∎

11institutetext: Roberta Amato 22institutetext: Dipartimento di Fisica e
Chimica - Emilio Segré, Università degli Studi di Palermo, via Archirafi, 36,
90123 Palermo, Italy
INAF-IASF Palermo, via Ugo La Malfa, 153, 90146 Palermo, Italy
IAAT, University of Tübingen, Sand 1, 72076 Tübingen, Germany
22email<EMAIL_ADDRESS>33institutetext: Sebastian Diebold
44institutetext: Alejandro Guzman 55institutetext: Emanuele Perinati
66institutetext: Chris Tenzer 77institutetext: Andrea Santangelo
88institutetext: IAAT, University of Tübingen, Sand 1, 72076 Tübingen, Germany
99institutetext: Teresa Mineo 1010institutetext: INAF-IASF Palermo, via Ugo La
Malfa, 153, 90146 Palermo, Italy

# Scattering efficiencies measurements of soft protons at grazing incidence
from an Athena Silicon Pore Optics sample

Roberta Amato  Sebastian Diebold  Alejandro Guzman  Emanuele Perinati  Chris
Tenzer  Andrea Santangelo  Teresa Mineo

(Received: date / Accepted: date)

###### Abstract

Soft protons are a potential threat for X-ray missions using grazing incidence
optics, as once focused onto the detectors they can contribute to increase the
background and possibly induce radiation damage as well. The assessment of
these undesired effects is especially relevant for the future ESA X-ray
mission Athena, due to its large collecting area. To prevent degradation of
the instrumental performance, which ultimately could compromise some of the
scientific goals of the mission, the adoption of ad-hoc magnetic diverters is
envisaged. Dedicated laboratory measurements are fundamental to understand the
mechanisms of proton forward scattering, validate the application of the
existing physical models to the Athena case and support the design of the
diverters. In this paper we report on scattering efficiency measurements of
soft protons impinging at grazing incidence onto a Silicon Pore Optics sample,
conducted in the framework of the EXACRAD project. Measurements were taken at
two different energies, $\sim$470 keV and $\sim$170 keV, and at four different
scattering angles between 0.6° and 1.2°. The results are generally consistent
with previous measurements conducted on eROSITA mirror samples, and as
expected the peak of the scattering efficiency is found around the angle of
specular reflection.

###### Keywords:

Soft protons X-ray background proton scattering grazing incidence angle X-ray
astronomy

##  1 Introduction

Orbiting X-ray observatories are subjected to the impact of charged particles
of different origins, e.g. cosmic rays, solar wind and solar events, electrons
and protons trapped in the Earth magnetosphere, etc. Amongst them, protons
with typical energies ranging from a few tens to a few hundreds of
kiloelectronvolts, the so-called ‘soft protons’ (SP), are scattered and
funneled towards the focal plane when impinging at grazing incidence on X-ray
optics. Once they reach the instruments at the focal plane, they significantly
contribute to the level of non-X-ray background (NXB), and potentially damage
the detectors in the most severe cases. The effects of SP were first observed
on the Chandra X-ray Observatory (Weisskopf et al. 2000) and on XMM-Newton
(Jansen et al. 2001), both launched in 1999. To this day SP affect the
operability of these X-ray missions by significantly reducing their useful
observing time and their duty cycles – for instance, the observing time of
XMM-Newton is reduced by ∼30–40 % (Ghizzardi et al. 2017).

SP constitute an even bigger issue for Athena (Advanced Telescope for High
Energy Astrophysics, Nandra et al. 2013), an L-class ESA mission, planned to
be launched in the early 2030s. The telescope will be equipped with Silicon
Pore Optics (SPO) (Willingale et al. 2013), wafers of silicon coated with
high-Z materials, stacked one upon the other and arranged to cover the whole
pupil of the telescope.

Athena will orbit around the second Lagrangian point L2111Currently, there are
strong suggestions in favour of an L1 orbit, where the particle environment is
better known, though highly variable of the Earth-Sun system, at a distance of
about 1.5 million kilometer from Earth in the anti-Sun direction. This orbit
lies in the tail of the Earth magnetosphere, where the particle environment is
extremely variable (Lotti et al. 2017). At present, the science driven
requirement for SP is that their flux at the focal plane has to be $<5\times
10^{-4}$ cts s-1 cm-2 keV-1 (corresponding to 10 % of the total NXB), in the
2–10 keV energy band, for 90 % of the observing
time222https://www.cosmos.esa.int/documents/400752/507693/Athena_SciRd_iss1v5.pdf.
In order to have a thorough estimate of the SP flux expected at the focal
plane, knowing the scattering efficiency of soft protons at grazing incidence
from Athena’s SPO is fundamental. To achieve this goal, we performed
experimental scattering efficiency measurements with a SPO sample.

Similar experimental measurements have already been performed for XMM-Newton
(Rasmussen et al. 1999) and eROSITA (Diebold et al. 2015, 2017) mirror
samples. Especially the measurements on eROSITA showed that the scattering is
always non-elastic, with energy losses of the order of a few tens of
kiloelectronvolts, depending on the energy of the incident beam. The
scattering efficiency always peaks close to the angle of specular reflection
and is higher for lower incident angles, while it is almost independent of the
energy of the impinging protons.

A model for the scattering efficiency of ions at grazing incidence from the
surface of a material was developed by Remizovich et al. (1980), in non-
elastic approximation. However, the model is generally valid for any surface
and contains some approximations, therefore a simple application of the
Remizovich formula to the eROSITA data did not exactly reproduce the
efficiencies measured in the laboratory (Amato et al. 2020). A fit of the
experimental data with the Remizovich formula led to the evaluation of the
parameter enclosing the micro-physics of the scattering, so that a new
analytical semi-empirical model that better reproduces the data from the
eROSITA mirror sample was derived (Amato et al. 2020). This model can be used
to assess the SP flux expected at the instrumental focal plane of the
satellite. The model can also be applied to Athena, provided that the proton
scattering properties of the SPO are experimentally determined.

In this publication, we present the first measurements of scattering
efficiencies of low-energy protons off a SPO sample. These experimental
activities were conducted in the framework of the EXACRAD (Experimental
Evaluation of Athena Charged Particle Background from Secondary Radiation and
Scattering in Optics) project funded by ESA. The paper is structured as
follows: we first describe the laboratory setup and the elements along the
beam line in Sect. 2; in Sect. 3 we illustrate how the scattering efficiency
is derived from the raw data; the new data are presented in Sect. 4 and are
compared to the data from eROSITA as well as to the semi-empirical model
mentioned above in Sect. 9; finally, we draw our conclusions in Sect. 5.

##  2 Experimental setup

The experiment was conducted at the 2.5 MV Van de Graaff accelerator at the
Goethe University (Riedberg Campus) in Frankfurt am Main. The setup of the
beamline, similar to that of Diebold et al. (2015, 2017), is given in Figs. 1,
2, and 3.

Figure 1: Schematic drawing (not in scale) of the beamline setup. The proton
beam enters the setup from the left-hand side. It encounters the pinhole
aperture (1 mm in diameter), the Al degrader foil (0.002 mm thick) and the
collimator. Inside the target chamber, the normalisation detector can be
lowered down to intercept the beam for the normalisation measurements. If the
normalisation detector is not in the line of the beam, then protons are
reflected from the SPO sample (in yellow) towards the detector chamber, where
they hit the central and lateral detectors. The incident angle $\theta_{0}$
between the line of the beam and the mirror varies with the inclination of the
target plate, while the scattering angle $\theta$ between the mirror and the
detectors in the detector chamber varies with the their height $h$. The
distance $d$ between the target plate and the vertical ax of the central
detectors is fixed to 942 mm. Figure 2: A CAD model of the beamline (same as
Diebold et al. 2015). The proton beam enters the setup from the right and
moves towards the left. The SPO sample is located in the target chamber, while
the detector is placed in the chamber at the end of the beamline (detector
chamber). A second detector (not shown in the picture) was placed next to the
central one, with an angular distance of $\sim 2^{\circ}$. Figure 3: Picture
of the beamline at the Van der Graaff accelerator facility. The beam direction
is from the right to the left. The position of the degrader foils and the
collimator inside the beamline are pointed out, as well as the detector and
target chambers.

Figure 4: Pictures of the SPO sample, with the reflecting surface up (left)
and down (right).

###  2.1 Beamline setup

Protons enter the beamline through a copper pinhole aperture of the diameter
of 1 mm, which reduces the size of the incoming beam to prevent pile-up and to
maintain reasonable rates on the detectors. Successively, the beam goes
through a 0.002 mm thick aluminium foil, which degrades the incoming beam
energy below the lower limit of the accelerator. The degraded beam enters, at
this point, a 78 cm long collimator, which directs part of the widened beam
directly to the target. Two further apertures are positioned at the entrance
and at the exit of the collimator, both with a diameter of 1 mm. This diameter
limits the smallest possible incident angle to $\sim$0.5∘. The apertures are
supported in their position by 2 mm aluminum plates, which absorb any proton
of the degraded beam not entering the apertures and being scattered by the
inner walls of the collimator and of the beamline itself.

The SPO target (Fig. 4), provided by
cosine333https://www.cosine.nl/cases/silicon-pore-optics-mirror-modules-spo-
for-astronomy/., consists of a 110 mm long single silicon wafer, 0.775 mm
thick, grooved in the bottom, and coated on top with a 10 nm of iridium and 7
nm of silicon carbide444Though iridium is the baseline coating material, a
low-Z overcoating is also considered to improve the reflectivity at lower
energies. Different low-Z materials and thicknesses are currently under
investigation.. It is located in an apposite chamber (hereafter called target
chamber) and mounted on a tiltable plate. The height of the target can be
adjusted by a set of screws underneath the plate. A linear manipulator is used
to change the inclination of the plate, i.e., of the incident angle
($\theta_{0}$). The pivoting point is is several centimeters below the line of
the beam, so that the target can be completely removed from the beam, allowing
for a determination of the primary beam position on the detector plane. The
manipulator is set below the target chamber and hence can be easily accessed
when the system is under vacuum.

Between the exit of the collimator and the target plate, a Passivated
Implanted Planar Silicon (PIPS) detector555The PIPS detectors used in this
experiment have a nominal depletion region of 0.1 mm and a lower energy
threshold of a few tens of keV. is mounted on a push-pull manipulator, at the
same height of the beamline. This detector is used to register the amount of
flux of the incident beam impinging on the target, useful to have
normalisation measurements. This detector will be called hereafter
’normalisation detector’. The push-pull manipulator permits a fast removal of
the detector, guaranteeing a measure of the impinging proton flux
($\Phi_{inc}$, cfr. Eq. 2) for each measure of the scattered beam (see Sect. 3
for the need of having frequent normalisation measurements). An aluminium
blind with an aperture of 3 mm is set on top of the normalisation detector to
avoid saturation. Lastly, downstream of the target chamber, a thick aluminum
sheet, with a slit of 3 cm height and 1 cm width, is installed a few
centimeters after the target plate. This window lets pass only the protons on
the line of the beam, while the sheet absorbs all the ones that have been
scattered by the inner walls or by the other elements in the target chamber.

At the end of the beamline, a second chamber (hereafter detector chamber)
hosts two more PIPS detectors, called ‘central detector’ and ‘lateral
detector’, respectively, used to register the on-axis and off-axis fluxes
($\Phi_{\text{scat}}(\theta_{0},\theta,\phi)$, cfr. Eq. 2) of the beam
scattered by the target. They are mounted on a second linear manipulator,
which allows a spatially resolved sampling of the scattered beam. The distance
between the center of the target plate and the detection plane is 942 mm. The
central detector is aligned with the beam, while the lateral detector is set
on the left. This configuration allows for a coverage of the scattered beam on
the the incident direction and at an azimuthal angle $\phi$ of
$1.97^{\circ}\pm 0.13^{\circ}$. On top of each detector there is a blind with
an aperture of a diameter of 1 mm for the central detectors and of 3 mm for
the lateral detector, respectively. They reduce the solid angle of the
detectors with respect to the mirror center to about $8\times 10^{-7}$ sr and
$2\times 10^{-5}$ sr for the central and lateral detector, respectively.

###  2.2 Data acquisition chain

The pulse signal produced by the PIPS is amplified and digitalised trough
several analogical/digital electronic components. A flow chart is given in
Fig. 5.

The PIPS detectors produce a pulse with an amplitude proportional to the
energy of the incident particle. The pulse signal from each PIPS goes through
its own pre-amplifier and amplifier and it is then digitalised by the Analog
to Digital Converter (ADC). The ADC receive the continuous signal (from 0 to
$\sim$10 V) from the three channels – one for each detector – and convert them
into discrete signals, distributing it into 8192 bins, with a resolution of
1.22 mV. The digitised signals are then passed to the histogramming memory,
which produces an histogram for each channel. Once the measurement is done,
the histograms are read out by the CAMAC module and are transferred to a
computer, which acquires and stores them as raw data files.

The process of digitalisation of the data within the ADC takes a certain time
(fractions of seconds), so that if a new signal comes within that time, it is
not registered. To account for this dead-time, a pulse generator, which
generates pulses at a fixed frequency, is connected to the ADC and to a
scaler, which counts the number of pulses produced by the pulse generator
during the acquisition time. The scaler is also fed to the CAMAC control
module. The difference between the reading of the counts from the ADC and that
from the scaler gives the dead-time correction factor (cfr. Eq. 4). The pulse
generator fed to the ADC constitutes another channel, so that the whole
acquisition systems consists of four channels, all working simultaneously, and
the scaler.

Figure 5: Data flow of the electronic chain for the acquisition of the
experimental data. The analogical signal from the PIPS detectors first goes
through a pre-amplifier and an amplifier, then it is converted into a digital
signal by the ADC, and finally it is stored in the histogramming memory.
Contemporary, a pulse genarator sends signal at a fixed frequency to the ADC
and to a scaler. The digitised signals are read out by a CAMAC controller
unit, which transmits them to a computer once the measurement is finished.

###  2.3 Alignment and angular calibration

The alignment of all the movable elements on the beamline, i.e., the pinhole
aperture, the slits, the normalisation detector, and the central detector, is
done by using a telescope previously aligned with the exit of the accelerator.

A 520 nm laser is employed to perform the angular calibration. The laser is
set right after the pinhole aperture and goes through all the slits. When the
target plate is down, the laser reaches the central detector in the detector
chamber. In this way, the zero of the beamline, corresponding to
$\theta=0^{\circ}$, can be established. This measurement gives also the
vertical offset on the linear manipulator of the central/lateral detectors.

To calibrate the incident and scattering angles, we use the property of the
mirror target to reflect optical light. Hence, we raise the target plate,
using its own manipulator, until the light is blocked. Then, we raise the
central detector till the laser beam is detected again. Assuming a specular
reflection, the angle subtended by the height $h$ of the manipulator will be
$\zeta=\theta+\theta_{0}=2\theta_{0}$, so that the incident angle can be
computed as:

$\theta_{0}=\frac{\zeta}{2}$ (1)

This operation is repeated several time, so that we end up with different
angles corresponding to different readings on the linear manipulator of the
target plate. The incident angle can then be determined with a simple linear
interpolation.

##  3 Efficiency definition and normalisation measurement

Figure 6: Geometrical scheme of the incident plane. The proton beam hits the
mirror sample, in the plane $xz$ with an incident angle $\theta_{0}$ and is
pseudo-reflected with a polar scattering angle $\theta$ and an azimuthal
scattering angle $\phi$.

In the laboratory system of reference, the scattering efficiency per unit
solid angle can be defined as:

$\eta(\theta_{0},\theta,\phi)=\frac{1}{\Omega(\theta,\phi)}\frac{\Phi_{\text{scat}}(\theta_{0},\theta,\phi)}{\Phi_{\text{inc}}}$
(2)

where $\theta_{0}$ is the incident angle, $\theta$ and $\phi$ are the polar
and azimuthal scattering angles (see Fig. 6), $\Phi_{scat}$ and $\Phi_{inc}$
are the scattered and incident proton count rates, and $\Omega(\theta,\phi)$
is the solid angle subtended by the detector.

The count rate of the scattered particles is given by the number of protons
$N_{\text{scat}}$ scattered by the SPO sample reaching the detectors in the
detector chamber divided by the integration time $\Delta t_{\text{scat}}$. In
a similar way, the count rate of the incident particles is given by the number
of particles $N_{\text{inc}}$ intercepted by the normalisation detector in
front of the mirror chamber divided by the integration time $\Delta
t_{\text{inc}}$. The number of counts of incident and scattered protons,
$N_{\text{inc}}$ and $N_{\text{scat}}$, is obtained by integrating the ADC
histograms. This number must be corrected for the dead-time of the ADC (cfr.
Sect. 2.2), so that the effective count rates can be expressed as:

$\Phi_{\text{scat}}(\theta_{0},\theta,\phi)=\alpha\,\frac{N_{\text{scat}}(\theta_{0},\theta,\phi)}{\Delta
t_{\text{scat}}}\quad,\quad\quad\Phi_{\text{inc}}=\alpha\,\frac{N_{\text{inc}}}{\Delta
t_{\text{inc}}}$ (3)

with the correction factor $\alpha$ given by:

$\alpha=\frac{N_{\text{scaler}}}{(N_{\text{pulser}})_{\text{ADC}}}$ (4)

where $N_{\text{scaler}}$ is the number of counts from the pulse generator as
read out from the scaler fed to the CAMAC controller module and
$(N_{\text{pulser}})_{\text{ADC}}$ is the number of pulses from the pulse
generator as read out from the ADC (see Fig. 5). For an ideal incoming proton
beam, the number of incident particles $N_{\text{inc}}$ is constant in time.
However, the beam exiting the Van de Graaff accelerator was not stable, with
fluctuations in the direction of the beamline varying in a time range from a
few to several tens of minutes. This made necessary to take normalisation
measurements before and after each scattering measurement and average them for
each scattering data point, so that:

$\frac{N_{\text{inc}}}{\Delta
t_{\text{inc}}}=\frac{1}{2}\left(\frac{N_{\mathrm{inc,1}}}{\Delta
t_{\mathrm{inc,1}}}+\frac{N_{\mathrm{inc,2}}}{\Delta
t_{\mathrm{inc,2}}}\right)$ (5)

where $N_{\mathrm{inc,1}}$ and $N_{\mathrm{inc,2}}$ are the counts in two
consecutive normalisation measurements with integration times $\Delta
t_{\mathrm{inc,1}}$ and $\Delta t_{\mathrm{inc,2}}$, respectively.

Concerning the uncertainties, the one on the scattering angle is given mainly
by the errors on the angular calibration, the detector aperture, and the
indeterminate position of the impact point of the beam on the mirror surface.
The uncertainty on the incident angle $\theta_{0}$ is dominated by the
dimension of the aperture on the central detector and by the length of the
target. It resulted in $\sim$$None$ for all the chosen scattering angles.
Lastly, the uncertainty on the scattering efficiency is mainly given by the
intrinsic fluctuation of the proton beam. Minor contributions are due to the
count statistics and to the error on the solid angle $\Omega(\theta,\phi)$.
The sum of this contributions results in statistical fluctuations of $\pm$20%
on the scattering efficiencies.

##  4 Results and discussion

We measured the scattering efficiency at two different energies (high- and
low-energy data sets, hereafter) and at four different incident angles:
$None$, $None$, $None$, and $None$, both on-axis and off-axis, the latter at
an angle $\phi$ of about 2∘. Each data set consists of on-axis and off-axis
scattering efficiencies. Results are shown in Figs. 7 and 8, where the
scattering efficiencies have been multiplied by the square of the incident
angle (as in Amato et al. 2020) and are displayed as a function of the
scattering angle divided by the incident one, i.e., $\Psi=\theta/\theta_{0}$.

For the high-energy data set (Fig. 8) we used a beam at
$\sim$$590\text{\,}\mathrm{keV}$ from the accelerator, which was degraded by
the Al foil down to 471$\pm$25 $\mathrm{keV}$. This energy value was chosen
mainly for purposes of comparison with the previous measurements on the
eROSITA mirror sample (Diebold et al. 2015, 2017, cfr. Fig. 9). The rationale
behind the low-energy value can be found in the work of Lotti et al. (2017),
who showed that the highest transmission efficiency of soft protons is
observed for those protons impacting the mirrors with 40-60 keV, for both
instruments on board of Athena. Hence, it is crucial to investigate the
scattering of soft protons at energies around and below 100 keV.
Unfortunately, the present setup could not reach such low energies, limiting
us to a proton beam with an energy of $\sim$340 $\mathrm{keV}$ at the exit of
the accelerator, degraded to 172$\pm$30 $\mathrm{keV}$ by the Al foil. In both
cases, the values of the incident energies were determined by simulations with
the software TRIM666The TRIM code is one of the SRIM (Stopping and Range of
Ions in Solids) group of programs, available at
http://www.srim.org/index.htm#HOMETOP. (TRansport of Ions in Matter, Ziegler
et al. 2010), already validated in Diebold et al. (2015).

Figure 7: Scattering efficiencies of the low-energy data set as a function of
the scattering angle $\Psi=\theta/\theta_{0}$, for the incident angels of
$None$, $None$, $None$, and $None$, for the on-axis (top panel) and off-axis
(bottom panel) configurations. Energy beam of 172$\pm$30 keV.

Figure 8: Scattering efficiencies of the high-energy data set as a function of
the scattering angle $\Psi$, for the incident angels of $None$, $None$,
$None$, and $None$, for the on-axis (top panel) and off-axis (bottom panel)
configurations. Energy beam of 471$\pm$25 keV.

As expected, the on-axis scattering efficiencies peak at the specular angle
($\psi\simeq 1$) and are consistent with each other within the uncertainties.
However, a higher spread is observed for the high-energy on-axis data set
(Fig. 8, top panel), with efficiencies ranging from 0.03 to 0.07 at the peak
of the distribution. Also the off-axis data show a significant spread, which
is expected in this case.

Overall, the maximum scattering efficiency values are $\sim$0.07 and
$\sim$0.02 for the on-axis and off-axis configurations, respectively, with the
low-energy data set showing slightly smaller efficiencies than the high-energy
one.

###  4.1 Comparison with the eROSITA measurements

Fig. 9 shows the eROSITA measurements (Diebold et al. 2015, 2017) overlapped
to the SPO data, for both the energies and the ox-axis and off-axis
configurations. Though the SPO efficiencies are systematically higher than the
eROSITA data, they are consistent within the error bars.

Figure 9: Comparison of the eROSITA scattering efficiencies (blue dots) with
the SPO ones (green dots for the low-energy set and red dots for the high-
energy set), for the on-axis (top panels) and off-axis (bottom panels) data.

Due to this consistency, we applied the semi-empirical model proposed in Amato
et al. (2020) to the scattering efficiencies of SPO, as shown in Fig. 10.

Overall, the model well reproduces the scattering efficiency of the low-energy
data set, but overestimates the efficiency of the high-energy data set by a
factor of $\sim$1.5 times. Nonetheless, it has to be borne in mind that at
this stage we simply overlapped the semi-empirical model developed for the
eROSITA to the new SPO experimental data. A more accurate model, specific for
SPO, can be obtained by fitting the data with the formula of Remizovich et al.
(1980) in non-elastic approximation, as in Amato et al. (2020), provided that
energy loss measurements are retrieved from the raw data.

Lastly, we group the efficiency values from the two data sets by the incident
angle, irrespective of the energy of the incident beam. Fig. 11 shows that the
data are perfectly consistent with each other and with the old eROSITA
measurements when grouped by the incident angle, without accounting for the
energy. Once again, the semi-empirical model derived for the eROSITA data is
overlapped with the data, resulting in a general acceptable agreement.

Figure 10: Comparison between the experimental scattering efficiency of SPO
(points) and the semi-empirical model developed from eROSITA data (solid
line), for the low-energy (green) and high-energy (red) data sets and for the
on-axis (top panels) and off-axis (bottom panels) measurements.

Figure 11: Scattering efficiencies from eROSITA mirror and the SPO samples,
grouped by different incident angles, irrespective of the energy of the
impinging beam. The solid line stand for the semi-empirical scattering model,
with the error on the resulting efficiency given by the green area.

##  5 Conclusions

Within the EXACRAD project, we measured for the first time the scattering
efficiency of a single wafer of SPO hit by low-energy protons at grazing
incidence. Measurements were performed at two different energies, of about
$470\text{\,}\mathrm{keV}$ and $170\text{\,}\mathrm{keV}$, and at four
different incident angles, $None$, $None$, $None$, and $None$, both on-axis
and at an off-axis angle of about 2∘.

Hereafter some major remarks:

  * •

the scattering efficiencies show the trend expected from Remizovich et al.
(1980) and from the experimental data on eROSITA (Diebold et al. 2015, 2017).
The on-axis data peak close to the specular reflection, while the off-axis
data show a peak shifted to higher $\Psi$; the off-axis data reach lower
efficiencies than the on-axis ones; higher incident angles resulted in higher
scattering efficiency;

  * •

the SPO data are generally consisted with the eROSITA data, though the high-
energy data set show a higher spread in efficiency;

  * •

as for the eROSITA data, the scattering efficiency very weakly depends on the
energy of the incident beam;

  * •

the semi-empirical model developed from eROSITA experimental data is able to
acceptably reproduce the low-energy data set, while it results in higher
efficiencies for the high-energy data set. The same model can be improved
specifically for SPO, with a direct fit of the experimental data, provided
that energy loss measurements are retrieved from the raw data.

As stated above, the experimental configuration used for this experiment is
not suited for measurements at energies below 100 keV, which are those
expected to contribute the most to the background of Athena (Lotti et al.
2018). Therefore, some possible changes to the setup could be explored in the
future to enable measurements at lower energy ranges.

The work here presented is only the first step towards a thorough estimation
of the SP flux expected at the focal plane of Athena. Indeed, after being
scattered by the optics, SP cross all the other elements along their path,
and, in particular, the filters located in front of each detector. As showed
in Lotti et al. (2018), the filters not only reduce the energy of the protons,
but also act as degraders, altering their original trajectories. Hence, a
thorough prediction of the overall transmission energy of SP can be achieved
only combining the processes of scattering from the optics, crossing of the
filters and energy release within the detectors. The latter two phenomena can
be investigated, for instance, by simulations, as done, e.g., by Fioretti et
al. (2018). If the estimated soft proton flux should still result higher than
the scientific requirement of $5\times 10^{-4}$ cts s-1 cm-2 keV-1, in the
2–10 keV energy band, for 90 % of the observing time (cfr. Sect. 1), then
appropriate solution should be adopted. Currently, the hypothesis of a
magnetic diverter specific for protons is under discussion and possible
designs are under investigation.

###### Acknowledgements.

The experiment was conducted under the ESA contract n°4000121062 /17/NL/LF.
Authors acknowledge S. Molendi, S. Lotti, and P. Nieminen as coordinators of
the EXACRAD project; M. Collon, for providing the SPO targets for the
experiment; S. Renner and S. Rieckert from the IAAT mechanical workshop for
contributing to the beamline setup; L. Schmidt and the colleagues of the Van
der Graaff accelerator facility for the onsite help. This version of the
article has been accepted for publication, after peer review but is not the
Version of Record and does not reflect post-acceptance improvements, or any
corrections. The Version of Record is available online at:
http://dx.doi.org/10.1007/s10686-021-09806-9.

## References

  * Amato et al. (2020) Amato R, Mineo T, D’Aı A, et al. (2020) Soft proton scattering at grazing incidence from X-ray mirrors: analysis of experimental data in the framework of the non-elastic approximation. Experimental Astronomy 49(3):115–140, DOI 10.1007/s10686-020-09657-w, 2003.07295
  * Diebold et al. (2015) Diebold S, Tenzer C, Perinati E, et al. (2015) Soft proton scattering efficiency measurements on x-ray mirror shells. Experimental Astronomy 39:343–365, DOI 10.1007/s10686-015-9451-4, 1504.01024
  * Diebold et al. (2017) Diebold S, Hanschke S, Perinati E, et al. (2017) Updates on experimental grazing angle soft proton scattering. In: Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 10397, p 103970W, DOI 10.1117/12.2272930
  * Fioretti et al. (2018) Fioretti V, Bulgarelli A, Molendi S, et al. (2018) Magnetic Shielding of Soft Protons in Future X-Ray Telescopes: The Case of the ATHENA Wide Field Imager. Astrophysical Journal 867(1):9, DOI 10.3847/1538-4357/aade99, 1808.09431
  * Ghizzardi et al. (2017) Ghizzardi S, Marelli M, Salvetti D, et al. (2017) A systematic analysis of the XMM-Newton background: III. Impact of the magnetospheric environment. Experimental Astronomy 44:273–285, DOI 10.1007/s10686-017-9554-1, 1705.04173
  * Jansen et al. (2001) Jansen F, Lumb D, Altieri B, et al. (2001) XMM-Newton observatory. I. The spacecraft and operations. Astronomy and Astrophysics 365:L1–L6, DOI 10.1051/0004-6361:20000036
  * Lotti et al. (2017) Lotti S, Mineo T, Jacquey C, et al. (2017) The particle background of the X-IFU instrument. Experimental Astronomy 44:371–385, DOI 10.1007/s10686-017-9538-1, 1705.04076
  * Lotti et al. (2018) Lotti S, Mineo T, Jacquey C, et al. (2018) Soft proton flux on ATHENA focal plane and its impact on the magnetic diverter design. Experimental Astronomy 45(3):411–428, DOI 10.1007/s10686-018-9599-9
  * Nandra et al. (2013) Nandra K, Barret D, Barcons X, et al. (2013) The Hot and Energetic Universe: A White Paper presenting the science theme motivating the Athena+ mission. arXiv e-prints 1306.2307
  * Rasmussen et al. (1999) Rasmussen A, Chervinsky J, Golovchenkor J (1999) Proton scattering off of XMM optics: XMM mirror and RGS grating sample. Rgs-col-cal-99009, Columbia Astrophysics Laboratory
  * Remizovich et al. (1980) Remizovich VS, Ryazanov MI, Tilinin IS (1980) Energy and angular distributions of particles reflected in glancing incidence of a beam of ions on the surface of a material. Soviet Journal of Experimental and Theoretical Physics 52:225
  * Weisskopf et al. (2000) Weisskopf MC, Tananbaum HD, Van Speybroeck LP, O’Dell SL (2000) Chandra X-ray Observatory (CXO): overview. In: Truemper JE, Aschenbach B (eds) X-Ray Optics, Instruments, and Missions III, Proceedings of the SPIE, vol 4012, pp 2–16, DOI 10.1117/12.391545, astro-ph/0004127
  * Willingale et al. (2013) Willingale R, Pareschi G, Christensen F, den Herder JW (2013) The Hot and Energetic Universe: The Optical Design of the Athena+ Mirror. arXiv e-prints arXiv:1307.1709, 1307.1709
  * Ziegler et al. (2010) Ziegler JF, Ziegler MD, Biersack JP (2010) SRIM - The stopping and range of ions in matter (2010). Nuclear Instruments and Methods in Physics Research B 268(11-12):1818–1823, DOI 10.1016/j.nimb.2010.02.091


# An inequality for the number of vertices with an interval spectrum in edge
labelings of regular graphs

N.N. Davtyan1, R.R. Kamalian2

###### Abstract

We consider undirected simple finite graphs. The sets of vertices and edges of
a graph $G$ are denoted by $V(G)$ and $E(G)$, respectively. For a graph $G$,
we denote by $\delta(G)$ and $\eta(G)$ the least degree of a vertex of $G$ and
the number of connected components of $G$, respectively. For a graph $G$ and
an arbitrary subset $V_{0}\subseteq V(G)$ $G[V_{0}]$ denotes the subgraph of
the graph $G$ induced by the subset $V_{0}$ of its vertices. An arbitrary
nonempty finite subset of consecutive integers is called an interval. A
function $\varphi:E(G)\rightarrow\\{1,2,\dots,|E(G)|\\}$ is called an edge
labeling of the graph $G$, if for arbitrary different edges $e^{\prime}\in
E(G)$ and $e^{\prime\prime}\in E(G)$, the inequality
$\varphi(e^{\prime})\neq\varphi(e^{\prime\prime})$ holds. If $G$ is a graph,
$x$ is its arbitrary vertex, and $\varphi$ is its arbitrary edge labeling,
then the set $S_{G}(x,\varphi)\equiv\\{\varphi(e)/e\in E(G),e\textrm{ is
incident with }x$} is called a spectrum of the vertex $x$ of the graph $G$ at
its edge labeling $\varphi$. If $G$ is a graph and $\varphi$ is its arbitrary
edge labeling, then $V_{int}(G,\varphi)\equiv\\{x\in
V(G)/\;S_{G}(x,\varphi)\textrm{ is an interval}\\}$. For an arbitrary
$r$-regular graph $G$ with $r\geq 2$ and its arbitrary edge labeling
$\varphi$, the inequality

$|V_{int}(G,\varphi)|\leq\bigg{\lfloor}\frac{3\cdot|V(G)|-2\cdot\eta(G[V_{int}(G,\varphi)])}{4}\bigg{\rfloor}.$

is proved.

Keywords: edge labeling, interval spectrum, regular graph, cubic graph.

Math. Classification: 05C15, 05C78

$\\\ ^{1}$Ijevan Branch of Yerevan State University, e-mail:
<EMAIL_ADDRESS>$\\\ ^{2}$The Institute for Informatics and Automation
Problems of NAS RA,
e-mail<EMAIL_ADDRESS>

We consider undirected simple finite graphs. The sets of vertices and edges of
a graph $G$ are denoted by $V(G)$ and $E(G)$, respectively. For a graph $G$,
we denote by $\delta(G)$ the least degree of a vertex of $G$. For any graph
$G$ we define a parameter $\eta(G)$ by the following way: if $G$ is empty then
$\eta(G)\equiv 0$, otherwise $\eta(G)$ is equal to the number of connected
components of $G$. If $G$ is a graph, $x\in V(G)$, $y\in V(G)$, then
$\rho_{G}(x,y)$ denotes the distance between the vertices $x$ and $y$ in $G$.
If $G$ is a graph, $x\in V(G)$, and $V_{0}\subseteq V(G)$, then
$\rho_{G}(x,V_{0})$ denotes the distance in the graph $G$ between its vertex
$x$ and the subset $V_{0}$ of its vertices. For a graph $G$ and an arbitrary
subset $V_{0}\subseteq V(G)$ $G[V_{0}]$ denotes the subgraph of the graph $G$
induced by the subset $V_{0}$ of its vertices.

For any graph $G$ and its arbitrary subgraph $H$, let us define the subgraph
$Surr[HinG]$ of the graph $G$ as follows:

$\begin{array}[]{l}V(Surr[HinG])\equiv\\{x\in V(G)/\;\rho_{G}(x,V(H))\leq
1\\},\\\ E(Surr[HinG])\equiv E(H)\cup\\{(x,y)\in E(G)/\;x\in
V(Surr[HinG])\backslash V(H),y\in V(H)\\}.\end{array}$

An arbitrary nonempty finite subset of consecutive integers is called an
interval. A function $\varphi:E(G)\rightarrow\\{1,2,\dots,|E(G)|\\}$ is called
an edge labeling of the graph $G$, if for arbitrary different edges
$e^{\prime}\in E(G)$ and $e^{\prime\prime}\in E(G)$, the inequality
$\varphi(e^{\prime})\neq\varphi(e^{\prime\prime})$ holds. For a graph $G$, the
set of all its edge labelings is denoted by $\tau(G)$.

If $G$ is a graph, $x\in V(G)$, $\varphi\in\tau(G)$, then the set
$S_{G}(x,\varphi)\equiv\\{\varphi(e)/e\in E(G),e\textrm{ is incident with }x$}
is called a spectrum of the vertex $x$ of the graph $G$ at its edge labeling
$\varphi$. If $G$ is a graph, $\varphi\in\tau(G)$, then
$V_{int}(G,\varphi)\equiv\\{x\in V(G)/\;S_{G}(x,\varphi)\textrm{ is an
interval}\\}$. The terms and concepts which are not defined can be found in
[1].

An upper bound for the cardinality of the set $V_{int}(G,\varphi)$ is obtained
in that cases when $G$ is a regular graph with $\delta(G)\geq 2$ and
$\varphi\in\tau(G)$.

First we recall the following

######  Proposition 1.

[2] Let $G$ be a graph with $\delta(G)\geq 2$. Let $\varphi\in\tau(G)$ and
$V_{int}(G,\varphi)\neq\emptyset$. Then $G[V_{int}(G,\varphi)]$ is a forest,
each connected component of which is a simple path.

######  Theorem 1.

If $G$ is a $r$-regular graph, $r\geq 2$, $\varphi\in\tau(G)$, then

$|V_{int}(G,\varphi)|\leq\bigg{\lfloor}\frac{r\cdot|V(G)|-2\cdot\eta(G[V_{int}(G,\varphi)])}{2\cdot(r-1)}\bigg{\rfloor}.$

Proof.  Let $\eta(G[V_{int}(G,\varphi)])=k$.

Case $\mathbf{1}$. $V_{int}(G,\varphi)=\emptyset$.

In this case the required inequality is the following evident one:

$0\leq\bigg{\lfloor}\frac{r\cdot|V(G)|}{2\cdot(r-1)}\bigg{\rfloor}.$

Case $\mathbf{2}$. $V_{int}(G,\varphi)\neq\emptyset$.

In this case $k\geq 1$. Since $\delta(G)=r\geq 2$, then, by the proposition 1,
$G[V_{int}(G,\varphi)]$ is a forest with $k$ connected components, each of
which is a simple path.

Let $P_{1},\dots,P_{k}$ be all connected components of the forest
$G[V_{int}(G,\varphi)]$.

It is not difficult to see that for $\forall i$, $1\leq i\leq k$, the equality
$|E(Surr[P_{i}inG])|=(r-1)\cdot|V(P_{i})|+1$ holds.

Let us also note that (if $k\geq 2$) for arbitrary integers $i^{\prime}$ and
$i^{\prime\prime}$ satisfying the inequality $1\leq
i^{\prime}<i^{\prime\prime}\leq k$, the relation
$E(Surr[P_{i^{\prime}}inG])\cap E(Surr[P_{i^{\prime\prime}}inG])=\emptyset$
holds.

Taking into account the evident relation
$(\bigcup_{i=1}^{k}E(Surr[P_{i}inG]))\subseteq E(G)$, we obtain

$|E(G)|=\frac{r\cdot|V(G)|}{2}\geq\bigg{|}\bigcup_{i=1}^{k}E(Surr[P_{i}inG])\bigg{|}=\sum_{i=1}^{k}|E(Surr[P_{i}inG])|=$
$=\sum_{i=1}^{k}((r-1)\cdot|V(P_{i})|+1)=k+(r-1)\cdot\sum_{i=1}^{k}|V(P_{i})|=k+(r-1)\cdot|V_{int}(G,\varphi)|,$
$|V_{int}(G,\varphi)|\leq\frac{1}{r-1}\cdot\bigg{(}\frac{r\cdot|V(G)|}{2}-k\bigg{)}=\frac{r\cdot|V(G)|-2k}{2\cdot(r-1)}.$

Consequently,

$|V_{int}(G,\varphi)|\leq\bigg{\lfloor}\frac{r\cdot|V(G)|-2k}{2\cdot(r-1)}\bigg{\rfloor}.$

The Theorem is proved.

######  Corollary 1.

If $G$ is a $r$-regular graph, $r\geq 2$, $\varphi\in\tau(G)$, then

$|V_{int}(G,\varphi)|\leq\bigg{\lfloor}\frac{r\cdot|V(G)|-2}{2\cdot(r-1)}\bigg{\rfloor}.$

.

######  Corollary 2.

If $G$ is a cubic graph, $\varphi\in\tau(G)$, then

$|V_{int}(G,\varphi)|\leq\bigg{\lfloor}\frac{3\cdot|V(G)|-2\cdot\eta(G[V_{int}(G,\varphi)])}{4}\bigg{\rfloor}.$

.

######  Corollary 3.

If $G$ is a cubic graph, $\varphi\in\tau(G)$, then

$|V_{int}(G,\varphi)|\leq\bigg{\lfloor}\frac{3\cdot|V(G)|-2}{4}\bigg{\rfloor}.$

.

## References

  * [1] D.B. West, Introduction to Graph Theory, Prentice-Hall, New Jersey, 1996.
  * [2] N.N. Davtyan, A.M. Khachatryan, R.R. Kamalian, On a subgraph induced at a labeling of a graph by the subset of vertices with an interval spectrum, Book of abstracts of the $8^{th}$ International Algebraic Conference in Ukraine. July 5–12(2011), Lugansk, pp. 61–62.


and $n_{1},\ldots,n_{s}$ with $n_{1}+\ldots+n_{s}=n$. The density of prime
numbers $p$ for which the reduction of $f$ modulo $p$ splits into $k$ factors
of degrees $f_{1},\ldots,f_{k}$ is equal to the number of permutations of the
roots of $f$ in the Galois group of $f$ consisting of $s$ cycles of lengths
$f_{1},\ldots,f_{s}$. In particular, $f$ splits into linear factors over
$\mathbb{Q}[x]$ if and only if its reduction modulo $p$ splits in
${\mathbb{F}}_{p}[x]$ into linear factors for almost all primes $p$.

This version was proven by Frobenius, while similar results were formulated by
Kronecker before. It is also an easy corollary of the Chebotarev density
theorem.

We now describe consequences of the hypothesis of the Grothendieck
$p$-curvature conjecture. They were already collected by Honda and we refer
for parts of the proof to his article. However, for the last assertion we give
a different proof. It compares the two algorithms obtained from the normal
form theorems in characteristics $0$ and $p$.

For an operator $L\in\mathbb{Q}[x][\partial]$ in characteristic $0$ and a
fixed prime $p$ we denote in the sequel by
${\overline{L}}=L_{p}\in{\mathbb{F}}_{p}[x][\partial]$ the reduction of $L$
modulo $p$, whenever this reduction is defined.

######  Proposition 4.16 (Honda).

Let $L\in\mathbb{Q}[x][\partial]$ be a differential operator with polynomial
coefficients over $\mathbb{Q}$. Assume that the induced equations
${\overline{L}}y=0$ modulo $p$ have an ${\mathbb{F}}_{p}[\\![x^{p}]\\!]$-basis
of power series solutions in ${\mathbb{F}}_{p}[\\![x]\\!]$, for almost all
primes $p$. Then

  1. (a)

The operator $L$ has a regular singularity at $0$.

  2. (b)

The local exponents of $L$ at $0$ are pairwise distinct rational numbers
$\rho_{i}\in\mathbb{Q}$.

  3. (c)

There exists a $\mathbb{Q}$-basis of Puiseux series solutions $y(x)$ of $Ly=0$
in $\sum_{\rho_{i}}x^{\rho_{i}}\mathbb{Q}[\\![x]\\!]$, where $\rho_{i}$ ranges
over the local exponents of $L$. In particular, this basis is independent of
the variables $t$ and $z_{i}$.

###### Proof.

For part (a) use [Hon81], Corollary p. 178, combined with Theorem. 4.6 and
Lemma 4.10 from above.

(b) See [Hon81], Thm. 2, p. 179, combined with Thms. 4.7 and 4.15 from above.
We provide here a variant of Honda’s proof. As a consequence of (a) there are
$n$ local exponents of $L$, counted with multiplicity, $n={\rm ord}\,L$.
Moreover, for almost all primes $p$, the local exponents of ${\overline{L}}$
have to be elements of the prime field ${\mathbb{F}}_{p}$. Indeed for any
local exponent $\rho\not\in{\mathbb{F}}_{p}$ we obtain using Theorem 3.17 a
solution of the form $t^{\rho}f(x)\in t^{\rho}{\mathbb{F}}[\\![x]\\!]$,
contradicting the existence of a basis of $n$ solutions of ${\overline{L}}y=0$
in ${\mathbb{F}}_{p}[\\![x]\\!]$.

It is then shown as in [Hon81] that the local exponents of $L$ are pairwise
incongruent modulo almost all primes $p$. The indicial polynomial $\chi_{L}$
of $L$ has coefficients in $\mathbb{Q}$ and its reduction modulo $p$ splits
into linear factors over ${\mathbb{F}}_{p}$ for almost all primes $p$. Thus,
by Theorem 4.15, $\chi_{L}$ splits into linear factors over $\mathbb{Q}$. It
follows that all local exponents of $L$ are rational. Assume now that two
local exponents are congruent modulo some $p$. Then their reduction modulo $p$
is a local exponent of ${\overline{L}}$ of multiplicity at least $2$. So,
Theorem 3.17 together with the remarks in section 4.2 upon avoiding the
variable $t$, yield a solution of ${\overline{L}}y=0$ of the form
$u^{-1}(x^{\rho}z_{1})$, where $u$ is the automorphism of the normal form
theorem in positive characteristic, Theorem 3.16. This solution now depends on
$z_{1}$, contradictory to the assumption. This proves (b).

(c) By Theorem 2.16 a basis of solutions of $Ly=0$ lies in

$\sum_{\rho_{i}}x^{\rho_{i}}\mathbb{Q}[\\![x]\\!][z],$

the sum varying over all local exponents $\rho_{i}$ of $L$. It remains to
prove that these solutions are independent of $z$. So assume the contrary: let
$f$ be a solution which depends on $z$. Without loss of generality we may
assume that

$f=u^{-1}(x^{\rho})=x^{\rho}(1+a_{1}(z)x+a_{2}(z)x^{2}+\ldots)$

for some local exponent $\rho\in\mathbb{Q}$ of $L$ and some
$a_{i}\in\mathbb{Q}[z]$. Let ${m}\in\mathbb{N}$ be the first index where
$a_{m}$ depends on $z$. We will construct from $f$ a solution $g$ of
${\overline{L}}y=0$, for a suitable prime $p$, which involves $z_{1}$-terms
which are not $p$-th powers. This will produce the required contradiction.

The construction of $g$ is, in fact, quite subtle. We have to run the two
normal form algorithms for the construction of $f$ and $g$ simultaneously in
characteristic $0$ and $p$ as long as no $z$ appears in characteristic $0$. At
the moment when $z$ occurs for the first time, say, in the computation of the
coefficient $a_{m}$ of $f$, a careful comparison ensures that $z_{1}$ shows up
also in the expansion of $g$ in the characteristic $p$ algorithm.

We choose the prime $p$ subject to the following conditions:

  * •

$p>n={\rm ord}\,L$;

  * •

There is a basis of solutions of ${\overline{L}}y=0$ in
${\mathbb{F}}_{p}[\\![x]\\!]$;

  * •

$p$ does not divide any of the denominators of the local exponents of $L$;

  * •

$p$ does not divide any of the denominators of the coefficients of
$a_{1},\ldots,a_{k}$.

Let $\Lambda$ be the set of positive integers $\ell$ smaller than ${m}$ such
that $\sigma:=\rho+\ell$ is a local exponent of ${\overline{L}}$. Here we
write $\sigma$ and $\rho$ for elements in $\mathbb{Q}$ as well as for the
representatives in $\\{0,1,\ldots,p-1\\}$ of their reduction modulo $p$. We
define

$g={\overline{u}}^{-1}\left(x^{\rho}\left(1+\sum_{\ell\in\Lambda}\overline{a_{\ell}}x^{\ell}\right)\right)=x^{\rho}(1+b_{1}(t,z)x+b_{2}(t,z)x^{2}+\ldots),$

where ${\overline{u}}$ is the automorphism of
$\mathcal{G}^{\rho}=x^{\rho}\sum_{k=0}^{\infty}\bigoplus_{\alpha\in\mathcal{A}_{k}}\Bbbk
z^{\alpha}x^{k}$ from the normal form theorem, Theorem 3.16, compare again to
the remarks on avoiding the variable $t$ in Section 4.2. The additional
summand $\sum_{\ell\in\Lambda}\overline{a_{\ell}}x^{\ell}$ in the inner
parenthesis of $g$ is required to make $f$ and $g$ coincide up to degree
${m}-1$.

We will show that $g$ is a solution of ${\overline{L}}y=0$ and that its
coefficient $b_{m}$ involves $z_{1}$. The first thing is easy since, by the
normal form theorem 3.16,

${\overline{L}}(g)=({\overline{L}}_{0}\circ{\overline{u}})(g)={\overline{L}}_{0}\left(x^{\rho}\left(1+\sum_{\ell\in\Lambda}\overline{a_{\ell}}x^{\ell}\right)\right)=0,$

for $\rho+\ell$ is a local exponent of ${\overline{L}}_{0}$ and hence
${\overline{L}}_{0}(x^{\rho+\ell})=0$. This proves ${\overline{L}}g=0$.

Next we prove inductively that $b_{\ell}=\overline{a_{\ell}}$ for
$\ell\leq{m}-1$, i.e., that the expansion of $g$ up to degree ${m}-1$ equals
the reduction of the respective expansion of $f$. This part is a bit
computational.

Write $T=L-L_{0}$ and ${\overline{T}}={\overline{L}}-{\overline{L}}_{0}$ as
earlier for the tails of $L$ and ${\overline{L}}$. We expand $T$ and
${\overline{T}}$ as sums of Euler operators

$T=T_{1}+\cdots+T_{r},$
${\overline{T}}={\overline{T}}_{1}+\cdots+{\overline{T}}_{r},$

Similarly, we define $S$ and $\SS$ as the inverses
$S=(L_{0}|_{\mathcal{H}})^{-1}$ and
$\SS=({\overline{L}}_{0}|_{\mathcal{\overline{H}}})^{-1}$ of $L_{0}$ and
${\overline{L}}_{0}$, respectively, on direct complements of their kernels, as
described in the normal form theorems, Theorems 2.15 and 3.16.

We now distinguish two cases.

(i) Assume first that $\rho+\ell$ is not a local exponent of $L$. Rewriting
the differential equations $Ly=0$ and ${\overline{L}}y=0$ as linear recursions
for the coefficients of the prospective solutions we obtain, for $\ell\leq
m-1$,

$a_{\ell}=S\left(\sum_{k=1}^{r}T_{k}\left(a_{\ell-k}x^{\rho+\ell-k}\right)\right),$

and

$b_{\ell}=\SS\left(\sum_{k=1}^{r}{\overline{T}}_{k}\left(b_{\ell-k}x^{\rho+\ell-k}\right)\right),$

where both sums in the parentheses are homogeneous of degree $\rho+\ell$ in
$x$. By induction on $\ell$ we may assume that
$b_{\ell-k}=\overline{a_{\ell-k}}$ equals the reduction of $a_{\ell-k}$ for
all $k=1,\ldots,r$. Hence this also holds for
$b_{\ell}=\overline{\alpha_{\ell}}$.

(ii) Assume now that $\rho+\ell$ is a local exponent of $L$. Here, the formula
for $b_{\ell}$ is different, by the very definition of $g$,

$b_{\ell}=\SS\left(\sum_{k=1}^{r}{\overline{T}}_{k}(b_{\ell-k}x^{\rho+\ell-k})\right)+\overline{a_{\ell}}.$

Now, as $\rho+\ell$ is a local exponent of $L$ and hence also of
${\overline{L}}$, the image $\SS(x^{\rho+\ell})$ will involve $z_{1}$.
Therefore, as $b_{\ell}$ does not involve $z_{1}$ by assumption, we get
${\overline{T}}_{k}(b_{\ell-k}x^{\rho+\ell-k})=0$. Hence again
$b_{\ell}=\overline{a_{\ell}}$ for all $\ell\leq{m}-1$.

This proves in both cases that $g$ is the reduction of $f$ modulo $p$ up to
degree ${m}-1$.

To finish the proof we will show that $b_{m}$ involves $z_{1}$. This will
produce the required contradiction. As $a_{m}$ depends on $z$ by assumption,
$\rho+{m}$ is necessarily a local exponent of $L$ and thus of
${\overline{L}}$. Hence $\SS(x^{\rho+{m}})$ will depend on $z_{1}$. Recall
that

$a_{m}=S\left(\sum_{k=1}^{r}T_{k}(a_{{m}-k}x^{\rho+{m}-k})\right),$

and

$b_{m}=\SS\left(\sum_{k=1}^{r}{\overline{T}}_{k}(b_{{m}-k}x^{\rho+{m}-k})\right).$

From the already established equalities $b_{\ell}=\overline{a_{\ell}}$ for
$\ell\leq{m}-1$ it follows that ${\overline{T}}_{k}(b_{{m}-k}x^{\rho+{m}-k})$
is the reduction modulo $p$ of $T_{k}(a_{{m}-k}x^{\rho+{m}-k})$. Now, if
$T_{k}(a_{{m}-k}x^{\rho+{m}-k})$ were $0$, its image $a_{m}$ under $S$ were
zero, which is excluded by the choice of ${m}$. So this term is non-zero. But
then it suffices to choose $p$ sufficiently large such that also the reduction
${\overline{T}}_{k}(b_{{m}-k}x^{\rho+{m}-k})$ is non-zero. Similarly as before
we then get that $b_{m}$ involves $z_{1}$, contradiction. $\square$

We illustrate the crucial step in the proof of (c) by an example.

######  Example 4.17.

The operator $L=x^{2}\partial^{2}-3x\partial-3x-x^{2}-x^{3}$ has the solution

$f(x)=u^{-1}(1)=1+a_{1}x+a_{2}x^{2}+\ldots=1-x+\frac{1}{2}x^{2}-\frac{1}{2}x^{3}-\frac{1}{2}x^{4}z+\ldots,$

so $a_{4}=-\frac{1}{2}z$ is the first coefficient which depends on $z$. Assume
that there was a full basis of solutions in ${\mathbb{F}}_{3}[\\![x]\\!]$. The
local exponents in characteristic $3$ are $0$ and $1$, so
$\Omega_{4}=\\{1,3\\}$. We compute, using $\overline{T}=x^{2}+x^{3}$ and
$\overline{L}_{0}=x^{2}\partial^{2}$, the expansion of the following solution

$\overline{u}_{3}^{-1}(1+2x+x^{3})=1+2x+2x^{2}+x^{3}+\ldots,$

which agrees with the reduction of $f$ up to order $3$. However, the next term
in the expansion is $S_{3}(x^{4})=x^{4}z_{1}$, so
$u_{3}^{-1}(1+2x+x^{3})\not\in{\mathbb{F}}_{3}[\\![x]\\!]$.

###  4.4 Outlook

If one wants to pursue the goal of proving the equivalence of the Grothendieck
$p$-curvature conjecture and the Bézivin conjecture, number theoretic
obstacles occur.

A power series $y(x)\in\mathbb{Q}[\\![x]\\!]$ is called globally bounded if
there is an integer $N$ such that $y(Nx)\in\mathbb{Z}[\\![x]\\!]$. In other
words, there are only finitely many prime numbers $p$ appearing in the
denominators of the coefficients of $y$ and they only grow geometrically. A
theorem of Eisenstein [Eis52] says that any algebraic power series is globally
bounded.

To prove that the validity of the Bézivin conjecture implies the validity of
the Grothendieck $p$-curvature conjecture it suffices to show that for a
linear differential equation $Ly=0$ whose reduction $L_{p}y=0$ has a full
basis of solutions in ${\mathbb{F}}_{p}[\\![x]\\!]$ the basis of solutions in
characteristic $0$ is globally bounded. For this it is natural to try to
compare the algorithms from the normal form theorems in characteristic $0$ and
$p$ further. Ideally, $p$ would not appear in the denominators of solutions in
characteristic $p$ if and only if there is a basis of solutions in
${\mathbb{F}}_{p}[\\![x]\\!]$ of $L_{p}y=0$, at least for almost all $p$.
However, the situation is not as easy as one might hope, as the following two
examples illustrate:

######  Example 4.18.

(i) The first example shows that for finitely many primes it may happen that a
full basis of solutions of the reduction of a linear differential equation
modulo $p$ exists, although $p$ appears in the denominator of one of the
solutions in characteristic $0$. The solution of $\partial-nx^{n-1}$ for
$n\in\mathbb{N}$ is $e^{x^{n}}$, a power series where each prime number
appears eventually in the denominators. However, for all prime numbers $p$
dividing $n$, the reduction of the equation modulo $p$ is an Euler equation
having the solution $1\in{\mathbb{F}}_{p}[\\![x]\\!]$. As this can happen only
for a finite number of primes, this does not contradict the Grothendieck
$p$-curvature conjecture.

(ii) The next example shows that to rule out the appearance of the prime
factor $p$ in the denominators of a solution of $Ly=0$ it is not sufficient to
work on the level of individual solutions associated to a local exponent and
its reduction. If possible at all, it has to take into account the existence
of a full basis of solutions.

The power series

$y(x)=\sum_{k=1}^{\infty}a_{k}x^{k}=\sum_{k=1}^{\infty}\frac{k(k+2)}{(k+1)}x^{k}=\frac{3}{2}x+\frac{8}{3}x^{2}+\frac{15}{4}x^{3}+\frac{24}{5}x^{4}+\ldots=\frac{\log(1-x)}{x}+\frac{x}{(x-1)^{2}}$

is annihilated by the third order operator

$L=x^{3}\partial^{3}+4x^{2}\partial^{2}+x\partial-1-(x^{4}\partial^{3}+8x^{3}\partial^{2}+13x^{2}\partial+3x).$

This operator $L$ is hypergeometric, i.e., $T=L-L_{0}$ is an Euler operator
with shift one. Moreover, $y$ is annihilated by the second order operator

$M=3x^{2}\partial^{2}+3x\partial-3+(x^{4}-4x^{3})\partial^{2}+(3x-12x^{2})\partial+x^{2}-4x,$

which is not hypergeometric. The operator $M$ is a right divisor of $L$, as
one verifies that

$\left(-\frac{1}{x-3}x\partial-\frac{1}{x-3}\right)M=L.$

Let us first concern ourselves with the operator $L$. Its local exponents are
$-1$ with multiplicity two and $1$ with multiplicity $1$. We have
$y=\frac{3}{2}\cdot u^{-1}(x)$, where $u$ is the automorphism described in the
normal form theorem in characteristic $0$. Moreover we compute
$u^{-1}(x^{-1})=x^{-1}$ and $u^{-1}(x^{-1}z)=x^{-1}z$. Thus a basis of
solutions of $Ly=0$ is given by $y,x^{-1}$ and $x^{-1}\log(x)$.

For all prime numbers $p$ the coefficient of $x^{p-2}$ in the expansion of $y$
is divisible by $p$, while the denominators of $a_{1},\ldots,a_{p-2}$ are not.
Thus

$y_{p}:=\sum_{k=1}^{p-2}a_{k}x^{k}$

is well defined in characteristic $p$ and a solution to the equation
$L_{p}y=0$. It is given as $u^{-1}_{p}(x)$ where $u_{p}$ is the automorphism
defined in the normal form theorem in characteristic $p$. The series $y$ is
not algebraic, as it is not globally bounded. In fact any prime number $p$
appears in the denominators of the coefficients $a_{i}$. However, the solution
in characteristic $p$ corresponding to the reduction of the local exponent $1$
is a genuine power series. Other linearly independent solutions in
characteristic $p$ are $x^{-1}$ and $x^{-1}z_{1}$. We see that in neither
characteristic there is a basis of power series solutions.

Let us now turn to the operator $M$, which has local exponents $-1$ and $1$ as
well, both with multiplicity $1$. A basis of solutions is given by $x^{-1}$
and $y$. This does not contradict the Grothendieck $p$-curvature conjecture,
as $y_{p}$ is not a solution of $M$. For $L$ the construction was very
dependent on the fact that the equation is hypergeometric, which is no longer
the case for $M$.

There still remain several questions about linear differential equations over
fields with positive characteristic. For linear differential equations with
holomorphic coefficients there is a criterion by Fuchs characterizing regular
singular points of an operator $L$ [Fuc66]. A point
$a\in\mathbb{P}^{1}_{\mathbb{C}}$ is at most a regular singularity of $L$ if
and only if there is a local basis of solutions of $Ly=0$, which grows at most
polynomially when approaching $a$. One would expect a similar criterion in
characteristic $p$: an $n$-dimensional vector space of solutions in
$\mathcal{R}$ over the constants $\mathcal{C}$ should suffice to conclude that
$0$ is a regular singular point of $L$. The needed framework could be provided
by adapting the solution theory of linear differential equations with
holomorphic coefficients and an irregular singularitiy at $0$ of N. Merkl,
described in section 2.4 to positive characteristic. More precisely, this
should allow the definition of a ring $\widetilde{\mathcal{R}}$ in which every
linear differential equations with an irregular singularity at $0$ in positive
characteristic has a basis of solutions. The corresponding criterion in
characteristic $p$ should then read: A linear differential equations $Ly=0$
admits a basis of solutions in $\mathcal{R}\subseteq\widetilde{\mathcal{R}}$
if and only if $0$ is a regular singularity of $L$

Moreover, the solutions of differential equations in $\mathcal{R}$ need to be
better understood. For example one would expect some kind of pattern in the
exponential function in positive characteristic discussed in Example 4.1.
However, no such structure seems obvious. Also the question of the
algebraicity of the constant term raised in Example 4.1 and Problem 4.2
deserves some attention and should be studied for the constant terms of
solutions of any equation.

In addition there is hope to extract information about the $p$-curvature of
linear differential operator $L$ in positive characteristics from the
description of a full basis of solutions in the differential extension
$\mathcal{R}$ of $\Bbbk$. In [BCS15] Bostan, Caruso and Schost describe an
algorithm on how to effectively compute the $p$-curvature of a differential
operator. They work over the ring $\Bbbk[\\![x]\\!]^{\rm dp}$ of series of the
form

$f=a_{0}+a_{1}\gamma_{1}(x)+a_{2}\gamma_{2}(x)+\ldots.$

The elements $\gamma_{i}(x)$ are formal variables, but should be thought of
$\frac{x^{i}}{i!}$. The multiplication on $\Bbbk[\\![x]\\!]^{\rm dp}$ is
consequently given by
$\gamma_{i}(x)\gamma_{j}(x)=\binom{i+j}{i}\gamma_{i+j}(x)$. In a suitable
extension extension of $\Bbbk[\\![x]\\!]^{\rm dp}$, accounting for local
exponents outside the prime field, they construct a basis of solutions of
$Ly=0$. From this basis they compute, passing to systems of first order
equations, the matrix representation of the $p$-curvature. A similar program
seems feasible working in $\mathcal{R}$ instead of $\Bbbk[\\![x]\\!]^{\rm
dp}$.

Finally, there remain, of course, the Grothendieck $p$-curvature conjecture
and the Bézivin conjecture. As Example 4.18 shows, the algorithms of the
normal form theorems in characteristic $p$ and $0$ show some unexpected
discrepancy. The hope that solutions of the reduction of differential
operators are reductions of solutions of the operator seems to be unfounded.
However, the phenomena shown require further investigation.

## References

  * [And04] Y. André  “Sur la conjecture des $p$-courbures de Grothendieck-Katz et un problème de Dwork”  In _Geometric Aspects of Dwork Theory_ , 2004, pp. 55–112
  * [BCS15] A. Bostan, X. Caruso and É. Schost  “A Fast Algorithm for Computing the p-Curvature”  In _ISSAC 2015_ ACM Press, 2015, pp. 69–76
  * [Béz91] J.-P. Bézivin  “Les suites $q$-récurrentes linéaires”  In _Compositio Mathematica_ 80.3, 1991, pp. 285–307
  * [Bos01] J.-B. Bost  “Algebraic leaves of algebraic foliations over number fields”  In _Publications mathématiques de l’I.H.É.S_ 93, 2001, pp. 161–221
  * [CC83] D. Chudnovsky and G. Chudnovsky  “Applications of Padé approximations to the Grothendieck conjecture on linear differential equations”  In _Number Theory_ 1135, Lecture Notes in Mathematics  Springer, 1983, pp. 52–100
  * [CDT21] F. Calegari, V. Dimitrov and Y. Tang  “The Unbounded Denominators Conjecture”, 2021  URL: https://arxiv.org/abs/2109.09040
  * [Cha02] L. Chambert-Loir  “Théorèmes d’algébricité en géométrie diophantienne. Séminaire Bourbaki, exp. 886.”  In _Astérisque_ 282, 2002, pp. 175–209
  * [Chr90] G. Christol  “Globally bounded solutions of differential equations”  In _Analytic Number Theory_ 1434, Lecture Notes in Mathematics  Springer, 1990, pp. 45–64
  * [Dwo90] B. Dwork  “Differential Operators with Nilpotent $p$-Curvature”  In _American Journal of Mathematics_ 112.5, 1990, pp. 749–786
  * [Eis52] G. Eisenstein  “Über eine allgemeine Eigenschaft der Reihen-Entwicklungen aller algebraischen Funktionen”, Berichte Königl. Preuss. Akad. Wiss. Berlin, 1852, pp. 441–443
  * [Fab85] E. Fabry  “Sur les intégrales des équations différentielles linéaires à coefficients rationnels”, 1885
  * [FK09] B. Farb and M. Kisin  “Rigidity, Locally Symmetric Varieties, and the Grothendieck–Katz Conjecture”  In _International Mathematics Research Notices_ 2009.22, 2009, pp. 4159–4167
  * [Fro73] G. Frobenius  “Ueber die Integration der linearen Differentialgleichungen durch Reihen”  In _Journal für die reine und angewandte Mathematik (Crelles Journal)_ 76, 1873, pp. 214–235
  * [Fro75] G. Frobenius  “Über die regulären Integrale der linearen Differentialgleichungen”  In _Journal für die reine und angewandte Mathematik (Crelles Journal)_ 80, 1875, pp. 317–333
  * [Fro96] G. Frobenius  “Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppe”, Berliner Sitzungsberichte, 1896, pp. 689–703
  * [Fuc66] L. Fuchs  “Zur Theorie der linearen Differentialgleichungen mit veränderlichen Coefficienten.”  In _Journal für die reine und angewandte Mathematik (Crelles Journal)_ 66, 1866, pp. 121–160
  * [Fuc68] L. Fuchs  “Zur Theorie der linearen Differentialgleichungen mit veränderlichen Coefficienten (Ergänzungen zu der im 66. Bande dieses Journals enthaltenen Abhandlung)”  In _Journal für die reine und angewandte Mathematik (Crelles Journal)_ 68, 1868, pp. 354–385
  * [GR71] H. Grauert and R. Remmert  “Analytische Stellenalgebren”  Springer, 1971
  * [Hon81] T. Honda  “Algebraic differential equations”  In _Symposia Mathematica_ 24, 1981, pp. 169–204
  * [Inc44] E.. Ince  “Ordinary Differential Equations”  Dover Publications, New York, 1944
  * [Jeo11] S. Jeong  “Calculus in positive characteristic $p$”  In _Journal of Number Theory_ 131, 2011, pp. 1089–1104
  * [Kat70] N. Katz  “Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin”  In _Publications Mathématiques de l’IHÉS_ 39, 1970, pp. 175–232
  * [Kat72] N. Katz  “Algebraic solutions of differential equations ($p$-curvature and the Hodge filtration)”  In _Inventiones mathematicae_ 18, 1972, pp. 1–118
  * [Kat82] N. Katz  “A conjecture in the arithmetic theory of differential equations”  In _Bulletin de la Société Mathématique de France_ 110, 1982, pp. 203–239
  * [Kro80] L. Kronecker  “Über die Irreductibiliät von Gleichungen”, Monatsberichte der Königlichen Preussischen Akademie der Wissenschaften zu Berlin, 1880, pp. 155–163
  * [Mai03] E. Maillet  “Sur les séries divergentes et les équations différentielles”  In _Annales Scientifiques de l’École Normale Supérieure_ 20, 1903, pp. 487–518
  * [Mat06] B. Matzat  “Differential equations and finite groups”  In _Journal of Algebra_ 300, 2006, pp. 673–686
  * [Mer22] N. Merkl  “Formal solutions of irregular singular differential equations”, 2022
  * [Mez11] M. Mezzarobba  “Autour de l’évaluation numériques des fonctions $D$-finies”, 2011
  * [Put96] M. Put  “Reduction modulo $p$ of differential equations”  In _Indagationes Mathematicae_ 7, 1996, pp. 367–387
  * [Sal19] B. Salvy  “Linear differential equations as a data structure”  In _Foundations of Computational Mathematics_ 19, 2019, pp. 1071–1112
  * [SP03] M. Singer and M. Put  “Galois Theory of Linear Differential Equations” 328, Grundlehren der mathematischen Wissenschaft  Springer, 2003
  * [Tho72] L.W. Thomé  “Zur Theorie der linearen Differentialgleichungen”  In _Journal für die reine und angewandte Mathematik (Crelles Journal)_ 74, 1872, pp. 193–217
  * [Tho73] L.W. Thomé  “Zur Theorie der linearen Differentialgleichungen (Fortsetzung 1)”  In _Journal für die reine und angewandte Mathematik (Crelles Journal)_ 75, 1873, pp. 265–291
  * [Tho73a] L.W. Thomé  “Zur Theorie der linearen Differentialgleichungen (Fortsetzung 2)”  In _Journal für die reine und angewandte Mathematik (Crelles Journal)_ 76, 1873, pp. 273–302

University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090,
Vienna, Austria
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