# Simulation of a Solar Jet Formed from an Untwisting Flux Rope
Interacting with a Null Point

Jiahao Zhu  School of Astronomy and Space Science and Key Laboratory for
Modern Astronomy and Astrophysics, Nanjing University, Nanjing 210023,
People’s Republic of China<EMAIL_ADDRESS>Yang Guo  School of Astronomy
and Space Science and Key Laboratory for Modern Astronomy and Astrophysics,
Nanjing University, Nanjing 210023, People’s Republic of China;
<EMAIL_ADDRESS>Mingde Ding  School of Astronomy and Space Science and Key
Laboratory for Modern Astronomy and Astrophysics, Nanjing University, Nanjing
210023, People’s Republic of China<EMAIL_ADDRESS>Brigitte Schmieder
LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université,
Université de Paris, 5 place Jules Janssen, 92190 Meudon, France  Centre for
Mathematical Plasma Astrophysics, Dept. of Mathematics, KU Leuven, 3001
Leuven, Belgium  University of Glasgow, School of Physics and Astronomy,
Glasgow, G128QQ, Scotland

###### Abstract

Coronal jets are eruptions identified by a collimated, sometimes twisted
spire. They are small-scale energetic events compared with flares. Using
multi-wavelength observations from the Solar Dynamics Observatory/Atmospheric
Imaging Assembly (SDO/AIA) and a magnetogram from Hinode/Spectro-Polarimeter
(Hinode/SP), we study the formation and evolution of a jet occurring on 2019
March 22 in the active region NOAA 12736. A zero-$\beta$ magnetohydrodynamic
(MHD) simulation is conducted to probe the initiation mechanisms and
appearance of helical motion during this jet event. As the simulation reveals,
there are two pairs of field lines at the jet base, indicating two distinct
magnetic structures. One structure outlines a flux rope lying low above the
photosphere in the north of a bald patch region and the other structure shows
a null point high in the corona in the south. The untwisting motions of the
observed flux rope was recovered by adding an anomalous (artificial)
resistivity in the simulation. A reconnection occurs at the bald patch in the
flux rope structure, which is moving upwards and simultaneously encounters the
field lines of the null point structure. The interaction of the two structures
results in the jet while the twist of the flux rope is transferred to the jet
by the reconnected field lines. The rotational motion of the flux rope is
proposed to be an underlying trigger of this process and responsible for
helical motions in the jet spire.

Solar magnetic fields(1503); Solar magnetic reconnection(1504);
Magnetohydrodynamics(1964); Solar activity(1475); Solar active region(1974)

##  1 Introduction

Coronal jets are one of the most common phenomena in the solar atmosphere.
They are typically collimated beams with helical motions in a transient
process. Modern observations of jets have already been taken through a wide
range of wavelengths from X-ray to extreme ultraviolet (EUV) (Shibata et al.,
1992; Cirtain et al., 2007; Patsourakos et al., 2008). Observations with high
spatio-temporal resolutions from the Atmospheric Imaging Assembly (AIA; Lemen
et al., 2012) on board the Solar Dynamics Observatory (SDO; Pesnell et al.,
2012) and simultaneous spectral observations from the Interface Region Imaging
Spectrograph (IRIS; De Pontieu et al., 2014) provide an opportunity to study
the origin and development of jets (Joshi et al., 2020a; Yang et al., 2020).
Although the origins and morphology of jets have been investigated for
decades, the results are still inconclusive (Liu et al., 2011a; Shibata et
al., 1994; Raouafi et al., 2016; Schmieder, 2022). Usually jets can be
classified into standard jets, which fit the standard reconnection picture,
and blowout jets, analogous to a miniature version of blowout explosion at
their bases (Moore et al., 2010). Coronal jets extend in length from tens of
Mm to about $400$ Mm and possess a speed from about $100$ km s-1 to $600$ km
s-1 (Nisticò et al., 2009; Paraschiv et al., 2010). They are not isolated
phenomena and are believed to be closely related to other activities like
coronal mass ejections, particle acceleration in the solar wind, and even to
coronal heating and so on (Nitta et al., 2006; Yu et al., 2014; Joshi et al.,
2020b). There are still many questions to be solved for solar coronal jets.

Magnetic reconnection is assumed to be responsible for explosive jet events in
the corona. Fan-spine structure is composed of a fan-like dome with a null
point connecting an outer spine to a remote region (Parnell et al., 1996;
Priest & Titov, 1996). The reconnection can easily occur at a null point and
transfer the plasma to the far end of the spine. When reconnection takes place
at a null point, energy stored in the magnetic field is continuously converted
into the kinetic energy and thermal energy of plasma. And then triggered by a
breakout structure, an intensive explosion or rapid motion of plasma can be
observed in the corona. In terms of the magnetic topology, fan-spine
structures (Liu et al., 2011b), eruption of magnetic flux ropes (Adams et al.,
2014; Zhu et al., 2017) and bald patches with a continuous build-up of
electric current (Guo et al., 2013; Schmieder et al., 2013) may easily trigger
jet events. Bald patch is a region where the magnetic field lines are
tangential to the photosphere and the direction of field lines points from
negative polarity to positive polarity (Titov et al., 1993). The magnetic
configuration of parasitic polarities situated in a large opposite polarity
tends to generate such topologies. In some studies, successive jets in coronal
holes exhibit a homologous behaviour of recurrence, resulting from impulsive
reconnection between closed and open field lines (Pariat et al., 2010).

Although spectral and EUV images have been widely used in the analysis of the
jet formation and evolution, the lack of simulation is still a restriction on
exploring the nature of jets (Schmieder, 2022). Recently, state-of-the-art
magnetohydrodynamic (MHD) models with high performance computing techniques
help to understand the origin of jets. Fang et al. (2014) simulated the
process of a twisted flux rope emerging in an open magnetic field and found
that magnetic reconnection is responsible for the explosive events. González-
Avilés et al. (2020) performed simulations of different parameters to probe
the impacts on the origin and morphology of the jets. They found the
significance of the resistivity and thermal conductivity in constructing the
jets. Wyper et al. (2019) simulated a helical jet caused by a sigmoidal flux
rope eruption, with artificial surface motions injecting free energy into the
active region. This model exhibits a coupled mechanism of the breakout
reconnection and ideal MHD instability in the process of jet eruption.
However, most of the simulations about jets mentioned above are ideal MHD
models controlled by artificial settings. Simulations utilizing magnetograms
from observations have not yet been widely studied. A force-free field is
widely used to study the stable magnetic configuration before eruptions
(Wiegelmann et al., 2006; Wiegelmann & Sakurai, 2021). Such fields can be
described by $\left(\nabla\times B\right)\times B=0$ and it becomes
$\nabla\times B=\alpha\left(r\right)B$ if considering the divergence-free
condition. When the torsional parameter $\alpha\left(r\right)$ equals to zero
or is a constant, the field is a potential field or a linear force-free field,
which can be solved in such particular situations. To further explore the
actual physical conditions, a non-linear force-free field (NLFFF) is solved
where $\alpha\left(r\right)$ varies with position. Although no general
analytic solutions exist because of the nonlinearity of the equation, various
algorithms have been proposed and implemented to solve this issue (Schrijver
et al., 2006; Metcalf et al., 2008). In this aspect, an NLFFF extrapolated
from the observed magnetogram and the data-driven simulation based on such a
field further help us to understand the magnetic structures of the active
region and nature of flux rope eruptions (Guo et al., 2019) including some
complex solar eruptions (Jiang et al., 2022). Jiang et al. (2016) simulated a
coronal magnetic field which transformed from the pre-eruptive to eruptive
state following a long-duration quasi-static evolution, and found the
establishment of a positive feedback between emerging flux and external
magnetic reconnection. Therefore, a promising method is to perform the data-
constrained or data-driven simulation that can reproduce the jet eruption with
helical motions as observed.

Recently, statistical studies on the origin of coronal jets draw a conclusion
that mini-filament eruptions play a critical role in triggering both standard
and blowout jets (Sterling et al., 2022; Baikie et al., 2022). The rotational
motion of a mini-filament along with its upward motion to the top of an arch
system can result in magnetic reconnection. The reconnection between the inner
and ambient field lines decreases the bound force imposed on the mini-filament
and thus leads to its eruption. The jet followed with the mini-filament
eruption shows a mixture of both cold and hot material and exhibits a helical
motion, often seen in coronal mass ejections (Sheeley & Wang, 2007) and active
region flares (Zhang et al., 2022). These characteristics suggest that the
relations between solar eruptions at different scales show quite similar
features. Besides, ideal experiments of magnetic reconnection have also been
performed. Farid et al. (2022) aimed to check the presence of a mini-filament.
They used the flux-rope insertion method to create an energized NLFFF model of
coronal jets and identified the consequent eruption as an untwisting jet.
Pariat et al. (2015) demonstrated that for getting a helical jet, it is a
primary requirement to store sufficient magnetic free energy and impulsively
release them. They thought the pre-eruption magnetic configuration might not
be so important. Wyper et al. (2017) on the other hand proposed that the
magnetic breakout is a universal model for solar eruptions including coronal
jets. However, current simulations can not spontaneously produce a mini-
filament structure directly from observed magnetograms.

In this paper, we analyze the motion of a jet through multi-wavelength
observations from AIA and IRIS. Then we simulate the active region with a
zero-$\beta$ MHD simulation using the Message Passing Interface Adaptive Mesh
Refinement Versatile Advection Code (MPI-AMRVAC; Keppens et al., 2003, 2012;
Porth et al., 2014; Xia et al., 2018) and find the structure of a null point
and a flux rope at the jet base. In Section 2, we show the observations of the
jet and depict its kinematic evolution. The extrapolation and simulation
methods are also described. Section 3 shows the results and Section 4 presents
a summary and discussion about our simulation. We show the importance of the
twisting flux rope in the reconnection of this jet event.

##  2 Observations and Methods

###  2.1 Instruments

SDO/AIA provides regular full-disk EUV observations at several different
wavelengths with a spatial resolution of $0.6^{\prime\prime}$ and a temporal
cadence of $12$ s. The AIA channels respond to a wide range of temperature,
(0.1–10 MK), and can be used to explore solar features over a wide range of
scales.

IRIS is widely used to detect the transition region between the chromosphere
and corona. It switches to different raster modes and obtains spectral data
with different resolutions. Also, it can provide high-resolution slit-jaw
images (SJIs) in several wavebands such as C II 1330 Å, as a supplement to the
SDO/AIA observations. The C II 1330 Å SJIs cover a specific region with a
field of view of $60^{\prime\prime}\times 68^{\prime\prime}$ and a pixel size
of $0.16^{\prime\prime}\times 0.32^{\prime\prime}$. We also adopt simultaneous
spectroscopic observations right above the reconnection site to analyze the
Doppler shift in this region. We use the Mg II k lines in this work, which
forms at chromospheric temperatures, and the SJI 1330 for the comparison of
the structures of AIA.

The magnetogram used in the magnetic field extrapolation and numerical
simulation comes from the Spectro-Polarimeter (SP; Lites et al., 2013) of the
Solar Optical Telescope (SOT; Suematsu et al., 2008; Shimizu et al., 2008;
Tsuneta et al., 2008; Ichimoto et al., 2008) on board Hinode (Kosugi et al.,
2007). It was obtained by scanning this region step-by-step along the
east–west direction in a fixed band of wavelengths centered on the Zeeman-
sensitive Fe I lines at 6302 Å. The observation started at 01:02:05 UT and
ended at 01:59:30 UT, right before the jet event which started at around 02:03
UT. The whole magnetogram covers a field of view of $270^{\prime\prime}\times
164^{\prime\prime}$ with a pixel size of $0.32^{\prime\prime}$ along the slit
and a scanning step size of $0.3^{\prime\prime}$. To compensate the stretch
caused by the long-time scanning, we scale its spatial resolution with the
assistance of the magnetogram from the Helioseismic and Magnetic Imager (HMI;
Scherrer et al., 2012; Schou et al., 2012) on board SDO by comparing the main
common features in the magnetograms, which have already been corrected for
projection effects. The pixel size of the HMI magnetogram is
$0.5^{\prime\prime}$. The position information between the instruments we used
is shown in Figure 1.

###  2.2 Observations

The jet event started at around 02:03 UT on 2019 March 22 in the active region
(AR) 12736 and lasted for several minutes. This event has been intensively
studied from an observational point of view (Yang et al., 2020; Joshi et al.,
2020a, 2021b, 2021a; Schmieder, 2022; Schmieder et al., 2022). Multi-waveband
observations from SDO/AIA recorded this intense eruption in all its wavebands.
We select four EUV images and one UV image provided by SDO/AIA as shown in
Figure 2, which reflect different temperatures in the chromosphere and corona.
At the EUV brightness peak time around 02:06 UT, the helical motion of the jet
around its axis can be recognized in most of the wavebands through the bright
and dark patterns. Meanwhile, IRIS observed the active region and one of its
slit position was just above the reconnection site of the jet base at the
early eruption phase. Figure 2 shows some observational features, such as the
shape of the jet base. An X-shaped reconnection site and a flux rope can also
be inferred by the heated brightening structures as shown in Figure 2. SOT/SP
on board Hinode obtained magnetic field on the photosphere from 01:02 UT to
01:59 UT by scanning this region with a spectrograph. A potential field can be
extrapolated when the magnetic field is in a stable or quasi-stable state.
Usually it is valid doing so before eruptions occur. As the magnetogram shows,
this magnetic field possesses a major positive polarity at northeast and a
major negative polarity at southwest, and several parasitic negative
polarities around the positive one. A fan-spine structure may easily form in
such a magnetic configuration around those parasitic polarities. In this case,
the fan lies above main polarities in the east and the outer spine connects to
the far side in the west region.

We focus on the small field of view of the jet base using the SDO/AIA full-
disk observations. To study the propagation of the twisting jet, we select a
slit along the direction of the jet indicated as the solid white line in
Figure 3(a) and present the time-distance map of the slit in Figure 3(b). The
figure shows the evolution of the jet spire, a bulk motion of bright plasma
with a velocity of $\sim 300$ km s-1 followed by a motion of dark structure at
around 02:04 UT. This suggests that the hot outflows set in through magnetic
reconnection at first, followed by the eruption of the cold material, which
may come from the flux rope in the jet base, and move along the reconnected
field lines. The curved pattern in Figure 3(b) also shows the helical motion
of the jet, indicating that a twist structure, probably a flux rope, may exist
previously at the jet base. To further look into the helical motion of the
jet, we draw five equi-distanced cuts perpendicular to the slit in Figure
3(a). Figure 4 displays the time-distance maps of the five cuts. The eruptive
motion of the bright and dark patterns shown by slices in two perpendicular
directions in both Figure 3(b) and Figure 4, as well as the recurrent features
labeled by dashed white lines in Figure 4, exhibit the helical motion of the
jet. The panels of Figure 4 in sequence also shows the separation between
these two patterns. The dark wave feature in the last panel of Figure 4, which
is sliced from the transverse direction, clearly shows an oscillatory motion
in the eruption.

IRIS performed spectral observations with a 4-step raster for the region
covering the reconnection site during the whole process of the jet event. At
the time of the jet eruption, the profile of the Mg II lines shows a broad
extension in both the red and blue wings, which indicates complex material
motions in the transition region. The presence of complex motions has been
separately verified by Joshi et al. (2020a). We adopt the moment method to
calculate the average Doppler velocity regardless of the irregular profile. We
mainly analyze the Mg II k lines of the first slit position, which is located
just above the reconnection site during the peak time. We thus obtain the
Doppler velocity from observations around the peak time and then compare it
with the numerical values calculated from the data-constrained simulation in
Section 3.

###  2.3 Magnetic field extrapolation and data-constrained simulation

Based on the vector magnetic field obtained from SOT/SP, we make an NLFFF
extrapolation to study the magnetic topology structure in AR 12736. To further
investigate the relationship between the magnetic structure and observational
morphology, we scale the spatial resolution of the SOT/SP magnetogram by
comparing it with the SDO/HMI magnetogram that has already been corrected for
the projection effects. Meanwhile, we resize the original data to reduce the
spatial resolution by twice for relieving the computational burden. The NLFFF
is reconstructed from the magneto-frictional module implemented by Guo et al.
(2016a, b) in MPI-AMRVAC. It can relax the potential field together with the
bottom boundary to NLFFF in the computation domain. The method can be applied
to multiple situations, such as Cartesian or spherical coordinates with
uniform grids or adaptive mesh refinement grids. We note that it is hard to
attain a completely force-free field relaxed from an observed magnetogram,
because the Lorentz force deduced from observations is not nicely balanced in
the photosphere due to the role of gravity (Zhu et al., 2016). So, the NLFFF
might not be in a complete equilibrium state initially.

Figures 5(a) and 5(b) show the vertical magnetic field and the distribution of
current density magnitude in the extrapolated field 1.2 Mm above the
photosphere. We zoom in the region of interest showing a current density iso-
surface of 0.03 A m-2 and the distribution of $J_{z}$ in Figures 5(c) and
5(d), respectively. The concentrated current reveals a strong magnetic flux
rope structure in the vicinity of the small parasitic negative polarities in
the north. It is the same structure detected in the HMI vector magnetograms by
Joshi et al. (2020a). And the elongated double-peaked $J_{z}$ pattern with
hooks confirms the existence of the flux rope (Aulanier & Dudík, 2019; Joshi
et al., 2020a). Figure 6(a) displays two magnetic structures in the
extrapolated field, a flux rope and a null point. Figure 6(b) shows the flux
rope lying above a bald patch region low in the photosphere. By tracing the
magnetic field lines of the flux rope, we find that they are actually a part
of the bald patch field lines. Figure 6(c) shows the null point in the south
above the flux rope. The X-shaped reconnection site is reproduced with the
spine connecting to a remote region in the west.

To probe the nature of the jet, we conduct a data-constrained MHD simulation
based on the result of the NLFFF extrapolation. The model we adopt is the same
as that in Guo et al. (2019), which solves the zero-$\beta$ MHD equations. For
the boundary conditions, we select the data-constrained case, which fixes the
bottom boundary, rather than the data-driven case, because of the limitation
of the scanned magnetogram from SOT/SP. The MHD equations are solved in the
dimensionless forms in MPI-AMRVAC. In this model, the gravity and gas pressure
gradient are omitted in the momentum equation, and the energy equation
including the heat conduction term and others is dropped entirely. In other
words, we only solve the continuity equation, momentum equation and magnetic
induction equation to get the density, velocity, and the magnetic field.

The initial condition is set to be the relaxed NLFFF and the initial density
distribution is calculated by assuming a temperature distribution of a
stepwise function of height. We set the computation grid to be $288\times
200\times 200$, namely $166\times 85\times 85$ Mm3 in real scale with the
resolution to be $0.58\times 0.42\times 0.42$ Mm3. Also, an anomalous
resistivity is added in this model to facilitate the occurrence of magnetic
reconnection. When the current density $J$ of a local point is higher than the
critical value $J_{c}$, the resistivity $\eta$ is set to be
$\eta=\eta_{0}\times({J-J_{c}})^{2}/{J_{c}}^{2}$ and otherwise to be zero.
Here, we set $\eta_{0}$ and $J_{c}$ to be $5\times 10^{-4}$ and $1000$ in
normalized units, respectively. The whole simulation time is about half an
hour. We output the snapshots of the simulation results every 24 s for further
comparisons with the AIA and IRIS observations.

##  3 Results

In the NLFFF extrapolation, we recognize a flux rope low in the photosphere
related to the north parasitic polarities, and an X-shaped current sheet high
in the corona in a south region between the two main polarities, which is also
the reconnection site as shown in Figure 6. At the early time of the
simulation, the eastern footpoint of the flux rope slowly drifts close to the
positive polarity in its southeast. Meanwhile, the main body of the flux rope
rotates around its axis counterclockwise and approaches the reconnection site
located high in the south region. The rotation slowly changes the magnetic
configuration and the kinetic energy of plasma is accumulated through the
continuous reconnection. When the field lines of the flux rope encounter the
X-shaped current sheet, they join the reconnection process with other field
lines within the sheet. Then, the explosive burst takes place and the jet with
helical motion comes out of the null point along the spine connecting to the
faraway side in the major negative polarity. At last, the field lines shrink
to a low magnetic arch structure after the burst. Thus, we can detect the hot
reconnection outflows and later the cold material from the flux rope,
corresponding to the bright and dark patterns in AIA observations,
respectively. The general evolution is revealed by four typical moments shown
in Figure 7.

To compare the simulation with the observations of SDO/AIA and IRIS, we
project the simulation to the line of sight and superpose it on the AIA
observations in Figure 8. Since the magnetogram from SOT/SP is scanned through
a period of time, we could not give an exact time that the simulation starts
from. Thus we use T0 to represent the start time and focus on the time
evolution of simulations and observations. The simulation matches the
observations quite well in the perspective of the triangular shape at the
base. The field lines of the flux rope fit with the anemone-like arcade and
the field lines at the null point display a triangular shape like the jet.
Note that only magnetic field lines are displayed in Figure 8, and the plasma
is assumed to move along the field lines. We find that the time-sequential
images match roughly synchronously with the real-time evolution revealed in
the observations. All these results indicate that the simulation reproduces
the observations to a satisfactory degree.

We further extract the velocity and density information from the simulation
snapshots. And then we analyze the Doppler velocity along the line of sight
and compare it with that from observations. The results are shown in Figure 9.
The Doppler velocity from the simulation is derived from the Gaussian fitting
based on the distribution function of density squared versus velocity along
the line of sight as shown by each white line in Figure 9(b). Note that every
white line is perpendicular to the IRIS slit and can be projected to one point
on the slit. Then we can obtain the velocity distribution from simulation at
these points along the slit. We discard the extremely large values compared
with the maximum absolute value from observations and further make a spatial
interpolation to the interval of observations. This is intended for
calculating the cross-correlation between Doppler velocities from observations
and simulations. Because of the lack of a lot of information about specific
matters in the zero-$\beta$ simulation, we take such an averaged method in
analyzing the Doppler velocities. For the observational result at a selected
time, we compare it with each simulation snapshot. And the best result with a
correlation coefficient of 0.71 is shown in Figure 9(e). The time difference
between the observation and simulation is about 6.5 minutes for this best
matched case, which is acceptable considering the simplified MHD equations
used in the simulation.

This jet event has already been studied by two different groups, as it was
mentioned in Section 2. Different opinions were proposed regarding the trigger
mechanism of the jet event. Yang et al. (2020) did an NLFFF extrapolation to
derive the main magnetic structure of this event. They confirmed a fan-spine
magnetic topology and found two flux ropes in the extrapolated results. Also,
they found a null point responsible for the reconnection site, which is the
same as that we find in this work. In their opinion, the breakout-type
reconnection caused by one of the flux ropes leads to the eruption of the jet.
By contrast, Joshi et al. (2020a) checked the magnetogram evolution prior to
the occurrence of the event and analyzed the vector magnetic field map in
comparison with an ideal MHD simulation. They found a flux rope along the
polarity inversion line in the north and the flux rope migrates towards the
south due to the photospheric motions, leading to the formation of a small
parasitic bipole. And the rotational motion of the flux rope was verified by
them through analysis on the IRIS spectra in their Figure 8(b). Though not so
obvious, the rotational characteristics can also be exhibited in our Figure
9(d). A strong extended bidirectional shifts at around $y=75$ and from $y=75$
blue shifts fade along the $y$ axis, while red shifts fade reverse the $y$
axis. They found that the bipole is also a bald patch region, above which a
current sheet forms. Thus the twist of the flux rope is transferred into the
jet through reconnection without direct eruptions. These two studies come into
a consensus that the helical motion of the jet comes from the pre-existing
twist in the flux rope. Nevertheless, the critical difference of these two
studies lies in how the jet erupts, namely, how is the twist transferred into
the jet. Such a difference could be ascribed to the different methods used in
these two studies, since the study by Yang et al. (2020) is based on the
magnetic field extrapolation together with a high-resolution observation from
NVST, while that of Joshi et al. (2020a) focuses on the evolution of the
magnetogram and comparisons with ideal MHD simulations. Neither of them
performed dynamic simulations about this active region. Thus, we make a
zero-$\beta$ MHD simulation in this work to probe how the jet is triggered.
The Doppler velocity of our simulation is in accordance with IRIS observations
from the aspect of correlation analysis. Specifically, our simulation shows
that the flux rope lies low at the bald patch and the null point is located
high in the corona. However, they are not far from each other when observed
from the viewing angle of the spacecraft. In that sense, it is hard to
distinguish their distinct features only according to the spectrum that is
actually an integration along the line of sight. Therefore, pure analysis of
observations might not lead to a solid conclusion. The combination of MHD
simulations and observations is needed for learning the physical nature of
solar eruptive events.

##  4 Summary and Discussion

With the simultaneous observations from AIA and IRIS, we analyze the evolution
of the jet in AR 12736 on 2019 March 22. We draw a time-slice map for the slit
along the jet axis and find that the jet is twisting around its axis as
propagating outwards at a speed of $\sim$ 300 km s-1. The jet erupts with hot
matter followed by cold matter. We suppose that they correspond to the hot
reconnection outflows through a null point and the cold material from the flux
rope. The helical motion comes from the untwisting of the flux rope.
Furthermore, we utilize an NLFFF to check the topological structure of the
magnetic field with the photospheric vector magnetic field from SOT/SP. A null
point structure in the south and a flux rope in the north of a bald patch
region are found to exist in the active region. Then, we adopt a zero-$\beta$
MHD simulation based on the results of the NLFFF extrapolation. It reveals
that the jet may be produced by the magnetic reconnection at the null point.
With the untwisting motion of the flux rope and slipping motion of its
footpoint, the inner field lines of the flux rope encounter and then reconnect
with the field lines originally situated at the null point. The reconnection
outflows propagate along the spine line connected to the major negative
polarity in the west region, which correspond to the bright and dark patterns
in the AIA images.

We experiment with both the SDO/HMI magnetogram and SOT/SP magnetogram and the
reconstruction of the NLFFF for further MHD simulations. An X-shaped current
sheet was found in both cases; however, the flux rope can only be found in the
SOT/SP case. Although we adopt the same magnetogram as that used in Yang et
al. (2020), the magnetic structures obtained are different in the two cases.
We attribute the different results to the usage of different extrapolation
methods. Moreover, in our trials without extra anomalous resistivity, the
rotational movement of the flux rope near the photosphere cannot be reproduced
in the simulation. The resistivity is artificially added in the regions where
the current density is large and thus the flux rope erupts with high
resistivity and reveals untwisting motions. This implies that the rotational
untwisting motion of the flux rope at the beginning comes from the
reconnection in the flux rope. Furthermore, the ideal MHD simulations
conducted by Wyper et al. (2018) show that minority-polarity intrusions may
add free energy into the field and can lead to both intermittent low-level
reconnection and explosive, high-energy-release reconnection above these
regions. Though the moving of the parasitic polarity is not considered in our
data-constrained simulation, this may partly account for the impacts of the
resistivity added in our experiments.

Our MHD simulation does not contain heat conduction and radiation process.
Therefore, the physical parameters we obtain in the numerical simulations are
not quantitatively accurate. Recently, Zhou et al. (2022) conducted an MHD
simulation of a sheared arcade configuration, in which a magnetic flux rope is
formed and erupts through reconnection. The flux rope in their simulation
rotates against its twist and transfers the twist into the surrounding field
lines, which resembles our results. However, further studies are needed for
clarifying the mechanism of the rotation of the flux rope leading to the
reconnection between the flux rope and the spine structure connected to the
remote side of the active region. In particular, we should focus on the role
of the anomalous resistivity in driving the reconnection in the inner part of
the flux rope and leading to its rotational motion around its axis and its
upward motion in the chromosphere. We should also improve the data-constrained
simulations, such as adding some specific magnetic structures and photospheric
motions in order to fit the observations more closely. On the other hand, we
also aim to carry out simulations with full MHD equations including heat
conduction and radiative cooling to learn the physics more accurately in such
jet events.

J.H.Z., Y.G. and M.D.D. are supported by the National Key Research and
Development Program of China (2022YFF0503004, 2021YFA1600504, and
2020YFC2201201) and NSFC (11773016 and 11733003). We thank the two anonymous
referees very much for their constructive suggestions. We deeply appreciate
the free access to the magnetogram and observation data provided by SDO/AIA,
SDO/HMI, SOT/SP and IRIS science teams. J.H.Z. thanks Ye Qiu and Ze Zhong for
their help in the work. Computational resources were provided by the High
Performance Computing Center (HPCC) at Nanjing University.

## References

  * Adams et al. (2014) Adams, M., Sterling, A. C., Moore, R. L., & Gary, G. A. 2014, ApJ, 783, 11, doi: 10.1088/0004-637X/783/1/11
  * Aulanier & Dudík (2019) Aulanier, G., & Dudík, J. 2019, A&A, 621, A72, doi: 10.1051/0004-6361/201834221
  * Baikie et al. (2022) Baikie, T. K., Sterling, A. C., Moore, R. L., et al. 2022, ApJ, 927, 79, doi: 10.3847/1538-4357/ac473e
  * Cirtain et al. (2007) Cirtain, J. W., Golub, L., Lundquist, L., et al. 2007, Science, 318, 1580, doi: 10.1126/science.1147050
  * De Pontieu et al. (2014) De Pontieu, B., Title, A. M., Lemen, J. R., et al. 2014, Sol. Phys., 289, 2733, doi: 10.1007/s11207-014-0485-y
  * Fang et al. (2014) Fang, F., Fan, Y., & McIntosh, S. W. 2014, ApJ, 789, L19, doi: 10.1088/2041-8205/789/1/L19
  * Farid et al. (2022) Farid, S. I., Savcheva, A., Tassav, S., & Reeves, K. K. 2022, ApJ, 938, 150, doi: 10.3847/1538-4357/ac8c2e
  * González-Avilés et al. (2020) González-Avilés, J. J., Guzmán, F. S., Fedun, V., & Verth, G. 2020, ApJ, 897, 153, doi: 10.3847/1538-4357/ab97b8
  * Guo et al. (2013) Guo, Y., Démoulin, P., Schmieder, B., et al. 2013, A&A, 555, A19, doi: 10.1051/0004-6361/201321229
  * Guo et al. (2016a) Guo, Y., Xia, C., & Keppens, R. 2016a, ApJ, 828, 83, doi: 10.3847/0004-637X/828/2/83
  * Guo et al. (2019) Guo, Y., Xia, C., Keppens, R., Ding, M. D., & Chen, P. F. 2019, ApJ, 870, L21, doi: 10.3847/2041-8213/aafabf
  * Guo et al. (2016b) Guo, Y., Xia, C., Keppens, R., & Valori, G. 2016b, ApJ, 828, 82, doi: 10.3847/0004-637X/828/2/82
  * Ichimoto et al. (2008) Ichimoto, K., Lites, B., Elmore, D., et al. 2008, Sol. Phys., 249, 233, doi: 10.1007/s11207-008-9169-9
  * Jiang et al. (2022) Jiang, C., Feng, X., Guo, Y., & Hu, Q. 2022, The Innovation, 3, 100236, doi: 10.1016/j.xinn.2022.100236
  * Jiang et al. (2016) Jiang, C., Wu, S. T., Feng, X., & Hu, Q. 2016, Nature Communications, 7, 11522, doi: 10.1038/ncomms11522
  * Joshi et al. (2020a) Joshi, R., Schmieder, B., Aulanier, G., Bommier, V., & Chandra, R. 2020a, A&A, 642, A169, doi: 10.1051/0004-6361/202038562
  * Joshi et al. (2021a) Joshi, R., Schmieder, B., Heinzel, P., et al. 2021a, A&A, 654, A31, doi: 10.1051/0004-6361/202141172
  * Joshi et al. (2021b) Joshi, R., Schmieder, B., Tei, A., et al. 2021b, A&A, 645, A80, doi: 10.1051/0004-6361/202039229
  * Joshi et al. (2020b) Joshi, R., Wang, Y., Chandra, R., et al. 2020b, ApJ, 901, 94, doi: 10.3847/1538-4357/abaf5a
  * Keppens et al. (2012) Keppens, R., Meliani, Z., van Marle, A. J., et al. 2012, Journal of Computational Physics, 231, 718, doi: 10.1016/j.jcp.2011.01.020
  * Keppens et al. (2003) Keppens, R., Nool, M., Tóth, G., & Goedbloed, J. P. 2003, Computer Physics Communications, 153, 317, doi: 10.1016/S0010-4655(03)00139-5
  * Kosugi et al. (2007) Kosugi, T., Matsuzaki, K., Sakao, T., et al. 2007, Sol. Phys., 243, 3, doi: 10.1007/s11207-007-9014-6
  * Lemen et al. (2012) Lemen, J. R., Title, A. M., Akin, D. J., et al. 2012, Sol. Phys., 275, 17, doi: 10.1007/s11207-011-9776-8
  * Lites et al. (2013) Lites, B. W., Akin, D. L., Card, G., et al. 2013, Sol. Phys., 283, 579, doi: 10.1007/s11207-012-0206-3
  * Liu et al. (2011a) Liu, C., Deng, N., Liu, R., et al. 2011a, ApJ, 735, L18, doi: 10.1088/2041-8205/735/1/L18
  * Liu et al. (2011b) Liu, W., Berger, T. E., Title, A. M., Tarbell, T. D., & Low, B. C. 2011b, ApJ, 728, 103, doi: 10.1088/0004-637X/728/2/103
  * Metcalf et al. (2008) Metcalf, T. R., De Rosa, M. L., Schrijver, C. J., et al. 2008, Sol. Phys., 247, 269, doi: 10.1007/s11207-007-9110-7
  * Moore et al. (2010) Moore, R. L., Cirtain, J. W., Sterling, A. C., & Falconer, D. A. 2010, ApJ, 720, 757, doi: 10.1088/0004-637X/720/1/757
  * Nisticò et al. (2009) Nisticò, G., Bothmer, V., Patsourakos, S., & Zimbardo, G. 2009, Sol. Phys., 259, 87, doi: 10.1007/s11207-009-9424-8
  * Nitta et al. (2006) Nitta, N. V., Reames, D. V., De Rosa, M. L., et al. 2006, ApJ, 650, 438, doi: 10.1086/507442
  * Paraschiv et al. (2010) Paraschiv, A. R., Lacatus, D. A., Badescu, T., et al. 2010, Sol. Phys., 264, 365, doi: 10.1007/s11207-010-9584-6
  * Pariat et al. (2010) Pariat, E., Antiochos, S. K., & DeVore, C. R. 2010, ApJ, 714, 1762, doi: 10.1088/0004-637X/714/2/1762
  * Pariat et al. (2015) Pariat, E., Dalmasse, K., DeVore, C. R., Antiochos, S. K., & Karpen, J. T. 2015, A&A, 573, A130, doi: 10.1051/0004-6361/201424209
  * Parnell et al. (1996) Parnell, C. E., Smith, J. M., Neukirch, T., & Priest, E. R. 1996, Physics of Plasmas, 3, 759, doi: 10.1063/1.871810
  * Patsourakos et al. (2008) Patsourakos, S., Pariat, E., Vourlidas, A., Antiochos, S. K., & Wuelser, J. P. 2008, ApJ, 680, L73, doi: 10.1086/589769
  * Pesnell et al. (2012) Pesnell, W. D., Thompson, B. J., & Chamberlin, P. C. 2012, Sol. Phys., 275, 3, doi: 10.1007/s11207-011-9841-3
  * Porth et al. (2014) Porth, O., Xia, C., Hendrix, T., Moschou, S. P., & Keppens, R. 2014, ApJS, 214, 4, doi: 10.1088/0067-0049/214/1/4
  * Priest & Titov (1996) Priest, E. R., & Titov, V. S. 1996, Philosophical Transactions of the Royal Society of London Series A, 354, 2951, doi: 10.1098/rsta.1996.0136
  * Raouafi et al. (2016) Raouafi, N. E., Patsourakos, S., Pariat, E., et al. 2016, Space Sci. Rev., 201, 1, doi: 10.1007/s11214-016-0260-5
  * Scherrer et al. (2012) Scherrer, P. H., Schou, J., Bush, R. I., et al. 2012, Sol. Phys., 275, 207, doi: 10.1007/s11207-011-9834-2
  * Schmieder (2022) Schmieder, B. 2022, Frontiers in Astronomy and Space Sciences, 9, 820183, doi: 10.3389/fspas.2022.820183
  * Schmieder et al. (2022) Schmieder, B., Joshi, R., & Chandra, R. 2022, Advances in Space Research, 70, 1580, doi: 10.1016/j.asr.2021.12.013
  * Schmieder et al. (2013) Schmieder, B., Guo, Y., Moreno-Insertis, F., et al. 2013, A&A, 559, A1, doi: 10.1051/0004-6361/201322181
  * Schou et al. (2012) Schou, J., Scherrer, P. H., Bush, R. I., et al. 2012, Sol. Phys., 275, 229, doi: 10.1007/s11207-011-9842-2
  * Schrijver et al. (2006) Schrijver, C. J., De Rosa, M. L., Metcalf, T. R., et al. 2006, Sol. Phys., 235, 161, doi: 10.1007/s11207-006-0068-7
  * Sheeley & Wang (2007) Sheeley, N. R., J., & Wang, Y. M. 2007, ApJ, 655, 1142, doi: 10.1086/510323
  * Shibata et al. (1994) Shibata, K., Nitta, N., Strong, K. T., et al. 1994, ApJ, 431, L51, doi: 10.1086/187470
  * Shibata et al. (1992) Shibata, K., Ishido, Y., Acton, L. W., et al. 1992, PASJ, 44, L173
  * Shimizu et al. (2008) Shimizu, T., Nagata, S., Tsuneta, S., et al. 2008, Sol. Phys., 249, 221, doi: 10.1007/s11207-007-9053-z
  * Sterling et al. (2022) Sterling, A. C., Moore, R. L., & Panesar, N. K. 2022, ApJ, 927, 127, doi: 10.3847/1538-4357/ac473f
  * Suematsu et al. (2008) Suematsu, Y., Tsuneta, S., Ichimoto, K., et al. 2008, Sol. Phys., 249, 197, doi: 10.1007/s11207-008-9129-4
  * Titov et al. (1993) Titov, V. S., Priest, E. R., & Demoulin, P. 1993, A&A, 276, 564
  * Tsuneta et al. (2008) Tsuneta, S., Ichimoto, K., Katsukawa, Y., et al. 2008, Sol. Phys., 249, 167, doi: 10.1007/s11207-008-9174-z
  * Wiegelmann et al. (2006) Wiegelmann, T., Inhester, B., & Sakurai, T. 2006, Sol. Phys., 233, 215, doi: 10.1007/s11207-006-2092-z
  * Wiegelmann & Sakurai (2021) Wiegelmann, T., & Sakurai, T. 2021, Living Reviews in Solar Physics, 18, 1, doi: 10.1007/s41116-020-00027-4
  * Wyper et al. (2017) Wyper, P. F., Antiochos, S. K., & DeVore, C. R. 2017, Nature, 544, 452, doi: 10.1038/nature22050
  * Wyper et al. (2019) Wyper, P. F., DeVore, C. R., & Antiochos, S. K. 2019, MNRAS, 490, 3679, doi: 10.1093/mnras/stz2674
  * Wyper et al. (2018) Wyper, P. F., DeVore, C. R., Karpen, J. T., Antiochos, S. K., & Yeates, A. R. 2018, ApJ, 864, 165, doi: 10.3847/1538-4357/aad9f7
  * Xia et al. (2018) Xia, C., Teunissen, J., El Mellah, I., Chané, E., & Keppens, R. 2018, ApJS, 234, 30, doi: 10.3847/1538-4365/aaa6c8
  * Yang et al. (2020) Yang, S., Zhang, Q., Xu, Z., et al. 2020, ApJ, 898, 101, doi: 10.3847/1538-4357/ab9ac7
  * Yu et al. (2014) Yu, H. S., Jackson, B. V., Buffington, A., et al. 2014, ApJ, 784, 166, doi: 10.1088/0004-637X/784/2/166
  * Zhang et al. (2022) Zhang, Y., Zhang, Q., Dai, J., Li, D., & Ji, H. 2022, Sol. Phys., 297, 138, doi: 10.1007/s11207-022-02072-8
  * Zhou et al. (2022) Zhou, Z., Jiang, C., Liu, R., et al. 2022, ApJ, 927, L14, doi: 10.3847/2041-8213/ac5740
  * Zhu et al. (2017) Zhu, X., Wang, H., Cheng, X., & Huang, C. 2017, ApJ, 844, L20, doi: 10.3847/2041-8213/aa8033
  * Zhu et al. (2016) Zhu, X., Wang, H., Du, Z., & He, H. 2016, ApJ, 826, 51, doi: 10.3847/0004-637X/826/1/51

Figure 1: (a) Overview of the jet event at 02:03 UT on 2019 March 22 by
SDO/AIA 171 Å. (b) Full-disk magnetogram observed by SDO/HMI from line of
sight at 02:00 UT, plotted in limits [-200,200] Gauss to enhance the image
contrast. The white squares in (a) and (b) outline the observation region
scanned by Hinode/SP. (c) The enlarged view of the white square shown in (a)
and the four red lines correspond to the slit positions of IRIS at the
corresponding time. (d) An enlarged view of the HMI magnetogram in the white
square shown in (b), plotted in limits [-1000,1000] Gauss.

The cyan dashed square is similar to that in Figure 5 regardless of projection
effects.

Figure 2: Multi-wavelength observations with (a)–(e) SDO/AIA and with (f) IRIS
of the jet at around 02:04 UT on 2019 March 22. The triangular jet base can be
clearly observed in all these wavelengths. The flux rope can be figured out by
the brightening structure in the north as shown in (a) and the X-shaped
current can be recognized in the south as shown in (f). The fan-spine
structure is outlined in (b), where the dashed line outlines the base shape of
the fan, and the solid lines outline the spine and several field lines
constituting the fan. The blue star in (f) represents the position of the null
point, where the spine connects the fan to the west remote region. The jet
event lasts for several minutes. A movie of the multi-wavelength observations
is attached to this figure. The movie traces the jet base on 2019 March 22
from 01:58 UT to 02:14 UT in all the wavelengths presented. Figure 3: (a)
SDO/AIA 171 Å image at 2:06 UT on 2019 March 22. The long white line
represents the slice used in the time-distance analysis, which is mainly along
the jet propagation direction. The five short slices perpendicular to the long
one are used to reveal the untwisting motion of the flux rope. (b) Time-
distance map of the slice along the jet propagation direction. The length of
this slice is $100^{\prime\prime}$. The bright pattern shows eruption of hot
plasma at a speed of 370 km s-1 and the dark pattern shows eruption of cold
plasma at a speed of 315 km s-1. Figure 4: Time-distance maps of the five
short slices perpendicular to the jet propagation direction. The panels from
top to bottom correspond to the slices 1 to 5 in Figure 3. The length of these
slices is $24^{\prime\prime}$ and the spacing is $10^{\prime\prime}$. The
white dashed lines indicate the helical motion of the jet event perpendicular
to the spire. These maps also reveal a transverse oscillation of the jet and
the mixture of hot and cold material propagating separately. Figure 5: (a)
Vertical magnetic field on the layer of 1.2 Mm above photosphere in the NLFFF
extrapolation. The cyan squares in panels (a) and (b) are also labeled in
Figure 1(d) regardless of projection effects and outline the field of view in
panels (c) and (d) where current mainly distributes. The black rectangles show
the main polarity and parasitic polarity in the region of interest. (b)
Distribution of the current density magnitude $J$ on the same layer. (c) An
iso-surface of 0.03 A m-2 of $J$ overlaid on the current density distribution.
The concentrated current shown by the iso-surface indicates the flux rope
structure. (d) Distribution of $J_{z}$ on the same layer as that in panel (a).
There exists an elongated double-peaked $J_{z}$ pattern, which reveals a long
flux rope structure along the polarity inversion line. The field of view has
been corrected from the projection effects, therefore the structures are more
elongated along the $x$ axis compared to Figure 1. Figure 6: (a) Sample
magnetic field lines in the NLFFF model showing different magnetic structures.
The background shows the vertical magnetic field $B_{z}$. The rectangle region
contains two main structures and is enlarged in panels (b) and (c) with a side
view. The orientations are labeled by the orientation axes in the small left
panels. (b) A flux rope structure colored by yellow field lines lying above a
bald patch near the photosphere in the north region. (c) A null point
structure in the X-shaped current sheet colored by red field lines in the
south region.  Figure 7: Magnetic field evolution in the zero-$\beta$
simulation. (a)–(d) Four moments of the simulation. Red field lines represent
the null point structure and the reconnection site. Cyan and yellow field
lines depict the flux rope structure and gray ones represent the ambient field
lines. (a) The initial magnetic field configuration, which is the same as that
in Figures 6(b) and 6(c). (b)–(c) The flux rope in the north region slowly
moves southwards, rotates counterclockwise and interacts with the field lines
connected to the null point. The reconnection outflow and flux rope material
move along the long spine of null point connected to a remote region in the
west. (d) Magnetic field lines after the rotation of the flux rope and
interactions between the flux rope and X-shaped current. The reconnection
process ends at this moment. A movie of the data-constrained MHD simulation is
attached to this figure. The movie shows the evolution of the simulation in 32
minutes from the local top view. Figure 8: (a)–(c) Simulation results
superposed on the SDO/AIA 171 Å images from the viewing angle of the observing
instruments at three moments. (d)–(f) Simulation results superposed on the
SDO/AIA 304 Å images from the viewing angle of the observing instruments at
three moments. The simulation time is expressed in the format as AIA and a T0
is used to represent the start time of the simulation. The field lines in our
simulation fit well with the arcade and the triangular shape of the jet in
observations. In general, the simulation results show an evolution roughly
synchronously with the observation. A movie of the results is attached to this
figure. The movie shows the evolution of the simulation in 32 minutes
superposed on simultaneous SDO/AIA 171 Å and 304 Å observations from the line-
of-sight view. Figure 9: Doppler velocities derived from the observations and
simulations. (a) Slit position shown in the simulation at a specific time
corresponding to that of observations. (b) Regions of the Gaussian fitting in
the direction of the line of sight. Each white line represents a fitting depth
of 20 Mm at each pixel in the simulation. (c) Slit position of IRIS at the
observation time under analysis. (d) The Mg II k spectrum from IRIS at 02:04
UT. (e) Doppler velocities calculated from both the observations (solid line)
and simulations (dashed line). Here we show the results with the highest
correlation coefficient of 0.71 for the corresponding times shown in the
panel.


# Asymptotic behaviour of integer programming and the v-function of a graded
filtration

Antonino Ficarra, Emanuele Sgroi  Antonino Ficarra, Department of mathematics
and computer sciences, physics and earth sciences, University of Messina,
Viale Ferdinando Stagno d’Alcontres 31, 98166 Messina, Italy
<EMAIL_ADDRESS>Emanuele Sgroi, Department of mathematics and computer
sciences, physics and earth sciences, University of Messina, Viale Ferdinando
Stagno d’Alcontres 31, 98166 Messina, Italy<EMAIL_ADDRESS>
###### Abstract.

The v-function of a graded filtration $\mathcal{I}=\\{I_{[k]}\\}_{k\geq 0}$ is
introduced. Under the assumption that $\mathcal{I}$ is Noetherian, we prove
that the v-function $\textup{v}(I_{[k]})$ is an eventually quasi-linear
function. This result applies to several situations, including ordinary
powers, and integral closures of ordinary powers, among others. As another
application, we investigate the asymptotic behaviour of certain integer
programming problems. Finally, we present the Macaulay2 package VNumber.

###### Key words and phrases:

v-number, primary decomposition, associated primes, monomial ideals

###### 2020 Mathematics Subject Classification:

Primary 13F20; Secondary 13F55, 05C70, 05E40.

## Introduction

Let $R$ be a finitely generated ${\mathbb{N}}$-graded algebra over a
Noetherian ring, and let ${\mathfrak{m}}$ be the maximal graded ideal of $R$.
We suppose that $R/{\mathfrak{m}}$ is an infinite field $K$. Let $I\subset R$
be a homogeneous ideal. Then, for any associated prime
$\mathfrak{p}\in\operatorname{Ass}(I)$, there exists a homogeneous element
$f\in R$ such that $(I:f)=\mathfrak{p}$. Following [11], we define the
$\operatorname{v}_{\mathfrak{p}}$-number of $I$ to be the least degree of such
an element. Whereas, the $\operatorname{v}$-number of $I$ is defined as
$\operatorname{v}(I)=\min_{\mathfrak{p}\in\operatorname{Ass}(I)}\operatorname{v}_{\mathfrak{p}}(I)$.
See also [4, 5, 13, 19, 29, 30, 31].

In [14], if $R$ is the standard graded polynomial ring $S$ over a field $K$
and $I$ is a graded ideal of $S$, we proved that $\operatorname{v}(I^{k})$ is
a linear function for $k\gg 0$, and
$\lim_{k\rightarrow\infty}\operatorname{v}(I^{k})/k$ is equal to the initial
degree of $I$. This result was shown independently by Conca [10] in the more
general frame of any graded ideal $I\subset R$. Further generalizations and
refinements of these results were considered in [16, 17, 26].

In this paper, we generalize the results in [10, 14, 16, 17, 26]. Let
$\mathcal{I}=\\{I_{[k]}\\}_{k\geq 0}$ be a Noetherian graded filtration of $R$
(see Section 1 for the precise definitions). In our main Theorem 1.3 we prove
that $\operatorname{v}(I_{[k]})$ is a quasi-linear function in $k$ for $k\gg
0$. This result applies to several situations. For instance, if
$I_{[k]}=I^{k}$ or $I_{[k]}=\overline{I^{k}}$ for all $k\geq 0$, where
$I\subset R$ is a graded ideal. See [12, Example 5.2] for other classes of
graded filtrations. In the case of a monomial ideal $I$ in the polynomial ring
$S$, the computation of the functions $\operatorname{v}_{\mathfrak{p}}(I^{k})$
and $\operatorname{v}_{\mathfrak{p}}(\overline{I^{k}})$ amounts to solve
certain asymptotic integer programming problems and determine their optimal
solutions. We explore these problem in Section 3. Let
$\operatorname{Ass}^{\infty}(\mathcal{I})$ be the set of those primes
$\mathfrak{p}\in\textup{Spec}(R)$ such that
$\mathfrak{p}\in\operatorname{Ass}(I_{[k]})$ for infinitely many $k$. In
Section 2, using Koszul homology and localization, we characterize the set
$\operatorname{Ass}^{\infty}(\mathcal{I})$. Section 4 contains a self
contained introduction to the Macaulay2 [18] package VNumber [15] and its main
features. We end the paper in Section 5 with a series of natural questions.

##  1\. The $\operatorname{v}$-function of a graded filtration

Let $R$ be a finitely generated ${\mathbb{N}}$-graded algebra over a
Noetherian ring, and let ${\mathfrak{m}}$ be the maximal graded ideal of $R$.
Suppose that $R/{\mathfrak{m}}$ is an infinite field. A graded filtration of
$R$ is a family $\mathcal{I}=\\{I_{[k]}\\}_{k\geq 0}$ of homogeneous ideals of
$R$ satisfying:

  1. (i)

$I_{[0]}=R$,

  2. (ii)

$I_{[k+1]}\subseteq I_{[k]}$ for all $k\geq 0$,

  3. (iii)

and $I_{[k]}I_{[\ell]}\subseteq I_{[k+\ell]}$ for all $k,\ell\geq 0$.

We use the notation $I_{[k]}$ to not confuse the ideal $I_{[k]}$ with the
$k$th homogeneous component $I_{k}$ of an homogeneous ideal $I\subset R$.

We say that $\mathcal{I}$ is Noetherian if the Rees algebra
$\mathcal{R(I)}=\bigoplus_{k\geq 0}I_{[k]}$ is Noetherian.

######  Lemma 1.1.

Let $\mathcal{I}=\\{I_{[k]}\\}_{k\geq 0}$ be a Noetherian graded filtration of
$R$. Then, there exists an integer $c>0$ such that

$\operatorname{Ass}(I_{[ck+j]})=\operatorname{Ass}(I_{[c(k+1)+j]})\ \ \
\textit{for all}\ k\gg 0,$

for all $j=0,\dots,c-1$.

###### Proof.

Since $\mathcal{R(I)}$ is a graded Noetherian ring, there exists an integer
$c$ such that the Veronese subalgebra

$\mathcal{R(I)}^{(c)}\ =\ \bigoplus_{k\geq 0}I_{[ck]}$

is standard graded (see [28] or [21, Theorem 2.1]). For each $j=0,\dots,c-1$,

$\mathcal{R(I)}_{(j)}\ =\ \bigoplus_{k\geq 0}I_{[ck+j]}$

is a finitely generated $\mathcal{R(I)}^{(c)}$-module.

Hence [1, Lemma 2.3] (which is the module version of [27, Proposition 2])
guarantees that

$\operatorname{Ass}(I_{[ck+j]})\ =\ \operatorname{Ass}(I_{[c(k+1)+j]}),$

for all $k\gg 0$ and all $j=0,\dots,c-1$.

We denote by $\operatorname{Ass}^{\infty}(\mathcal{I})$ the set of those
primes $\mathfrak{p}\in\textup{Spec}(R)$ such that
$\mathfrak{p}\in\operatorname{Ass}(I_{[k]})$ for infinitely many $k$. Using
the notation in the above lemma,
$\mathfrak{p}\in\operatorname{Ass}^{\infty}(\mathcal{I})$ if and only if
$\mathfrak{p}\in\operatorname{Ass}(I_{[ck+j]})$ for some
$j\in\\{0,\dots,c-1\\}$ and all $k\gg 0$.

######  Remark 1.2.

If the Rees algebra $\mathcal{R(I)}$ is standard graded, then in Lemma 1.1 we
have $c=1$ and we obtain that
$\operatorname{Ass}(I_{[k]})=\operatorname{Ass}(I_{[k+1]})$ for all $k\gg 0$.

A numerical function $f:{\mathbb{Z}}_{\geq 0}\rightarrow{\mathbb{Z}}_{\geq 0}$
is called a quasi-linear function (of period $c$) if there exist a positive
integer $c$ and linear functions $f_{i}(k)=a_{i}k+b_{i}$, $i=0,\dots,c-1$,
such that $f(k)=f_{i}(k)$ if $k\equiv i$ (mod $c$). The aim of this section is
to prove:

######  Theorem 1.3.

Let $\mathcal{I}=\\{I_{[k]}\\}_{k\geq 0}$ be a Noetherian graded filtration of
$R$. Then $\operatorname{v}(I_{[k]})$ is a quasi-linear function in $k$, for
$k\gg 0$.

We call the function $\operatorname{v}(I_{[k]})$ the
$\operatorname{v}$-function of the graded filtration $\mathcal{I}$.

In order to prove the theorem, we need some auxiliary lemmata. We begin with
the following result of Conca [10, Lemma 1.2]. Let $M$ be a graded $R$-module.
We denote by $\alpha(M)=\min\\{d:M_{d}\neq 0\\}$ and
$\omega(M)=\max\\{d:(M/\mathfrak{m}M)_{d}\neq 0\\}$ the initial degree and the
final degree of $M$, respectively.

######  Lemma 1.4.

Let $I\subset R$ be a graded ideal, and let
$\mathfrak{p}\in\operatorname{Ass}(I)$ be an associated prime of $I$. Let
$X_{\mathfrak{p}}=\\{\mathfrak{p}_{1}\in\operatorname{Ass}(I):\mathfrak{p}\subsetneq\mathfrak{p}_{1}\\}$
and $\mathfrak{q}$ be the product of the elements in $X_{\mathfrak{p}}$ if
$X_{\mathfrak{p}}$ is non-empty, otherwise let $\mathfrak{q}=R$. Then,

$\operatorname{v}_{\mathfrak{p}}(I)\ =\
\alpha\Big{(}\frac{(I:\mathfrak{p})}{I:(\mathfrak{p}+\mathfrak{q}^{\infty})}\Big{)}.$

Here $\mathfrak{q}^{\infty}=\bigcup_{k\geq 0}(\mathfrak{q}:\mathfrak{m}^{k})$.

The following technical lemma will be needed in a moment.

######  Lemma 1.5.

Let $T=\bigoplus_{k\geq 0}T_{k}$ be a Noetherian graded ring, and
$\mathfrak{q}$ be a $\mathfrak{p}$-primary ideal such that $\mathfrak{p}$
contains $T_{+}=\bigoplus_{k>0}T_{k}$. Then $\mathfrak{q}$ contains $T_{t}$
for all $t\gg 0$.

###### Proof.

Since $T$ is Noetherian, there exist elements $f_{i}\in T_{d_{i}}$, for
$i=1,\dots,m$, such that $T$ is finitely generated by
$\\{f_{1},\dots,f_{m}\\}$ as a $T_{0}$-module. Hence, each homogeneous element
$f\in T_{\ell}$ of degree $\ell$ is of the form

$f=\sum_{j_{1},\dots,j_{m}}a_{j_{1},\dots,j_{m}}f_{1}^{j_{1}}\cdots
f_{m}^{j_{m}},$ (1)

where $a_{j_{1},\dots,j_{m}}\in T_{0}$ and
$\deg(f)=j_{1}d_{1}+\dots+j_{m}d_{m}=\ell$.

Since $T$ is Noetherian, $\mathfrak{p}^{b}\subseteq\mathfrak{q}$ for some
integer $b$. By our assumption $T_{+}\subseteq\mathfrak{p}$. Hence
$f_{i}^{b}\in\mathfrak{p}^{b}\subseteq\mathfrak{q}$ for all $1\leq i\leq m$.
Let $f\in T_{t}$ as expressed in (1) for some $t$. Then
$\deg(f)=j_{1}d_{1}+\dots+j_{m}d_{m}=t$ for all $j_{1},\dots,j_{m}$ such that
$a_{j_{1},\dots,j_{m}}\neq 0$. This equation implies that if $t$ is big
enough, then for all $j_{1},\dots,j_{m}$ such that $a_{j_{1},\dots,j_{m}}\neq
0$ at least one index $j_{i}$ must be bigger than $b$. This shows that for all
$t\gg 0$ we have $f\in\mathfrak{q}$, and so $T_{t}\subseteq\mathfrak{q}$.

A crucial property which we need for the proof of Theorem 1.3 is an analogue
of the so-called Ratliff property which assures that we have
$(I^{k+1}:I)=I^{k}$ for all $k\gg 0$, if $I$ is an ideal of a Noetherian
domain $R$ [28, Corollary 4.2]. This property was pivotal in the proofs of
[10, Theorem 1.1] and [14, Theorem 3.1]. The following result is probably
well-known in the context of graded filtrations, but due to the lack of a
suitable reference, we provide a proof.

######  Lemma 1.6.

Let $\mathcal{I}=\\{I_{[k]}\\}_{k\geq 0}$ be a Noetherian graded filtration of
$R$. Then, there exists an integer $t\geq 0$ such that

$(I_{[k+1]}:I_{[1]})\cap I_{[t]}\ =\ I_{[k]}$

for all $k\gg 0$.

###### Proof.

The proof is very similar to that of [25, Proposition 8.5.7]. Notice that the
associated graded ring $\operatorname{gr}(\mathcal{I})=\bigoplus_{k\geq
0}(I_{[k]}/I_{[k+1]})$ is a Noetherian ${\mathbb{N}}$-graded ring, for it is a
quotient of the Noetherian ring $\mathcal{R(I)}$. Let
$0=\mathfrak{q}_{1}\cap\dots\cap\mathfrak{q}_{r}$ be a primary decomposition
of the zero ideal of $\operatorname{gr}(\mathcal{I})$. Let $\mathfrak{p}_{i}$
be the radical of $\mathfrak{q}_{i}$. Then $\mathfrak{p}_{i}$ is a prime
ideal. Without loss of generality, we may assume that $\mathfrak{p}_{i}$
contains all elements of $\operatorname{gr}(\mathcal{I})$ of positive degree
for $1\leq i\leq s$, while $\mathfrak{p}_{s+1},\dots,\mathfrak{p}_{r}$ do not.
By Lemma 1.5 we can find an integer $t$ such that $\mathfrak{q}_{i}$ contains
$\operatorname{gr}(\mathcal{I})_{k}=I_{[k]}/I_{[k+1]}$ for all $k\geq t$ and
all $1\leq i\leq s$.

By Prime Avoidance, we can find an homogeneous element
$x+I_{[d+1]}\in\operatorname{gr}(\mathcal{I})_{d}$ of positive degree $d$,
with $x\in I_{[d]}$, that is not contained in
$\mathfrak{p}_{s+1},\dots,\mathfrak{p}_{r}$. Since $R/{\mathfrak{m}}$ is an
infinite field, by [25, Theorem A.1.2] we may assume that $d=1$.

We now prove that $(I_{[k+1]}:I_{[1]})\cap I_{[t]}\ =\ I_{[k]}$ for all $k\gg
0$. Indeed, it is clear that $I_{[k]}$ is contained in
$(I_{[k+1]}:I_{[1]})\cap I_{[t]}$ for $k\gg 0$. For the opposite inclusion, we
prove that $(I_{[k+1]}:x)\cap I_{[t]}\subseteq I_{[k]}$ for all $k\geq t$.
Since $(I_{[k+1]}:I_{[1]})\cap I_{[t]}\subseteq(I_{[k+1]}:x)\cap I_{[t]}$,
this will imply the desired opposite inclusion.

Let $k\geq t$ and suppose for a contradiction that there exists
$y\in(I_{[k+1]}:x)\cap I_{[t]}$ such that $y\notin I_{[k]}$. Since
$R=I_{[0]}\supseteq I_{[1]}\supseteq\cdots\supseteq I_{[k]}\supseteq\cdots$
and $y\notin I_{[k]}$, there exists $p$ the largest integer such that $y\in
I_{[p]}$. Then $t\leq p<k$. In $\operatorname{gr}(\mathcal{I})$ we have
$(x+I_{[2]})(y+I_{[p+1]})=0\subseteq\mathfrak{q}_{j}$ for $s+1\leq j\leq r$.
Since $x+I_{[2]}\notin\mathfrak{p}_{j}$ for $s+1\leq j\leq r$, we see that
$y+I_{[p+1]}\in\mathfrak{q}_{s+1}\cap\dots\cap\mathfrak{q}_{r}$. On the other
hand, since $p\geq t$, we have that
$y+I_{[p+1]}\in\mathfrak{q}_{1}\cap\dots\cap\mathfrak{q}_{s}$. Thus
$y+I_{[p+1]}\in\bigcap_{i=1}^{r}\mathfrak{q}_{i}=0$. This means that $y\in
I_{[p+1]}$, against the meaning of $p$. It follows that $(I_{[k+1]}:x)\cap
I_{[t]}\subseteq I_{[k]}$, as desired.

######  Lemma 1.7.

Let $\mathcal{I}=\\{I_{[k]}\\}_{k\geq 0}$ be a Noetherian graded filtration of
$R$, and let $\mathfrak{p}\in\operatorname{Ass}^{\infty}(\mathcal{I})$. Then
$I_{[1]}\subseteq\mathfrak{p}$.

###### Proof.

Let $\mathfrak{p}\in\operatorname{Ass}^{\infty}(\mathcal{I})$. Then
$I_{[k]}\subseteq\mathfrak{p}$ for some $k$. By property (iii) of a graded
filtration, we have $I_{[1]}^{k}\subseteq I_{[k]}\subseteq\mathfrak{p}$.
Taking the radicals the conclusion follows.

For a bigraded module $M=\bigoplus_{d,k\geq 0}M_{(d,k)}$, we set
$M_{(*,k)}=\bigoplus_{d\geq 0}M_{(d,k)}$.

###### Proof of Theorem 1.3.

As in Lemma 1.1, let $c>0$ such that $\mathcal{R(I)}^{(c)}=\bigoplus_{k\geq
0}I_{[ck]}$ is standard graded. We set
$\mathcal{I}_{(0)}=\\{I_{[ck]}\\}_{k\geq 0}$ and
$\mathcal{I}_{(j)}=\\{R=I_{[0]}\\}\cup\\{I_{[c(k-1)+j]}\\}_{k\geq 1}$ for
$0<j\leq c-1$. Then each $\mathcal{I}_{(j)}$ is a Noetherian graded
filtration. Lemma 1.1 implies that for each $j\in\\{0,\dots,c-1\\}$,

$\operatorname{Ass}(I_{[ck+j]})\ =\ \operatorname{Ass}(I_{[c(k+1)+j]})$

for all $k\gg 0$. Hence, for each $j\in\\{0,\dots,c-1\\}$, and all $k\gg 0$,

$\operatorname{v}(I_{[ck+j]})\ =\
\min_{\mathfrak{p}\in\operatorname{Ass}^{\infty}(\mathcal{I}_{(j)})}\operatorname{v}_{\mathfrak{p}}(I_{[ck+j]}).$

Thus to prove the theorem, it is enough to show that
$\operatorname{v}_{\mathfrak{p}}(I_{[ck+j]})$ is an eventually linear function
in $k$, for all
$\mathfrak{p}\in\operatorname{Ass}^{\infty}(\mathcal{I}_{(j)})$ and all
$j=\\{0,\dots,c-1\\}$.

Fix $j\in\\{0,\dots,c-1\\}$. We set
$\mathcal{J}=\mathcal{I}_{(j)}=\\{J_{[k]}\\}_{k\geq 0}$. Let
$\mathfrak{p}\in\operatorname{Ass}^{\infty}(\mathcal{J})$. Then
$\mathfrak{p}\in\operatorname{Ass}(J_{[k]})$ for all $k\gg 0$. Let
$\mathfrak{q}$ be the product of the elements in the set
$X_{\mathfrak{p}}=\\{\mathfrak{p}_{1}\in\operatorname{Ass}^{\infty}(\mathcal{J}):\mathfrak{p}\subsetneq\mathfrak{p}_{1}\\}$
if $X_{\mathfrak{p}}$ is non-empty, otherwise let $\mathfrak{q}=R$. Let
$\mathcal{R}^{\prime}=\mathcal{R}(\mathcal{J})(1)=\bigoplus_{k\geq 1}J_{[k]}$,
that is the Rees algebra of $\mathcal{J}$ shifted by degree one. Consider the
module

$\textup{Soc}_{\mathfrak{p}}^{*}(\mathcal{J})\ =\ \frac{(\mathcal{R}^{\prime}\
:_{\mathcal{R}(\mathcal{J})}\
\mathfrak{p}\mathcal{R}(\mathcal{J}))}{(\mathcal{R}^{\prime}\
:_{\mathcal{R}(\mathcal{J})}\
(\mathfrak{p}+\mathfrak{q}^{\infty})\mathcal{R}(\mathcal{J}))}.$ (2)

Here, if $J$ is an ideal of $R$, then $J\mathcal{R}(\mathcal{J})$ is the
extension of $J$ in $\mathcal{R}(\mathcal{J})$. Notice that
$\textup{Soc}_{\mathfrak{p}}^{*}(\mathcal{J})$ is a finitely generated
bigraded $\mathcal{R(I)}^{(c)}$-module. Moreover, by construction,

$\textup{Soc}_{\mathfrak{p}}^{*}(\mathcal{J})_{(*,k)}\ =\
\frac{(J_{[k+1]}:{\mathfrak{p}})\cap
J_{[k]}}{(J_{[k+1]}:(\mathfrak{p}+\mathfrak{q}^{\infty}))\cap J_{[k]}}.$

By Lemma 1.7 we have $J_{[1]}\subseteq\mathfrak{p}$. Applying Lemma 1.6 there
exists $t\geq 0$ such that $(J_{[k+1]}:J_{[1]})\cap J_{[t]}=J_{[k]}$ for all
$k\gg 0$. On the other hand if $k\geq t$, then $J_{[k]}\subseteq J_{[t]}$.
Therefore, for $k\gg 0$, $(J_{[k+1]}:{\mathfrak{p}})\cap
J_{[k]}\subseteq(J_{[k+1]}:J_{[1]})\cap J_{[t]}=J_{[k]}$ and likewise
$(J_{[k+1]}:(\mathfrak{p}+\mathfrak{q}^{\infty}))\cap J_{[k]}\subseteq
J_{[k]}$. We deduce that

$\textup{Soc}_{\mathfrak{p}}^{*}(\mathcal{J})_{(*,k)}\ =\
\frac{(J_{[k+1]}:{\mathfrak{p}})}{J_{[k+1]}:(\mathfrak{p}+\mathfrak{q}^{\infty})}.$
(3)

for all $k\gg 0$. Applying Lemma 1.4 we then deduce that for all $k\gg 0$

$\operatorname{v}_{\mathfrak{p}}(J_{[k]})\ =\
\alpha(\textup{Soc}_{\mathfrak{p}}^{*}(\mathcal{J})_{(*,k-1)}).$ (4)

Since $\textup{Soc}_{\mathfrak{p}}^{*}(\mathcal{J})_{(*,k-1)}$ is a finitely
generated $\mathcal{R(I)}^{(c)}$-module and $\mathcal{R(I)}^{(c)}$ is standard
graded, equation (4) and variation of [8, Theorem 8.3.4] (see also [14,
Proposition 2.5]) imply that $\operatorname{v}_{\mathfrak{p}}(J_{[k]})$ is a
linear function in $k$ for all $k\gg 0$, as desired.

Let $\mathfrak{p}\in\operatorname{Ass}^{\infty}(\mathcal{I})$. If follows from
the proof of Theorem 1.3 that there exists $c$ such that
$\operatorname{Soc}_{\mathfrak{p}}^{*}(\mathcal{I})_{(*,c(k+1)+j)}=I_{[c]}\operatorname{Soc}_{\mathfrak{p}}^{*}(\mathcal{I})_{(*,ck+j)}$
for all $k\gg 0$ and all $j=0,\dots,c-1$. Hence, we obtain the following
consequences of Theorem 1.3 and equation (3). The reader may confront this
property with [14, Propositions 2.7 and 3.2].

######  Corollary 1.8.

Under the same assumptions and notation of Theorem 1.3, for any
$\mathfrak{p}\in\operatorname{Ass}^{\infty}(\mathcal{I})$, there exist $c>0$
and $j\in\\{0,\dots,c-1\\}$ such that

$(I_{[c(k+1)+j]}:\mathfrak{p})\ =\ I_{[c]}(I_{[ck+j]}:\mathfrak{p})$

for all $k\gg 0$. In particular,

$\lim_{k\rightarrow\infty}\frac{\operatorname{v}_{\mathfrak{p}}(I_{[ck+j]})}{k}\
\in\ \\{d:(I_{[c]}/{\mathfrak{m}}I_{[c]})_{d}\neq 0\\}.$

######  Remark 1.9.

In this section, we focused on the $\operatorname{v}$-function
$\operatorname{v}(I_{[k]})$, and proved its asymptotic quasi-linearity. This
result can be generalized, with the obvious modifications and under the
analogue suitable assumptions as in Theorem 1.3, to show that the
$\operatorname{v}$-functions $\operatorname{v}(I_{[k]}M/I_{[k+1]}N)$ and
$\operatorname{v}(M/I_{[k+1]}N)$ are quasi-linear for $k\gg 0$, where
$\mathcal{I}=\\{I_{[k]}\\}_{k\geq 0}$ is a Noetherian graded filtration, $M$
is a finitely generated graded $R$-module, and $N$ is a graded submodule of
$M$. See also [16] and [17].

##  2\. When $\mathfrak{p}\in\operatorname{Ass}^{\infty}(\mathcal{I})$?

Let $\mathcal{I}=\\{I_{[k]}\\}_{k\geq 0}$ be a Noetherian graded filtration of
$R$. Given $\mathfrak{p}\in\operatorname{Spec}(R)$, when do we have that
$\mathfrak{p}\in\operatorname{Ass}^{\infty}(\mathcal{I})$? The next theorem
answers this question. For the proof of this result, we recall some basic
facts.

Let $(R,\mathfrak{m},K)$ be a Noetherian local ring, $I\subset R$ be an ideal,
and $M$ be a finitely generated $R$-module. Then, all minimal system of
generators of $I$ have the same lenght, namely
$\mu(I)=\dim_{K}(I/\mathfrak{m}I)$. Let $\mu(I)=m$, ${\bf
f}:f_{1},\dots,f_{m}$ be a minimal system of generators of $I$, and let
$H_{i}({\bf f};M)$ be the $i$th Koszul homology of ${\bf f}$ with respect to
$M$. Then [9, Proposition 1.6.21] implies that if ${\bf g}:g_{1},\dots,g_{m}$
is another minimal system of generators of $I$, we have $H_{i}({\bf f};M)\cong
H_{i}({\bf g};M)$ for all $i$. In other words, the Koszul homology modules do
not depend upon the choice of a minimal system of generators of $I$.
Therefore, we write $H_{i}(I;M)$ to denote $H_{i}({\bf f};M)$, where ${\bf f}$
can be any minimal system of generators of $I$.

######  Theorem 2.1.

Under the assumptions and notation of Theorem 1.3, for a prime ideal
$\mathfrak{p}\in\operatorname{Spec}(R)$, the following conditions are
equivalent:

  1. (a)

$\mathfrak{p}\in\operatorname{Ass}^{\infty}(\mathcal{I})$.

  2. (b)

$\mathfrak{p}\in\operatorname{Ass}(I_{[k]})$ for infinitely many $k$.

  3. (c)

The Krull dimension $\dim
H_{\mu(\mathfrak{p}R_{\mathfrak{p}})-1}(\mathfrak{p}R_{\mathfrak{p}}\,;\,\mathcal{R}(\mathcal{I})_{\mathfrak{p}})$
is positive.

###### Proof.

That (a) and (b) are equivalent is by definition of the set
$\operatorname{Ass}^{\infty}(\mathcal{I})$.

Before proving that (b) and (c) are equivalent, we notice the following fact.
By localizing at $\mathfrak{p}$, we see that
$\mathfrak{p}\in\operatorname{Ass}(I_{[k]})$ if and only if
$\mathfrak{p}R_{\mathfrak{p}}\in\operatorname{Ass}((I_{[k]})_{\mathfrak{p}})$.
As $R_{\mathfrak{p}}$ is again a Noetherian ring, then
$\mathfrak{p}R_{\mathfrak{p}}\in\operatorname{Ass}((I_{[k]})_{\mathfrak{p}})$
if and only if there exists $f\in R_{\mathfrak{p}}$ such that
$((I_{[k]})_{\mathfrak{p}}:f)=\mathfrak{p}R_{\mathfrak{p}}$. This implies that
$((I_{[k]})_{\mathfrak{p}}:\mathfrak{p}R_{\mathfrak{p}})/(I_{[k]})_{\mathfrak{p}}\neq
0$. Conversely, if
$((I_{[k]})_{\mathfrak{p}}:\mathfrak{p}R_{\mathfrak{p}})/(I_{[k]})_{\mathfrak{p}}\neq
0$ and $g+(I_{[k]})_{\mathfrak{p}}$ is a non-zero element belonging to this
module, then $((I_{[k]})_{\mathfrak{p}}:g)$ is a proper ideal that contains
$\mathfrak{p}R_{\mathfrak{p}}$. Since $\mathfrak{p}R_{\mathfrak{p}}$ is the
unique maximal ideal of $R_{\mathfrak{p}}$, we see that
$((I_{[k]})_{\mathfrak{p}}:g)=\mathfrak{p}R_{\mathfrak{p}}$ and so
$\mathfrak{p}R_{\mathfrak{p}}\in\operatorname{Ass}((I_{[k]})_{\mathfrak{p}})$.

Summarizing, $\mathfrak{p}\in\operatorname{Ass}(I_{[k]})$ if and only if
$((I_{[k]})_{\mathfrak{p}}:\mathfrak{p}R_{\mathfrak{p}})/(I_{[k]})_{\mathfrak{p}}\neq
0$.

Now we prove that (b) and (c) are equivalent. By the previous fact,
$\mathfrak{p}\in\operatorname{Ass}(I_{[k]})$ for infinitely many $k$ if and
only if
$((I_{[k]})_{\mathfrak{p}}:\mathfrak{p}R_{\mathfrak{p}})/(I_{[k]})_{\mathfrak{p}}\neq
0$ for infinitely many $k$.

Notice that $\mathcal{R(I)}_{\mathfrak{p}}$ is a Noetherian graded ring.
Indeed, localization preserves the Noetherian property. Whereas,
$\mathcal{R(I)}_{\mathfrak{p}}=\bigoplus_{k\geq 0}(I_{[k]})_{\mathfrak{p}}$ is
graded, because applying localization to the property (iii) of $\mathcal{I}$
yields
$(I_{[k]})_{\mathfrak{p}}(I_{[\ell]})_{\mathfrak{p}}\subseteq(I_{[k+\ell]})_{\mathfrak{p}}$
for all $k,\ell\geq 0$. Hence, the Koszul homology
$H_{\mu(\mathfrak{p}R_{\mathfrak{p}})-1}(\mathfrak{p}R_{\mathfrak{p}};\mathcal{R}(\mathcal{I})_{\mathfrak{p}})$
is a graded $\mathcal{R}(\mathcal{I})_{\mathfrak{p}}$-module whose $k$th
graded component is
$H_{\mu(\mathfrak{p}R_{\mathfrak{p}})-1}(\mathfrak{p}R_{\mathfrak{p}};(I_{[k]})_{\mathfrak{p}})=((I_{[k]})_{\mathfrak{p}}:\mathfrak{p}R_{\mathfrak{p}})/(I_{[k]})_{\mathfrak{p}}$.
Thus (b) holds if and only infinitely many graded components of
$H_{\mu(\mathfrak{p}R_{\mathfrak{p}})-1}(\mathfrak{p}R_{\mathfrak{p}};\mathcal{R}(\mathcal{I})_{\mathfrak{p}})$
are non-zero. But this is clearly equivalent to condition (c).

In the case of the ordinary powers of monomial ideals of a standard graded
polynomial ring with coefficients over a field, this result was firstly noted
by Bayati, Herzog and Rinaldo [2]. Furthermore, in such a situation one can
replace ordinary localization with monomial localization. See [2, Section 2]
for the details.

##  3\. Integer Programming and the $\operatorname{v}$-number

In this section, we show how the computation of the
$\operatorname{v}$-function of certain graded filtrations of monomial ideals
is related to integer programming problems.

Let $k$ be an integer and let ${\bf a}=(a_{1},\dots,a_{n}),{\bf
b}=(b_{1},\dots,b_{n})\in{\mathbb{Z}}^{n}$ be integral vectors. We set $k{\bf
a}=(ka_{1},\dots,ka_{n})$ and ${\bf a+b}=(a_{1}+b_{1},\dots,a_{n}+b_{n})$. The
modulus of ${\bf a}$ is defined as $|{\bf a}|=a_{1}+\dots+a_{n}$. We denote by
${\bf a}[i]=a_{i}$ the $i$th component of ${\bf a}$. Let ${\bf
e}_{1},\dots,{\bf e}_{n}$ be the canonical basis of ${\mathbb{Z}}^{n}$. That
is, ${\bf e}_{i}[j]=1$ for $i=j$ and ${\bf e}_{i}[j]=0$ if $i\neq j$. We
denote by ${\bf 0}$ the vector $(0,0,\dots,0)\in{\mathbb{Z}}^{n}$.

Let ${\bf A}=\\{{\bf a}_{1},\dots,{\bf a}_{m}\\}$ and ${\bf B}=\\{{\bf
b}_{1},\dots,{\bf b}_{\ell}\\}\subseteq\\{{\bf e}_{1},\dots,{\bf e}_{n}\\}$ be
two finite collections of integral vectors of ${\mathbb{Z}}^{n}_{\geq 0}$. Let
$k\geq 1$. We define $\mathcal{P}_{k}({\bf A})$ to be the following set of
lattice points of ${\mathbb{Z}}^{n}$:

$\mathcal{P}_{k}({\bf A})=\mathcal{P}_{k}({\bf a}_{1},\dots,{\bf
a}_{m})=\big{\\{}\\!\sum_{i=1}^{m}k_{i}{\bf a}_{i}+\sum_{i=1}^{n}h_{i}{\bf
e}_{i}\ :\ k_{i},h_{i}\geq 0,\ \sum_{i=1}^{m}k_{i}\geq k\big{\\}}.$

For any $k\geq 1$, we consider the following integer program:

$P_{{\bf A,B},k}\ :\ \ \begin{cases}\textup{minimize}&{\bf
c}\in{\mathbb{Z}}^{n},\\\ \textup{subject to}&{\bf c}+{\bf
d}\in\mathcal{P}_{k}({\bf A})\ \textup{if and only if}\ {\bf
d}\in\mathcal{P}_{1}({\bf B}).\end{cases}$

We say that a solution ${\bf c}$ of $P_{{\bf A,B},k}$, if it exists, is an
optimal solution if its module $|{\bf c}|$ is minimum among all solutions of
the problem.

Let $K$ be a field and $S=K[x_{1},\dots,x_{n}]$. For a non-empty subset $A$ of
${\mathbb{Z}}_{\geq 0}^{n}$, let $I({\bf A})\subset S$ be the monomial ideal
generated by ${\bf x^{a}}=x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$, ${\bf
a}=(a_{1},\dots,a_{n})\in{\bf A}$. Then $I({\bf A})^{k}=I(\mathcal{P}_{k}({\bf
A}))$ for all $k\geq 1$. Notice that $I({\bf A})\neq S$ if and only if ${\bf
0}\notin{\bf A}$.

By Brodmann [6], if $I\subset S$ is a homogeneous ideal, the sets
$\operatorname{Ass}(I^{k})$ stabilize, that is:
$\operatorname{Ass}(I^{k})=\operatorname{Ass}(I^{k+1})$ for all $k\gg 0$. We
denote by $\operatorname{Ass}^{\infty}(I)$ the common sets
$\operatorname{Ass}(I^{k})$ for $k\gg 0$.

######  Theorem 3.1.

Let ${\bf A},{\bf B}\subset{\mathbb{Z}}_{\geq 0}^{n}$ be non-empty finite
subsets such that ${\bf 0}\notin{\bf A}$ and ${\bf B}\subseteq\\{{\bf
e}_{1},\dots,{\bf e}_{n}\\}$. Then, the following conditions are equivalent.

  1. (a)

The integer program $P_{{\bf A,B},k}$ has a solution for all $k\gg 0$.

  2. (b)

$I({\bf B})=(x_{i}:{\bf e}_{i}\in B)\in\operatorname{Ass}(I({\bf A})^{k})$ for
all $k\gg 0$.

  3. (c)

The Krull dimension $\dim\operatorname{Soc}^{*}_{I({\bf B})}(\\{I({\bf
A})^{k}\\}_{k\geq 0})$ is positive.

Furthermore, if any of the equivalent conditions hold, then for all $k\gg 0$
the modulus of the optimal solution of $P_{{\bf A,B},k}$ is a linear function
of the form $ak+b$, where $a,b\in{\mathbb{Z}}$ are suitable integers. In
particular, $a\in\\{|{\bf a}|\ :\ {\bf a}\in{\bf A}\\}$.

###### Proof.

Fix a field $K$ and an integer $k\geq 1$. Then, there exists a solution ${\bf
c}\in{\mathbb{Z}}_{\geq 0}^{n}$ of the integer program $P_{{\bf A,B},k}$ if
and only if we have ${\bf x^{c}}{\bf x^{d}}\in I(\mathcal{P}_{k}({\bf
A}))=I({\bf A})^{k}$ only when ${\bf x^{d}}\in I({\bf B})$. That is, there
exists a solution ${\bf c}\in{\mathbb{Z}}_{\geq 0}^{n}$ of the integer program
$P_{{\bf A,B},k}$ if and only if $(I({\bf A})^{k}:{\bf x^{c}})=I({\bf B})$.
Thus (a) is equivalent to condition (b). Let $\mathcal{I}=\\{I({\bf
A})^{k}\\}_{k\geq 0}$. By Lemma 1.4, (b) is equivalent to the fact that
$\operatorname{Soc}^{*}_{I({\bf B})}(\mathcal{I})_{(*,k)}$ is non-zero for
infinitely many $k$. This is equivalent to condition (c).

Finally, suppose that (a) or (b) or (c) holds. For $k\gg 0$ the modulus $|{\bf
c}|$ of the optimal solution ${\bf c}$ of $P_{{\bf A,B},k}$ is the
$\operatorname{v}_{I({\bf B})}$-number $\operatorname{v}_{I({\bf B})}(I({\bf
A})^{k})$. By Theorem 1.3, (see also [14, Theorem 3.1] or [10, Theorem 1.1])
$\operatorname{v}_{I({\bf B}))}(I({\bf A})^{k})=ak+b$ for all $k\gg 0$. By
[10, Remark 1.3], (see also [14, Proposition 3.2]) we have $a\in\\{\deg({\bf
x^{a}}):{\bf x^{a}}\in\mathcal{G}(I({\bf A}))\\}\subseteq\\{|{\bf a}|\ :\ {\bf
a}\in{\bf A}\\}$, as desired.

As before, let ${\bf A}=\\{{\bf a}_{1},\dots,{\bf a}_{m}\\}$ and ${\bf
B}=\\{{\bf b}_{1},\dots,{\bf b}_{\ell}\\}\subseteq\\{{\bf e}_{1},\dots,{\bf
e}_{n}\\}$ be two finite collections of integral vectors of
${\mathbb{Z}}^{n}_{\geq 0}$. Let $k\geq 1$. We define
$\overline{\mathcal{P}}_{k}({\bf A})$ to be the set of lattice points of the
convex hull in ${\mathbb{R}}^{n}$ of the set $\mathcal{P}({\bf A})$. That is,
we set $\overline{\mathcal{P}}_{k}({\bf A})\ =\
\textup{conv}(\mathcal{P}_{k}({\bf A}))\cap{\mathbb{Z}}_{\geq 0}^{n}$. By [20,
Corollary 1.4.3], $I(\overline{\mathcal{P}}_{k}({\bf A}))$ is equal to the
integral closure of $\overline{I(\mathcal{P}_{k}({\bf A}))}$. Hence
$I(\overline{\mathcal{P}}_{k}({\bf A}))=\overline{I({\bf A})^{k}}$ for all
$k\geq 1$.

For any ${\bf A}=\\{{\bf a}_{1},\dots,{\bf a}_{m}\\}$, ${\bf B}=\\{{\bf
b}_{1},\dots,{\bf b}_{\ell}\\}\subseteq\\{{\bf e}_{1},\dots,{\bf e}_{n}\\}$
and $k\geq 1$, we consider the following integer program:

$\overline{P}_{{\bf A,B},k}\ :\ \ \begin{cases}\textup{minimize}&{\bf
c}\in{\mathbb{Z}}^{n},\\\ \textup{subject to}&{\bf c}+{\bf
d}\in\overline{\mathcal{P}}_{k}({\bf A})\ \textup{if and only if}\ {\bf
d}\in\mathcal{P}_{1}({\bf B}).\end{cases}$

It is well-known that $\mathcal{I}=\\{\overline{I^{k}}\\}_{k\geq 0}$ is a
Noetherian graded filtration, if $I\subset S$ is a homogeneous ideal. By
Brodmann [6], or [27], the sets $\operatorname{Ass}(\overline{I^{k}})$
stabilize: that is
$\operatorname{Ass}(\overline{I^{k}})=\operatorname{Ass}(\overline{I^{k+1}})$
for all $k\gg 0$. We set
$\operatorname{Ass}^{\infty}(\\{\overline{I^{k}}\\}_{k\geq
0})=\overline{\textup{Ass}}^{\infty}(I)$. As in Theorem 3.1 one can show that

######  Theorem 3.2.

Let ${\bf A},{\bf B}\subset{\mathbb{Z}}_{\geq 0}^{n}$ be non-empty finite
subsets such that ${\bf 0}\notin{\bf A}$ and ${\bf B}\subseteq\\{{\bf
e}_{1},\dots,{\bf e}_{n}\\}$. Then, the following conditions are equivalent.

  1. (a)

The integer program $\overline{P}_{{\bf A,B},k}$ has a solution for all $k\gg
0$.

  2. (b)

$I({\bf B})=(x_{i}:{\bf e}_{i}\in
B)\in\overline{\operatorname{Ass}}^{\infty}(I({\bf A}))$.

  3. (c)

The Krull dimension $\dim\operatorname{Soc}^{*}_{I({\bf
B})}(\\{\overline{I({\bf A})^{k}}\\}_{k\geq 0})$ is positive.

Furthermore, if any of the equivalent conditions hold, then for all $k\gg 0$
the modulus of the optimal solution of $\overline{P}_{{\bf A,B},k}$ is a
quasi-linear function.

##  4\. The package VNumber

In this section we give a self contained introduction to the Macaulay2 [18]
package VNumber [15] and demonstrate its main features.

Let $I\subset S=K[x_{1},\dots,x_{n}]$ be a homogeneous ideal, and let
$\mathfrak{p}\subset S$ be a prime ideal. We set
$\operatorname{Ass}^{\infty}(I)=\operatorname{Ass}^{\infty}(\\{I^{k}\\}_{k\geq
0})$ and by $\operatorname{Max}^{\infty}(I)$ we denote the set of primes of
$\operatorname{Ass}^{\infty}(I)$ which are maximal with respect to the
inclusion. In the next table we collect the functions available in the
package. Except for the functions reesMap$(I)$, vNumberP$(I,\mathfrak{p})$ and
vNumber$(I)$, all other functions require that $I$ is a monomial ideal.

Functions | Description
---|---
reesMap$(I)$ |  Computes the Rees map of $I$
stablePrimes$(I)$ |  Computes $\operatorname{Ass}^{\infty}(I)$
stableMax$(I)$ |  Computes $\operatorname{Max}^{\infty}(I)$
isStablePrime$(I,\mathfrak{p})$ |  Checks if $\mathfrak{p}\in\operatorname{Ass}^{\infty}(I)$
vNumberP$(I,\mathfrak{p})$ |  Computes $\operatorname{v}_{\mathfrak{p}}(I)$
vNumber$(I)$ |  Computes $\operatorname{v}(I)$
soc$(I,\mathfrak{p})$ |  Computes $\operatorname{Soc}_{\mathfrak{p}}^{*}(I)$
vFunctionP$(I,\mathfrak{p})$ |  Computes $\operatorname{v}_{\mathfrak{p}}(I^{k})=a_{\mathfrak{p}}k+b_{\mathfrak{p}}$ for $k\gg 0$
vFunction$(I)$ |  Computes $\operatorname{v}(I^{k})=ak+b$ for $k\gg 0$
Table 1. List of the functions of VNumber.

Let $I\subset S$ be a homogeneous ideal and let $f_{1},\dots,f_{m}$ be any
minimal homogeneous generating set of $I$. Then
$\mathcal{R}(I)=\bigoplus_{k\geq 0}I^{k}$ may be computed as follows. First,
to keep track of the natural bidegree of $\mathcal{R}(I)$ we introduce the
variable $t$ and we replace $\mathcal{R}(I)$ by $\bigoplus_{k\geq
0}I^{k}t^{k}$. Then, we can consider the bigraded map

$\varphi_{I}\ :\ T=K[x_{1},\dots,x_{n},y_{1},\dots,y_{m}]\rightarrow
S[t]=K[x_{1},\dots,x_{n},t]$

defined by setting $\varphi(x_{i})=x_{i}$ for $1\leq i\leq n$ and
$\varphi(y_{j})=f_{j}t$ for $1\leq j\leq m$, and with
$\textup{bideg}(x_{i})=(1,0)$ and $\textup{bideg}(y_{j})=(\deg f_{j},1)$. We
call $\varphi_{I}$ the Rees map of $I$ (induced by $f_{1},\dots,f_{m}$).
Notice that $\varphi_{I}$ depends on the choice of a minimal homogeneous
generating set of $I$. However, no matter which minimal system of homogeneous
generators of $I$ we choose, we always have

$\mathcal{R}(I)\ \cong\ T/\ker\varphi_{I}.$

The function reesMap$(I)$ allows to compute the map $\varphi_{I}$, for some
choice of the minimal homogeneous generating set of $I$.

The functions stablePrimes$(I)$, stableMax$(I)$,
isStablePrime$(I,\mathfrak{p})$ are based on Theorem 2.1 which says that
$\mathfrak{p}\in\operatorname{Ass}^{\infty}(I)$ if and only if the Krull
dimension of the module
$H_{\mu(\mathfrak{p}R_{\mathfrak{p}})-1}(\mathfrak{p}R_{\mathfrak{p}}\,;\,\mathcal{R}(I)_{\mathfrak{p}})$
is positive. At the moment, for the function isStablePrime$(I,\mathfrak{p})$
it is required that $I$ is a monomial ideal, so that one can replace ordinary
localization by monomial localization as shown in [2, Section 2]. Indeed our
code for this function is just the one given in [3]. On the other hand, if $I$
is a homogeneous ideal (not necessarily monomial) while Macaulay2 [18] can
compute the module
$H_{\mu(\mathfrak{p}R_{\mathfrak{p}})-1}(\mathfrak{p}R_{\mathfrak{p}}\,;\,\mathcal{R}(I)_{\mathfrak{p}})$,
in most cases it is not able to compute its Krull dimension.

To compute vNumberP$(I,\mathfrak{p})$ and vNumber$(I)$ the Lemma 1.4 of Conca
is used. In this case $I$ can be any homogeneous ideal of $S$. When $I$ is a
monomial ideal, the function soc$(I,\mathfrak{p})$ computes the module
$\operatorname{Soc}_{\mathfrak{p}}^{*}(I)$ using equation (2) and the function
reesMap$(I)$. The computation of the $\operatorname{v}$-function
vFunction$(I)$ and of the $\operatorname{v}_{\mathfrak{p}}$-function
vFunctionP$(I,\mathfrak{p})$ employs the proof of [10, Theorem 1.2] or also
Theorem 1.3.

The next example illustrates how to use the package.

`i1: loadPackage "VNumber"`

`i2: S = QQ[x_1..x_6];`

`i3: I = ideal(x_1*x_2,x_1*x_3,x_2*x_3,x_2*x_4,x_3*x_4,x_4*x_5,x_5*x_6);`

`i4: stablePrimes I`

`o4:`
$\\{\texttt{ideal}{}\left(x_{5},\,x_{3},\,x_{2}\right),\texttt{ideal}{}\left(x_{5},\,x_{4},\,x_{2},\,x_{1}\right),\texttt{ideal}{}\left(x_{5},\,x_{4},\,x_{3},\,x_{1}\right),$

``
$\texttt{ideal}{}\left(x_{6},\,x_{4},\,x_{2},\,x_{1}\right),\texttt{ideal}{}\left(x_{6},\,x_{4},\,x_{3},\,x_{1}\right),\texttt{ideal}{}\left(x_{6},\,x_{4},\,x_{3},\,x_{2}\right),$

``
$\texttt{ideal}{}\left(x_{1},\,x_{2},\,x_{3},\,x_{4},\,x_{6}\right),\texttt{ideal}{}\left(x_{1},\,x_{2},\,x_{3},\,x_{4},\,x_{5}\right),\texttt{ideal}{}\left(x_{1},\,x_{2},\,x_{3},\,x_{4},\,x_{5},\,x_{6}\right)\\}$

`i5: P = (stablePrimes I)#1;`

`i6: vFunctionP(I,P)`

`o6: {2,0}`

`i7: vFunction I`

`o7: {2,-1}`

##  5\. Open questions

Let $\mathcal{I}=\\{I_{[k]}\\}_{k\geq 0}$ be a Noetherian graded filtration of
$R$.

######  Question 5.1.

Is it true that $\operatorname{v}(I_{[k]})\leq\operatorname{reg}(I_{[k]})$ for
all $k\gg 0$?

######  Problem 5.2.

Find an example of non Noetherian graded filtration
$\mathcal{I}=\\{I_{[k]}\\}_{k\geq 0}$ of $R$ whose $\operatorname{v}$-function
$\operatorname{v}(I_{[k]})$ is not an eventually quasi-linear function in $k$.
Does the limit $\lim_{k\rightarrow\infty}\operatorname{v}(I_{[k]})/k$ exists?
If so, what it is equal to?

This problem was recently addressed in [31] where an example of a non
Noetherian graded filtration $\mathcal{I}$ whose $\operatorname{v}$-function
$\operatorname{v}(I_{[k]})$ is not an eventually quasi-linear function is
provided.

Suppose now that $R$ is the standard graded polynomial ring
$S=K[x_{1},\dots,x_{n}]$ over an infinite field $K$. Let $I\subset S$ be a
homogeneous ideal, and let $\mathcal{I}=\\{I^{k}\\}_{k\geq 0}$ be the $I$-adic
filtration. Let $\mathfrak{p}\in\operatorname{Ass}^{\infty}(I)$. While we know
that $\operatorname{v}_{\mathfrak{p}}(I^{k})$ and $\operatorname{v}(I^{k})$
are eventually linear functions [14, Theorem 3.1], their initial behaviour is
quite mysterious. In particular, we do not know if they can have any given
number of strict local maxima. Due to experimental evidence, we ask the
following

######  Question 5.3.

a

  1. (a)

Is it true that
$\operatorname{v}_{\mathfrak{p}}(I^{k})<\operatorname{v}_{\mathfrak{p}}(I^{k+1})$
for all $\mathfrak{p}\in\operatorname{Ass}^{\infty}(I)$ and all $k\geq 1$?

  2. (b)

Is it true that $\operatorname{v}(I^{k})<\operatorname{v}(I^{k+1})$ for all
$k\geq 1$?

In [14, Theorem 4.1] we proved that
$\lim_{k\rightarrow\infty}\operatorname{v}(I^{k})/k=\alpha(I)$. Thus
$\operatorname{v}(I^{k})=\alpha(I)k+b$ for all $k\gg 0$ and a certain integer
$b\in{\mathbb{Z}}$. Since
$\operatorname{v}(I^{k})=\max_{\mathfrak{p}\in\operatorname{Ass}^{\infty}(I)}\operatorname{v}_{\mathfrak{p}}(I^{k})$
for all $k\gg 0$, and each $\operatorname{v}_{\mathfrak{p}}(I^{k})$ is an
eventually linear function in $k$, there exists at least one prime
$\mathfrak{p}\in\operatorname{Ass}^{\infty}(I)$ such that
$\lim_{k\rightarrow\infty}\operatorname{v}_{\mathfrak{p}}(I^{k})/k=\alpha(I)$.
In view of this observation and some experimental evidence, we expect that

######  Conjecture 5.4.

Let $I\subset S$ be a homogeneous ideal. For all
$\mathfrak{p}\in\operatorname{Max}^{\infty}(I)$, we have

$\lim_{k\rightarrow\infty}\frac{\operatorname{v}_{\mathfrak{p}}(I^{k})}{k}\ =\
\alpha(I).$

Assume that $I\subset S$ is a proper monomial ideal. By [13, Proposition 2.2]
for all $k\gg 0$, $\operatorname{v}(I^{k})=ak+b$ where $a=\alpha(I)\geq 1$ and
$b\geq-1$. By [12, Theorem 5.5], $\operatorname{reg}(I^{k})=ck+d$ where $c\geq
1$ and $d\geq 0$. In view of Question 5.1 we ask the following

######  Question 5.5.

Let $a,b,c,d$ integers such that $a\geq 1$, $b\geq-1$, $c\geq 1$, $d\geq 0$
and $ak+b\leq ck+d$ for all $k\gg 0$. Can we find a monomial ideal $I$ in some
polynomial ring $S$ such that $\operatorname{v}(I^{k})=ak+d$ and
$\operatorname{reg}(I^{k})=ck+d$ for all $k\gg 0$.

In [5], the authors introduced the $\operatorname{v}$-stability index of a
graded ideal $I\subset S$ as

$\textup{v-stab}(I)\ =\ \min\\{t\ :\ \operatorname{v}(I^{k})=\alpha(I)k+b,\
\textit{for all}\ k\geq t,\ \textit{and some}\ b\in{\mathbb{Z}}\\}.$

Let $\mathfrak{p}\in\operatorname{Ass}^{\infty}(I)$. We define the
$\operatorname{v}_{\mathfrak{p}}$-stability index of $I\subset S$ as the
integer

$\operatorname{v}_{\mathfrak{p}}\\!\textup{-stab}(I)\ =\ \min\\{t\ :\
\operatorname{v}_{\mathfrak{p}}(I^{k})=ak+b,\ \textit{for all}\ k\geq t,\
\textit{and some}\ a,b\in{\mathbb{Z}}\\}.$

Recall that by $\textup{astab}(I)$ one denotes the integer

$\textup{astab}(I)\ =\ \min\\{t\ :\
\operatorname{Ass}(I^{k})=\operatorname{Ass}^{\infty}(I),\ \textit{for all}\
k\geq t\\}.$

Whereas, for $\mathfrak{p}\in\operatorname{Ass}^{\infty}(I)$ we define the
integer

$\textup{astab}_{\mathfrak{p}}(I)\ =\ \min\\{t\ :\
\mathfrak{p}\in\operatorname{Ass}(I^{k}),\ \textit{for all}\ k\geq t\\}.$

Notice that
$\max_{\mathfrak{p}\in\operatorname{Ass}^{\infty}(I)}\textup{astab}_{\mathfrak{p}}(I)=\textup{astab}(I)$
and
$\textup{astab}_{\mathfrak{p}}(I)\leq\operatorname{v}_{\mathfrak{p}}\\!\textup{-stab}(I)$.
On the other hand,

$\textup{vstab}(I)\ \leq\
\max_{\mathfrak{p}\in\operatorname{Ass}^{\infty}(I)}\operatorname{v}_{\mathfrak{p}}\\!\textup{-stab}(I)$

and the inequality may be strict.

For instance, for any edge ideal $I(G)$ having linear resolution,
$\textup{vstab}(I(G))=1$. Indeed, by [13, Theorem 5.1] we have
$\operatorname{v}(I(G)^{k})=2k-1$ for all $k\geq 1$. However, in most cases
$\operatorname{Ass}(I(G))\neq\operatorname{Ass}^{\infty}(I(G))$. Hence, for
some $\mathfrak{p}\in\operatorname{Ass}^{\infty}(I(G))$,
$\operatorname{v}_{\mathfrak{p}}\\!\textup{-stab}(I(G))\geq\textup{astab}_{\mathfrak{p}}(I(G))\geq
2>1=\textup{vstab}(I(G))$. For a simple example, consider
$I(G)=(x_{1}x_{2},x_{1}x_{3},x_{2}x_{3})$, the edge ideal of the complete
graph on three vertices. Notice that $I(G)$ has linear resolution. The package
VNumber [15] reveals that
$\operatorname{Ass}(I(G))=\\{(x_{1},x_{2}),(x_{1},x_{3}),(x_{2},x_{3})\\}$ and
$\operatorname{Ass}(I(G)^{k})=\operatorname{Ass}(I(G))\cup\\{(x_{1},x_{2},x_{3})\\}$
for all $k\geq 2$. So that
$\operatorname{v}_{(x_{1},x_{2},x_{3})}\\!\textup{-stab}(I(G))=2$.

This example also shows that $\textup{vstab}(I)$ can be smaller than
$\textup{astab}(I)$. On the other hand, if $I$ is a monomial ideal of
$K[x,y]$, by [14, Corollary 5.3] we have $\textup{astab}(I)=1$, while
$\textup{vstab}(I)\geq 1$ can be bigger than $\textup{astab}(I)$. Thus, there
is no comparison between the stability indices $\textup{astab}(I)$ and
$\textup{vstab}(I)$.

Again, let $I\subset S$ be a graded ideal. By Brodmann [7],
$\operatorname{depth}I^{k}=\operatorname{depth}I^{k+1}$ for all $k\gg 0$. We
denote by $\textup{dstab}(I)$ the least integer $t$ such that
$\operatorname{depth}I^{k}=\operatorname{depth}I^{k+1}$ for all $k\geq t$.
Finally, let $\operatorname{reg}(I^{k})=ck+d$ for all $k\gg 0$. We denote by
$\textup{rstab}(I)$ the least integer $t$ such that
$\operatorname{reg}(I^{k})=ck+d$ for all $k\geq t$.

######  Question 5.6.

Given positive integer $a,d,r,v\geq 1$, can we find a graded ideal $I$ in some
polynomial ring $S$ such that

$\textup{astab}(I)=a,\ \ \ \textup{dstab}(I)=d,\ \ \ \textup{rstab}(I)=r,\ \ \
\textup{vstab}(I)=v?$

Let $I\subset S$ be a monomial ideal. In [23], Le Tuan Hoa bounded the index
$\textup{astab}(I)$ of $I$ using exquisite techniques from combinatorics and
asymptotic linearity of integer programming. For similar recent results, see
the papers [22, 24].

######  Problem 5.7.

Let $I\subset S$ be a monomial ideal. Using integer programming techniques,
determine good upper bounds for the $\operatorname{v}$-stability index
$\textup{vstab}(I)$ of $I$.

Acknowledgment. We thank A. Vanmathi and P. Sarkar for their careful reading
of the paper and pointing out some inaccuracies in a previous version of the
manuscript. Lemma 1.5 owes their existence to them.

## References

  * [1] S. H. Ahn, Asymptotic primes and asymptotic grade on modules, J. Algebra 174 (1995), 980-998.
  * [2] S. Bayati, J. Herzog, G. Rinaldo, On the stable set of associated prime ideals of a monomial ideal, Arch. Math., 98(2012), 213-217.
  * [3] S. Bayati, J. Herzog, G. Rinaldo, A routine to compute the stable set of associated prime ideals of a monomial ideal, http://ww2.unime.it/algebra/rinaldo/stableset/, 2011.
  * [4] P. Biswas, M. Mandal, A study of v-number for some monomial ideals, 2023, arXiv preprint arXiv:2308.08604.
  * [5] P. Biswas, M. Mandal, K. Saha, Asymptotic behaviour and stability index of v-numbers of graded ideals, 2024, preprint arXiv:2402.16583.
  * [6] M. Brodmann, Asymptotic stability of $\textup{Ass}(M/I^{n}M)$, Proc. Am. Math. Soc., 74(1979), 16–18
  * [7] M. Brodmann, The asymptotic nature of the analytic spread, Math. Proc. Cambridge Philos. Soc., 86 (1979), 35–39.
  * [8] W. Bruns, A. Conca, C. Raicu, M. Varbaro, Determinants, Gröbner bases and cohomology. Springer, 2022.
  * [9] W. Bruns, J. Herzog, Cohen–Macaulay rings, Cambridge University Press, 1998.
  * [10] A. Conca. A note on the v-invariant, 2023, arXiv:2401.00022.
  * [11] S.M. Cooper, A. Seceleanu, S.O. Tohăneanu, M. Vaz Pinto, R.H. Villarreal. Generalized minimum distance functions and algebraic invariants of geramita ideals. Advances in Applied Mathematics, 112:101940, 2020.
  * [12] A. De Stefani, J. Montaño, L. Núñez-Betancourt. Frobenius methods in combinatorics. São Paulo Journal of Mathematical Sciences 17.1 (2023): 387-429.
  * [13] A. Ficarra, Simon’s conjecture and the $\operatorname{v}$-number of monomial ideals, 2023, preprint https://arxiv.org/abs/2309.09188
  * [14] A. Ficarra, E. Sgroi, Asymptotic behaviour of the $\operatorname{v}$-number of homogeneous ideals, 2023, preprint https://arxiv.org/abs/2306.14243
  * [15] A. Ficarra, E. Sgroi, VNumber, Macaulay2 Package available at https://github.com/EmanueleSgroi/VNumber, 2024.
  * [16] L. Fiorindo, A theorem on the asymptotic behaviour of the generalised $\operatorname{v}$-number, 2024, preprint arxiv:2401.17815.
  * [17] D. Ghosh, On the asymptotic v-numbers of graded modules, 2024, preprint arxiv:2401.16358.
  * [18] D. R. Grayson, M. E. Stillman. Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2.
  * [19] G. Grisalde, E. Reyes, R.H. Villarreal. Induced matchings and the v-number of graded ideals. Mathematics, 9(22), 2021.
  * [20] J. Herzog, T. Hibi. _Monomial ideals_ , Graduate texts in Mathematics 260, Springer, 2011.
  * [21] J. Herzog, T. Hibi, N.V. Trung, Symbolic powers of monomial ideals and vertex cover algebras, Advances in Mathematics 210 (2007), no. 1, 304–322.
  * [22] C. Heuberger, J. Rath, R. Rissner, Stabilization of associated prime ideals of monomial ideals – Bounding the copersistence index, 2023, preprint arXiv:2310.13431.
  * [23] L. T. Hoa, Stability of associated primes of monomial ideals. Vietnam. J. Math. 34 (2006), 473-487.
  * [24] L. T. Hoa, Asymptotic behavior of integer programming and the stability of the Castelnuovo–Mumford regularity. Math. Program. 193 (2022), no. 1, Ser. A, 157–194.
  * [25] C. Huneke, I. Swanson. Integral closure of ideals, rings, and modules. Vol. 336. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2006.
  * [26] N. Kotal, K. Saha, On the v-number of Gorenstein ideals and Frobenius powers, 2023, preprint arxiv:2311.04136.
  * [27] S. McAdam, P. Eakin, The asymptotic Ass, J. Algebra, 61 (1979), 71-81.
  * [28] L.J. Ratliff, Notes on essentially powers filtrations. Michigan Math. J. 26(3) (1979), 313–324.
  * [29] K. Saha, The $\operatorname{v}$-number and Castelnuovo-Mumford regularity of cover ideals of graphs, 2023, preprint arXiv:2308.09996
  * [30] K. Saha, I. Sengupta, The $\operatorname{v}$-number of monomial ideals. J Algebr Comb 56, 903–927 (2022) https://doi.org/10.1007/s10801-022-01137-y
  * [31] A. Vanmathi, P. Sarkar, $v$-numbers of filtrations of homogeneous ideals, 2024, preprint https://arxiv.org/abs/2403.09175.