).$

###  2.3. The relative Frobenius action on local cohomology

In this subsection, we recall the notion of relative Frobenius action on local
cohomology which was introduced in [14] by Polstra and Quy in study
$F$-nilpotent rings. Let $K\subseteq I$ be ideals of $R$. The Frobenius
endomorphism $F:R/K\to R/K$ can be factored as composition of two natural
maps:

$R/K\to R/K^{[p]}\twoheadrightarrow R/K,$

where the second map is the natural project map. We denote the first map by
$F_{R}:R/K\to R/K^{[p]}$, $F_{R}(a+K)=a^{p}+K^{[p]}$ for all $a\in R$. The
homomorphism $F_{R}$ induces the relative Frobenius actions on local
cohomology $F_{R}:H^{i}_{I}(R/K)\to H^{i}_{I}(R/K^{[p]})$ via Čech complexes.

######  Definition 2.7.

  1. (1)

We define the relative Frobenius closure of the zero submodule of
$H^{i}_{I}(R/K)$ with respect to $R$ as follows

$0^{F_{R}}_{H^{i}_{I}(R/K)}=\\{\eta\mid F^{e}_{R}(\eta)=0\in
H^{i}_{I}(R/K^{[p^{e}]})\text{ for some }e\gg 0\\}.$

  2. (2)

If there is an integer $e$ such that

$0^{F_{R}}_{H^{i}_{I}(R/K)}=\operatorname{Ker}(H^{i}_{I}(R/K)\xrightarrow{F^{e}_{R}}H^{i}_{I}(R/K^{[p^{e}]})),$

then we call the smallest of such integers the Hartshorne-Speiser-Lyubeznik
number of $H^{i}_{I}(R/K)$ with respect to $R$, denoted by
$\mathrm{HSL}_{R}(H^{i}_{I}(R/K))$. And convention that
$\mathrm{HSL}_{R}(H^{i}_{I}(R/K))=\infty$ if we have no such integer.

######  Lemma 2.8.

Let $(R,\mathfrak{m})$ be a Noetherian local ring, and $I$ an ideal of $R$.
Then

$0^{F_{R}}_{H_{\mathfrak{m}}^{0}(R/I)}=I^{F}\cap(I:\mathfrak{m}^{\infty})/I.$

###### Proof.

Pick $a+I\in 0^{F_{R}}_{H_{\mathfrak{m}}^{0}(R/I)}$. We have $a\in
I:\mathfrak{m}^{\infty}$ and there exists an integer $e$ such that
$F^{e}(a+I)=a^{p^{e}}+I^{[p^{e}]}=0\in R/I^{[p^{e}]}$. Equivalently, $a+I\in
I^{F}\cap(I:\mathfrak{m}^{\infty})/I$. ∎

##  3\. Main result

Throughout this section, let $(R,\mathfrak{m})$ be a Noetherian local ring of
prime characteristic $p>0$ and of dimension $d$, and $t\leq d$ an integer such
that $H_{\mathfrak{m}}^{j}(R)/0^{F}_{H^{j}_{\mathfrak{m}}(R)}$ has finite
length for all $j<t$. Let $x_{1},\ldots,x_{t}$ be a filter regular sequence.
Set $I_{i}=(x_{1},\ldots,x_{i})$ for all $i\leq t$ and
$I=(x_{1},\ldots,x_{t})$.

######  Lemma 3.1.

Let $(R,\mathfrak{m})$ be a local ring of prime characteristic $p>0$ and of
dimension $d$, and $t\leq d$ an integer such that
$H_{\mathfrak{m}}^{j}(R)/0^{F}_{H^{j}_{\mathfrak{m}}(R)}$ has finite length
for all $j<t$. Let $n_{0}$ be an non-negative integer such that
$\mathfrak{m}^{n_{0}}H_{\mathfrak{m}}^{j}(R)/0^{F}_{H^{j}_{\mathfrak{m}}(R)}=0$
for all $j<t$. Then for every filter regular sequence $x_{1},\ldots,x_{t}$ we
have

$\mathfrak{m}^{2^{i}n_{0}}H^{j}_{\mathfrak{m}}(R/I_{i})\subseteq
0^{F_{R}}_{H^{j}_{\mathfrak{m}}(R/I_{i})}$

for all $i\leq t$ and all $j<t-i$.

###### Proof.

We proceed by induction on $i$. The case $i=0$ is nothing to do. Suppose $i>0$
and the assertion holds true for $i-1$. For each $e\geq 1$ we consider
commutative diagram

$\begin{CD}0\longrightarrow
R/(I_{i-1}:x_{i})@>{x_{i}}>{}>R/I_{i-1}@>{}>{}>R/I_{i}@>{}>{}>0\\\
@V{(F^{e}_{R})^{\prime}}V{}V@V{F^{e}_{R}}V{}V@V{F^{e}_{R}}V{}V(\star)\\\
0\longrightarrow
R/(I_{i-1}^{[p^{e}]}:x_{i}^{p^{e}})@>{x_{i}^{p^{e}}}>{}>R/I_{i-1}^{[p^{e}]}@>{}>{}>R/I_{i}^{[p^{e}]}@>{}>{}>0\end{CD}$

where the first vertical map is the composition

$R/(I_{i-1}:x_{i})\xrightarrow{F^{e}_{R}}R/(I_{i-1}:x_{i})^{[p^{e}]}\twoheadrightarrow
R/(I_{i-1}^{[p^{e}]}:x_{i}^{p^{e}}).$

Because $x_{1}^{p^{e}},\ldots,x_{t}^{p^{e}}$ is a filter regular sequence,
$(I_{i-1}^{[p^{e}]}:x_{i}^{p^{e}})/I_{i-1}^{[p^{e}]}$ has finite length. Thus

$H^{j+1}_{\mathfrak{m}}(R/(I_{i-1}^{[p^{e}]}:x_{i}^{p^{e}}))\cong
H^{j+1}_{\mathfrak{m}}(R/(I_{i-1}:x_{i})^{[p^{e}]})\cong
H^{j+1}_{\mathfrak{m}}(R/I_{i-1}^{[p^{e}]})$

for all $j\geq 0$ and for all $e\geq 0$. Therefore the diagram $(\star)$
induces the following commutative diagram with exact rows

$\begin{CD}\cdots\xrightarrow{x_{i}}H_{\mathfrak{m}}^{j}(R/I_{i-1})@>{}>{}>H_{\mathfrak{m}}^{j}(R/I_{i})@>{\delta}>{}>H_{\mathfrak{m}}^{j+1}(R/I_{i-1})\\\
@V{F^{e}_{R}}V{}V@V{F^{e}_{R}}V{}V@V{F^{e}_{R}}V{}V\\\
\cdots\xrightarrow{x_{i}^{p^{e}}}H_{\mathfrak{m}}^{j}(R/I_{i-1}^{[p^{e}]})@>{\alpha}>{}>H_{\mathfrak{m}}^{j}(R/I_{i}^{[p^{e}]})@>{\beta}>{}>H_{\mathfrak{m}}^{j+1}(R/I_{i-1}^{[p^{e}]}).\end{CD}$

Pick any $u\in H^{j}_{\mathfrak{m}}(R/I_{i})$, and
$x,y\in\mathfrak{m}^{2^{i-1}n_{0}}$. Then $\delta(u)\in
H^{j+1}_{\mathfrak{m}}(R/I_{i-1})$. By the inductive hypothesis $x\delta(u)\in
0^{F_{R}}_{H_{\mathfrak{m}}^{j+1}(R/I_{i-1})}$, so there is an integer $e$
such that

$0=F^{e}_{R}(x\delta(u))=\beta(F^{e}_{R}(xu)).$

Hence there exists $v\in H_{\mathfrak{m}}^{j}(R/I_{i-1}^{[p^{e}]})$ such that
$\alpha(v)=F^{e}_{R}(xu)$. Moreover, $y^{p^{e}}\in\mathfrak{m}^{2^{i-1}n_{0}}$
so $y^{p^{e}}v\in 0^{F_{R}}_{H_{\mathfrak{m}}^{j}(R/I_{i-1}^{[p^{e}]})}$ by
using the induction for the sequence $x_{1}^{p^{e}},\ldots,x_{i-1}^{p^{e}}$.
Therefore,

$\alpha(y^{p^{e}}v)=y^{p^{e}}.F^{e}_{R}(xu)=F^{e}_{R}(yxu)\in
0^{F_{R}}_{H_{\mathfrak{m}}^{j}(R/I_{i}^{[p^{e}]})}.$

Leading to $xyu\in 0^{F_{R}}_{H_{\mathfrak{m}}^{j}(R/I_{i})}$. Hence
$xy\in\operatorname{Ann}_{R}(H_{\mathfrak{m}}^{j}(R/I_{i})/0^{F_{R}}_{H^{j}_{\mathfrak{m}}(R/I_{i})})$,
and so

$\mathfrak{m}^{2^{i}n_{0}}\subseteq\operatorname{Ann}_{R}(H_{\mathfrak{m}}^{j}(R/I_{i})/0^{F_{R}}_{H^{j}_{\mathfrak{m}}(R/I_{i})})$

for all $i\leq t$ and $j<t-i$. The proof is complete. ∎

######  Remark 3.2.

Using the notation in Lemma 3.1, and suppose
$x_{1}\ldots,x_{t}\in\mathfrak{m}^{2^{t}n_{0}}$. Then we have

$\operatorname{Im}(H^{j}_{\mathfrak{m}}(R/I_{i-1}:x_{i})\xrightarrow{x_{i}}H^{j}_{\mathfrak{m}}(R/I_{i-1}))\subseteq
0^{F_{R}}_{H^{j}_{\mathfrak{m}}(R/I_{i-1})}$

for all $i\leq t$ and $j\leq t-i$. Indeed the case $1\leq j\leq t-i$ follows
from Lemma 3.1 and the fact $H^{j}_{\mathfrak{m}}(R/I_{i-1}:x_{i})\cong
H^{j}_{\mathfrak{m}}(R/I_{i-1})$. For $j=0$ we use Lemma 3.1 and the
surjective map $H^{0}_{\mathfrak{m}}(R/I_{i-1})\twoheadrightarrow
H^{0}_{\mathfrak{m}}(R/I_{i-1}:x_{i})$. In particular, every relative
nilpotent element of the induced Frobenius action on
$\mathrm{Coker}(H^{j}_{\mathfrak{m}}(R/I_{i-1}:x_{i})\xrightarrow{x_{i}}H^{j}_{\mathfrak{m}}(R/I_{i-1}))$
is an image of some element in $0^{F_{R}}_{H^{j}_{\mathfrak{m}}(R/I_{i-1})}$.

Using the above Remark and the argument of [16, Proof of the main theorem] and
[13, Theorem 3.1] we obtain the following whose proof is left to the reader.

######  Proposition 3.3.

Let $(R,\mathfrak{m})$ be a local ring of prime characteristic $p>0$ and of
dimension $d$, and $t\leq d$ an integer such that
$H_{\mathfrak{m}}^{j}(R)/0^{F}_{H^{j}_{\mathfrak{m}}(R)}$ has finite length
for all $j<t$. Then there exists a non-negative integer $e_{0}$ such that for
every filter regular sequence $x_{1},\ldots,x_{t}$ we have

$\mathrm{HSL}_{R}(H_{\mathfrak{m}}^{0}(R/(x_{1},\ldots,x_{t})))\leq\sum\limits_{k=0}^{t}\binom{t}{k}\mathrm{HSL}(H_{\mathfrak{m}}^{k}(R))+e_{0}.$

######  Remark 3.4.

If $x_{1},\ldots,x_{t}$ is a regular sequence then $H^{j}_{\mathfrak{m}}(R)=0$
for all $j<t$, and the number $e_{0}$ in the previous result can be chosen as
zero. Thus we have
$\mathrm{HSL}_{R}(H_{\mathfrak{m}}^{0}(R/(x_{1},\ldots,x_{t})))\leq\mathrm{HSL}(H^{t}_{\mathfrak{m}}(R))$.

The following is known to experts.

######  Lemma 3.5.

Let $(R,\mathfrak{m})$ be an $F$-finite local ring of dimension $d$. Let
$\mathfrak{p}\in\mathrm{Spec}(R)$ a prime ideal. Then for all $i\leq d$ we
have $\mathrm{HSL}(H^{i-\dim
R/\mathfrak{p}}_{\mathfrak{p}R_{\mathfrak{p}}}(R_{\mathfrak{p}}))\leq\mathrm{HSL}(H^{i}_{\mathfrak{m}}(R))$.
In particular, $\mathrm{HSL}(R_{\mathfrak{p}})\leq\mathrm{HSL}(R)$.

###### Proof.

Since $(R,\mathfrak{m})$ is $F$-finite, it is an image of some regular local
ring $(S,\mathfrak{n})$ by Gabber [4]. Suppose that $\dim S=n$. For each
$i\leq d$, by the definition we have $\mathrm{HLS}(H^{i}_{\mathfrak{m}}(R))$
is the smallest non-negative integer $e$ such that

$\mathrm{Ker}(H^{i}_{\mathfrak{m}}(R)\xrightarrow{F^{e}}H^{i}_{\mathfrak{m}}(F^{e}_{*}R))=\mathrm{Ker}(H^{i}_{\mathfrak{m}}(R)\xrightarrow{F^{e+1}}H^{i}_{\mathfrak{m}}(F^{e+1}_{*}R)).$

By the local duality theorem we have

$\mathrm{Coker}(\mathrm{Ext}^{n-i}_{S}(F^{e}_{*}R,S)\xrightarrow{(F^{e})^{\vee}}\mathrm{Ext}^{n-i}_{S}(R,S))=\mathrm{Coker}(\mathrm{Ext}^{n-i}_{S}(F^{e+1}_{*}R,S)\xrightarrow{(F^{e+1})^{\vee}}\mathrm{Ext}^{n-i}_{S}(R,S)).$

Let $P$ be the preimage of $\mathfrak{p}$ in $S$. Taking localization we have

$\mathrm{Coker}(\mathrm{Ext}^{n-i}_{S_{P}}(F^{e}_{*}R_{\mathfrak{p}},S_{P})\to\mathrm{Ext}^{n-i}_{S_{P}}(R_{\mathfrak{p}},S_{P}))=\mathrm{Coker}(\mathrm{Ext}^{n-i}_{S_{P}}(F^{e+1}_{*}R_{\mathfrak{p}},S_{P})\to\mathrm{Ext}^{n-i}_{S_{P}}(R_{\mathfrak{p}},S_{P})).$

Applying the local duality theorem for
$(R_{\mathfrak{p}},\mathfrak{p}R_{\mathfrak{p}})$ we have

$\mathrm{Ker}(H^{i-\dim
R/\mathfrak{p}}_{\mathfrak{p}R_{\mathfrak{p}}}(R_{\mathfrak{p}})\xrightarrow{F^{e}}H^{i-\dim
R/\mathfrak{p}}_{\mathfrak{p}R_{\mathfrak{p}}}(F^{e}_{*}R_{\mathfrak{p}}))=\mathrm{Ker}(H^{i-\dim
R/\mathfrak{p}}_{\mathfrak{p}R_{\mathfrak{p}}}(R_{\mathfrak{p}})\xrightarrow{F^{e+1}}H^{i-\dim
R/\mathfrak{p}}_{\mathfrak{p}R_{\mathfrak{p}}}(F^{e+1}_{*}R_{\mathfrak{p}})).$

Hence $\mathrm{HSL}(H^{i-\dim
R/\mathfrak{p}}_{\mathfrak{p}R_{\mathfrak{p}}}(R_{\mathfrak{p}}))\leq e$. The
proof is complete. ∎

We are ready to prove the main result of this paper.

######  Theorem 3.6.

Let $(R,\mathfrak{m})$ be a local ring of dimention $d$ and of prime
characteristic $p>0$, and $t\leq d$ an integer such that
$H_{\mathfrak{m}}^{j}(R)/0^{F}_{H^{j}_{\mathfrak{m}}(R)}$ has finite length
for all $j<t$. Then there exists an integer $C$ such that for any filter
regular sequence $x_{1},\ldots,x_{t}$ we have
$\mathrm{Fte}(x_{1},\ldots,x_{t})\leq C$.

###### Proof.

By passing to the completion we can assume that $(R,\mathfrak{m})$ is
complete. We next use the $\Gamma$-construction of Hochster and Huneke [7] to
obtain a faithfully flat extension $R\to R^{\Gamma}$ such that $R^{\Gamma}$ is
$F$-finite and $\mathfrak{m}R^{\Gamma}$ is the maximal ideal of $R^{\Gamma}$.
Notice that the induced Frobenius actions on
$H^{j}_{\mathfrak{m}}(R)/0^{F}_{H^{j}_{\mathfrak{m}}(R)}$ are injecitve for
all $j\geq 0$. By [3, Lemmas 2.9 and 4.3] we can choose a sufficiently small
$\Gamma$ such that the Frobenius actions on
$R^{\Gamma}\otimes_{R}H^{j}_{\mathfrak{m}}(R)/0^{F}_{H^{j}_{\mathfrak{m}}(R)}$
are injecitve for all $j\geq 0$. Thus
$0^{F}_{H^{j}_{\mathfrak{m}}(R^{\Gamma})}\cong
R^{\Gamma}\otimes_{R}0^{F}_{H^{j}_{\mathfrak{m}}(R)}$. Since
$(x_{1},\ldots,x_{t})^{F}R^{\Gamma}\subseteq((x_{1},\ldots,x_{t})R^{\Gamma})^{F}$,
it is enough to prove the requirement for $R^{\Gamma}$. Therefore we can
assume henceforth that $(R,\mathfrak{m})$ is $F$-finite.

Let $c$ be the upper bound in Proposition 3.3 and $h:=\mathrm{HSL}(R)$. Let
$I=(x_{1},\ldots,x_{t})$, and pick any $a\in I^{F}\setminus I$. Suppose $\dim
R/(I:a)=s$. We will prove $\dim R/(I^{[p^{hs}]}:a^{p^{hs}})\leq 0$ by
induction on $s$. There is nothing to do if $s=0$. Suppose $s>0$, and let
$\mathfrak{p}$ be any minimal prime of $R/(I:a)$ with $\dim R/\mathfrak{p}=s$.
Then $aR_{\mathfrak{p}}\in(IR_{\mathfrak{p}})^{F}$ by [15, Lemma 3.3]. By the
minimality we have $a+IR_{\mathfrak{p}}\in
H^{0}_{\mathfrak{p}R_{\mathfrak{p}}}(R_{\mathfrak{p}}/IR_{\mathfrak{p}})$.
Since $x_{1},\ldots,x_{t}$ becomes a regular sequence in $R_{\mathfrak{p}}$ we
have
$\mathrm{HSL}_{R_{\mathfrak{p}}}(H^{0}_{\mathfrak{p}R_{\mathfrak{p}}}(R_{\mathfrak{p}}/IR_{\mathfrak{p}}))\leq\mathrm{HSL}(H^{t}_{\mathfrak{p}R_{\mathfrak{p}}}(R_{\mathfrak{p}}))\leq
h$ by Remark 3.4 and Lemma 3.5. Therefore
$F^{h}_{R_{\mathfrak{p}}}(a+IR_{\mathfrak{p}})=0\in
R_{\mathfrak{p}}/I^{[p^{h}]}R_{\mathfrak{p}}$, and so
$a^{p^{h}}R_{\mathfrak{p}}\in I^{[p^{h}]}R_{\mathfrak{p}}$. Hence
$R_{\mathfrak{p}}/(I^{[p^{h}]}:a^{p^{h}})R_{\mathfrak{p}}=0$ for all
$\mathfrak{p}\in\mathrm{Min}(R/(I:a))$ with $\dim R/\mathfrak{p}=s$. Thus
$\dim R/(I^{[p^{h}]}:a^{p^{h}})\leq s-1$. The claim now follows from the
inductive hypothesis for $I^{[p^{h}]}$ and $a^{p^{h}}$.

On the other hand $s\leq d-t$, so we always have $\dim
R/(I^{[p^{(d-t)h}]}:a^{p^{(d-t)h}})\leq 0$. Moreover, if $\dim R/(I:a)\leq 0$,
then $\bar{a}=a+I\in
I^{F}\cap(I:\mathfrak{m}^{\infty})/I=0^{F_{R}}_{H_{\mathfrak{m}}^{0}(R/I)}$ by
Lemma 2.8. By Proposition 3.3 we have $F^{c}_{R}(\bar{a})=0$. Thus
$a^{p^{c}}\in I^{[p^{c}]}$. Putting all together we have $a^{p^{(d-t)h+c}}\in
I^{[p^{(d-t)h+c}]}$ for all $a\in I^{F}$, so $\mathrm{Fte}(I)\leq(d-t)h+c$.
The proof is complete. ∎

######  Acknowledgement .

The authors are grateful to the referee for carefully reading of the paper and
valuable suggestions and comments. The first author was funded by Vingroup
Joint Stock Company and supported by the Domestic Master/ PhD Scholarship
Programme of Vingroup Innovation Foundation (VINIF), Vingroup Big Data
Institute (VINBIGDATA).

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