# Temporal and spatial characteristics of hard X-ray sources in flare model
with a vertical current sheet

Alexander N. Shabalin Ioffe Institute, Russian Academy of Sciences, Saint-
Petersburg 194021, Russia  Evgeniia P. Ovchinnikova Ioffe Institute, Russian
Academy of Sciences, Saint-Petersburg 194021, Russia  Yuri E. Charikov Ioffe
Institute, Russian Academy of Sciences, Saint-Petersburg 194021, Russia

###### Abstract

We analyzed changes in the height of the coronal hard X-ray (HXR) source for
flares SOL2013-05-13T01:50 and SOL2013-05-13T15:51. Analysis of the Reuven
Ramaty High Energy Solar Spectroscopic Imager data revealed the downward
motion of the HXR source and the separation of the sources by energy and
height. In the early stages of the flares, a negative correlation was found
between the HXR source area in the corona and HXR flux. For the
SOL2013-05-13T15:51 event, an increasing trend in the time delay spectra at
the footpoints was obtained. For both events, the spectra of the time delays
in the coronal HXR source showed a decreasing trend across the energies in
certain flare phases. To interpret the observed phenomena, we considered a
flare model of collapsing traps and calculated the distribution functions of
accelerated electrons along the magnetic loop using a nonstationary
relativistic kinetic equation. This approach considers betatron and Fermi
first-order acceleration mechanisms. The increasing trend of the time delay
spectra at the footpoints was explained by the high mirror ratio in the
magnetic loop and betatron acceleration mechanism. The observed features in
the spatial and temporal behavior of the HXR sources, such as the negative
correlation between the HXR source area and HXR flux, can be interpreted by
the collapsing trap model.

Solar flares(1496) — Active solar corona(1988) — Solar active region magnetic
fields(1975) — Solar energetic particles(1491) — Solar x-ray emission(1536) —
Solar extreme ultraviolet emission(1493)

##  1 Introduction

According to observations in the extreme ultraviolet (EUV) range and model
calculations, magnetic field configurations with a vertical current sheet in
the corona are possible in active regions of the solar atmosphere (Warren et
al., 2018; Gary et al., 2018; Jiang et al., 2018). In the standard flare model
CSHKP (Carmichael-Sturrock-Hirayama-Kopp-Pneuman), the primary acceleration of
charged particles occurs in the regions of the cusp and current sheet (Lin &
Forbes, 2000; Zharkova et al., 2011; Chen et al., 2020) above the soft X-ray
loops. The ejection of collisionless plasma from the magnetic reconnection
region is accompanied by the transformation of magnetic loops initially
extended in the direction of the current sheet to a configuration close to a
dipole (magnetic relaxation). The magnetic loops that relax in this manner
undergo longitudinal and transverse contraction, which leads to the further
acceleration of electrons through betatron acceleration and Fermi first-order
acceleration (Somov & Kosugi, 1997). Magnetic field dynamic results in spatial
and temporal features of hard X-ray (HXR) and radio sources in flares. For
example, the relaxing magnetic field in the cusp explains the position of the
HXR source above the loop (Masuda et al., 1994, 1995) and its brightness
(Fletcher & Martens, 1998; Shabalin et al., 2022).

Analysis of the relative motion of X-ray sources at the footpoints of the
flaring loop and looptop allows us to determine the regions of magnetic field
reconnection and acceleration of charged particles (Bogachev et al., 2005;
Krucker et al., 2005; Somov et al., 2005). In the works (Sui & Holman, 2003;
Sui et al., 2004; Liu et al., 2004, 2008; Veronig et al., 2006; Joshi et al.,
2007), an observed decrease in the height of the X-ray source in some solar
flares was noted at an early stage of the HXR flux rise. Sui et al. (2004)
noted a decrease in the height of the coronal HXR sources at the beginning of
the flare for three M-class flares that occurred on April 14–16, 2002. To
explain the downward motion of the HXR sources, the authors proposed two
mechanisms: one is associated with the relaxation of magnetic fields in the
cusp region formed during magnetic reconnection, and the other is related to a
change in the type of magnetic field reconnection from slow to Petschek
regime. The structure of the magnetic field in the cusp region and the
evolution of this structure in the collapsing trap model (hereinafter, the
CollTr model) allows us to expect an altitude change of the HXR source, which
can be caused by loop shrinkage (Somov & Kosugi, 1997). In models, loop
contraction is associated with the ”unshearing” of magnetic loops during
energy dissipation. This process results in the converging motion of the
footpoints and downward movement of the X-ray looptop source (Ji et al., 2007;
Zhou et al., 2013). In (Li & Gan, 2005; Zaitsev & Stepanov, 2020), the
shrinking of the loop at 34 GHz is explained by a current decrease due to an
increase in resistance caused by the plasma instability of Rayleigh-Taylor
within the chromospheric footpoints of the magnetic loop.

In addition to changes in the height of the HXR sources, the relaxation
process of the magnetic fields in the CollTr model can manifest itself in an
affect the hardness of the X-ray and radio spectra, as well as the time delay
spectra. Changes in the hardness of the energy spectrum in the X-ray range
during solar flares have been extensively studied in many papers. These
changes are the most commonly reported soft-hard-soft (SHS) patterns (Parks &
Winckler, 1969; Kane & Anderson, 1970; Benz, 1977; Brown & Loran, 1985; Lin &
Schwartz, 1987; Fletcher & Hudson, 2002). In addition to the SHS pattern, a
soft-hard-harder (SHH) (Frost & Dennis, 1971; Cliver et al., 1986; Kiplinger,
1995; Lysenko et al., 2019) and hard-soft-hard (HSH) (Shao & Huang, 2009)
patterns have also been observed. According to the authors, possible reasons
for the formation of these patterns may include a change in the efficiency of
the accelerator, the presence of several stages of acceleration, the effects
of particle transfer in the loop (owing to the return current), and the
influence of turbulent modes in the regions of acceleration and radiation.
Grigis & Benz (2004) showed that elementary flare bursts also exhibit SHS
patterns. According to (Battaglia & Benz, 2006), an analysis of spatially
separated sources at the top and footpoints showed that the SHS pattern was
observed separately in the spectra of each source. The authors concluded that
this SHS pattern was generated in the accelerator. The CollTr model assumes a
change in the magnetic field configuration associated with the relaxation of
the magnetic loops (collapse). This process can affect the formation of the
spectrum of accelerated electrons and, hence, the dynamics of the X-ray
spectrum.

The relaxation process of magnetic fields can manifest itself in the influence
on the dynamic spectrum of the time delays of HXR radiation. Charikov et al.
(2015) demonstrated that the time delay spectra (TD spectra) depend on the
energy and pitch-angle distributions of the accelerated electrons and the
parameters of the magnetic field and plasma. Typically, the analysis of TD
spectra in the HXR range is performed using full-Sun observational data (Bai &
Ramaty, 1979; Aschwanden et al., 1995; Bespalov et al., 1987; Tsap et al.,
2015). In the full-Sun dataset, researchers encounter superimposed HXR flux
originating from the chromospheric and coronal regions of the flare arcade.
However, the analysis of HXR fluxes from the local regions of magnetic loops
on the millisecond timescale, which is preferable for calculating time delays,
is complicated because of poor counting statistics.

This study aims to validate the CollTr model based on HXR radiation data and
the features of its evolution in the region below the vertical current sheet.
Such manifestations may include a decrease or absence of an increase in the
height of the HXR sources, a negative correlation between the area of the
coronal source and the HXR flux, and features in the energy spectra and TD
spectra, particularly at the initial stage of flare development. The observed
features of HXR radiation were substantiated by numerically solving the
relativistic nonstationary kinetic equation for accelerated electrons
propagating in flaring magnetic loops. In particular, we emphasize that in the
simulations, in addition to the transport processes, Coulomb losses and
diffusion, return current, and magnetic mirroring, the equation also considers
the influence of the magnetic field dynamics (collapse) in the initial phase
of the flare loop relaxation, which leads to betatron and Fermi first-order
acceleration. Sections 2–4 present the results of the spatial and temporal
analyses of the HXR fluxes for the solar events SOL2013-05-13T01:50 and
SOL2013-05-13T15:51. The results of modeling the propagation of accelerated
electrons in a magnetic loop and the radiation generated by them in the HXR
range are presented in sections 5 and 6. The results of the observations and
simulations are discussed in Section 7.

##  2 Spatial characteristics of flare hard X-rays

Figure 1: RHESSI images at different times of the flares SOL2013-05-13T01:50
in panel (a) and SOL2013-05-13T15:51 in panel (b). SDO/AIA images of
SOL2013-05-13T01:50 flare at 193 Å in the early phase and during decay (c,e).
The same applies to panels (d) and (f) for SOL2013-05-13T15:51 flare at 131 Å.
All the EUV images were superimposed by the RHESSI contours at different
energy bands. Table 1: Correlation coefficients between the HXR light curves
and areas of the HXR sources. Velocities of coronal HXR sources.

|  | Coefficient of correlation |  |  | Velocity of coronal source, km s-1
---|---|---|---|---|---
|  | 6-10 keV | 10-16 keV | 16-20 keV | 20-25 keV |  |  | 6-10 keV | 10-16 keV | 16-20 keV | 20-25 keV
SOL2013-05-13T01:50 | interval 1 | -0.5$\pm 0.2$ | -0.7$\pm 0.1$ | -0.7$\pm 0.1$ | – |  | interval 1 | -4.9 | -7.7 | -12.1 | -14.9
| interval 2 | -0.5$\pm 0.3$ | – | -0.8$\pm 0.1$ | -0.8$\pm 0.1$ |  | interval 2 | 9.9 | 11.9 | 14.0 | 16.2
SOL2013-05-13T15:51 | interval 1 | -0.5$\pm 0.3$ | -0.7$\pm 0.2$ | -0.6$\pm 0.3$ | – |  | interval 1 | 0.6 | -7.9 | -19.9 | -23.5
| interval 2 | +0.7$\pm 0.2$ | +0.6$\pm 0.3$ | +0.7$\pm 0.2$ | +0.7$\pm 0.2$ |  | interval 2 | 0.6 | 9.1 | 3.1 | -1.3
| interval 3 | -0.4$\pm 0.3$ | -0.8$\pm 0.1$ | -0.6$\pm 0.2$ | -0.6$\pm 0.2$ |  | interval 3 | 11.6 | 14.8 | 17.4 | 22.1

Note. — The negative velocity values correspond to the downward motion of the
coronal source.

Figure 2: RHESSI normalized time profiles, HXR source height, and area of HXR
sources for (a) SOL2013-05-13T01:50, (b) SOL2013-05-13T15:51. The gray
vertical stripes mark the intervals with a change in the attenuator state,
excluded from the study.

We present observations of the solar observatory, The Reuven Ramaty High
Energy Solar Spectroscopic Imager (RHESSI) (Lin et al., 2002), of two X-class
flares from the same active region, AR11748. The X-1.7 GOES class flare
SOL2013-05-13T01:50 is partially occulted by a solar limb. During this flare,
a single HXR source was observed in the corona prior to the main peak (Figure
1a). Six minutes before flare onset, a precursor was observed in the extreme
ultraviolet range and soft X-rays, which were studied in detail by (Shen et
al., 2017). A second flare, SOL2013-05-13T15:51, X-2.8 GOES class, occurred 14
hours later. Three HXR sources were observed during this flare: two in the
chromosphere and one in the corona (Figure 1b). In both events, flare arcades
were visible in ultraviolet data (Figure 1c–1f) obtained with The Atmospheric
Imaging Assembly on the Solar Dynamics Observatory SDO/AIA (Lemen et al.,
2012). In panels 1d and 1e, the cusp and the thin structure above it can be
observed in the EUV range. The long thin feature is considered as a possible
current sheet position. The RHESSI contours in the energy bands of 6–12 and
12–25 keV coincide with the EUV cusp area in both flares. For flare
SOL2013-05-13T15:51, the 50–100 keV contours coincided with the footpoints of
the EUV loops (panels 1b, 1d, and 1f). The images in panels 1c and 1d were
obtained during the initial phase of the flux growth, whereas those in panels
1e and 1f were obtained during the decay phase.

RHESSI CLEAN images with an eight-second integration time were used to analyze
the height and area of HXR sources. The 4F–8F detectors were utilized for
image generation. The HXR source area was estimated as 50% intensity contour
for SOL2013-05-13T01:50 and 70% intensity contour for SOL2013-05-13T15:51. The
position of the HXR source was determined as the distance from the center of
the solar disk to the HXR centroid in the plane of the sky. The time
evolutions of the flux, height, and area of the coronal HXR source are shown
in Figure 2.

During the initial phase of the SOL2013-05-13T01:50 flare, the altitude of the
HXR source decreased by 1 arcsec over a period of 4 minutes. At the subsequent
time interval, the height of the HXR source increased by 2 arcsec over a
period of 3 minutes, and the HXR flux significantly increased. The average HXR
source velocities for different energy bands are listed in Table 1. The
velocities of the HXR sources in the downward and upward directions increased
with energy. Negative velocity values corresponded to a decrease in the
altitude of the coronal HXR sources, whereas positive values corresponded to a
height increase. It is worth noting that the centroids of the HXR sources with
higher energies are located higher than those of the HXR sources with lower
energies (Figure 2a).

The obtained HXR area values were subjected to correlation analyses with the
HXR flux in the energy ranges 6–10, 10–16, 16–20, and 20–25 keV. Cross-
correlation analysis of the coronal HXR source area and the HXR flux revealed
a negative correlation. This was observed in interval 1 with decreasing height
and in the subsequent interval 2 with increasing height of the HXR sources.
The correlation coefficients are listed in Table 1.

In the SOL2013-05-13T15:51 flare, the change in the height of the HXR source
exhibited step-like behavior (Figure 2b, middle panel). At the onset of the
flare, there was a slight increase in the HXR flux, which coincided with a
decrease in the height of the HXR source. In addition, the area of the HXR
source showed an inverse correlation with the HXR flux increment. After the
flare onset, the increase in the HXR flux ceases and is replaced by relatively
stable radiation over a period of approximately 4 minutes. During the initial
time phases at intervals 1 and 2 (Figure 2, middle panel on the right), the
area values of the HXR sources displayed substantial fluctuations around a
mean value of approximately 170 arcsec2. The area of the HXR source during
second time interval was positively correlated with the HXR flux, whereas the
height of the HXR source above the solar limb did not exhibit a clear tendency
to increase or decrease. This was followed by a rapid increase in the HXR flux
(interval 3). At the beginning of this phase, within the interval 15:59–16:00
UT, the attenuators passed a switch. After the attenuators were switched, a
sharp change in the source height was observed. It is possible that a brighter
HXR source emerged, which was located approximately 8 arcsec lower. The
emergence of this HXR source likely occurred during the transition of the
flare to its explosive phase, and the location of this source differed within
the considered magnetic configuration. A subsequent increase in the HXR flux
was accompanied by an increase in the height of the HXR source above the solar
limb. In this phase, the area of the HXR source showed an inverse correlation
with the increase in the HXR flux. Observation of the flare revealed the
separation of the HXR source centroids at various energy ranges throughout the
event. The HXR sources observed at higher energies corresponded to greater
heights above the solar limb, as shown in the middle panels of Figure 2b. It
should be noted that the velocities of the downward and upward motions
differed for different energy channels and had higher values at higher
energies, as in the SOL2013-05-13T01:50 flare.

Thus, a negative correlation between the area of the corresponding HXR source
and the increase in HXR flux was observed during the time intervals with an
increase in the HXR flux in the considered flares. A positive correlation
between the area and HXR flux corresponded to the time interval in which the
area and HXR flux did not change significantly (Table 1).

Figure 3: The normalized HXR fluxes and spectral indices for flares
SOL2013-05-13T01:50 (top panel) and SOL2013-05-13T15:51 (bottom panel).
Colored rectangles indicate the beginning and end times of the SHS patterns.
Figure 4: TD spectra (top panel) and HXR fluxes (bottom panel) for the
SOL2013-05-13T01:50 event obtained from GBM/Fermi and RHESSI.  Figure 5: TD
spectra in time windows 1-4 and HXR fluxes (left panel) for the
SOL2013-05-13T15:51 event, obtained with RHESSI in the 28.0-42.8 keV channel
for HXR sources at the looptop and both footpoints of the flare arcade.

##  3 Energy spectra of hard X-ray radiation

Figure 2 and Table 1 show the downward motion of the HXR sources during the
early phases of both flares. Hereafter, the energy and time delay spectral
analyses focus on this initial phase. The upper panel of Figure 3 shows the
normalized HXR flux in the 46–55 keV energy range and the power-law spectrum
index for the initial stage of the SOL2013-05-13T01:50 event obtained from the
4 s resolution RHESSI data. The spectral indices were derived using a two-
power function approximation of the emission spectrum in the OSPEX (Schwartz
et al., 2002) within the 25–100 keV energy range, where the effect of the
thermal plasma was insignificant. During the time interval presented in the
top panel of Figure 3, the break energy $E_{break}$ varies within the range of
38–55 keV. Hence, the spectral index corresponds to an energy range of 25–45
keV. The spectral index beyond the break energy was not provided because it
was determined to be unreliable owing to insufficient photons in the coronal
source at energies greater than 45 keV. At the early stage of HXR flux growth,
the count rates at attenuator positions A0 and A1 are subject to the pileup
effect (Smith et al., 2002). For the HXR flux shown in Figure 3, the pileup
effect was considered for the SOL2013-05-13T01:50 flare using the RHESSI
software. Alternatively, the pileup effect can be incorporated into a set of
fitting functions in the OSPEX software. No qualitative changes were observed
for this particular event when the pileup effect was considered using the
OSPEX software. The colored rectangles in Figure 3 indicate the detected SHS
patterns across varying time scales ranging from 16 s to approximately 120 s.

The lower panel of Figure 3 displays the HXR fluxes across the six energy
channels, ranging from 25 keV to 52 keV, for the SOL2013-05-13T15:51 event.
Analyzing the spectra from the RHESSI data for local sources with a time
resolution of 4 s poses a significant challenge. This is especially the case
when examining the spectrum of a coronal source in the presence of
chromospheric HXR sources. Given our primary interest in the early flare
stage, the analysis becomes even more complex because of the relatively small
HXR flux. Therefore, the bottom panel of Figure 3 presents the spectral index
derived from the full-Sun RHESSI data. Single-temperature thermal and power-
law functions were employed to approximate the HXR spectra. The time interval
near 15:56 was excluded from the analysis because of extremely low HXR flux.
The time interval near 15:59:30 (light rectangle) was excluded from the
analysis because the data were considered invalid owing to the switching of
the RHESSI attenuators. The colored rectangles indicate the SHS-type
dependencies between the HXR flux and the spectral index (black dashed curve).
It is also worth noting that the spectral index curve exhibited a downward
trend overall. As a result, the average values of the power-law spectrum index
decrease, corresponding to the progressive hardening of the HXR radiation
spectrum or the soft-hard-harder (SHH) pattern.

##  4 Time delay spectra of hard X-ray emission

Figure 4 shows the TD spectra for the SOL2013-05-13T01:50 event. During this
event, the coronal source was predominantly observed, whereas the footpoints
of the arcade remained essentially invisible (see Figure 1a). Therefore, data
from full-Sun observations can be used to analyze the coronal source for this
event. TD spectra were derived through cross-correlation analysis of high-
resolution HXR radiation across various energy channels for a coronal source.
We utilized 4 s full-Sun RHESSI data (channels 26.6–34.6, 34.6–45.2,
45.5–58.9, 58.9–76.7 keV) and cspec data (detector no. 5) from the Gamma-ray
Burst Monitor on the Fermi Gamma-ray Space Telescope (GBM/Fermi) (Meegan et
al., 2009). The time resolution varied from 4.096 s at the onset of the flare
to 1.024 s later. The GBM/Fermi data were integrated from 29 to 6 channels in
the 25–90 keV energy range. In the top panel of Figure 4, the energy values
correspond to the midpoints of the analyzed energy ranges. To estimate the
error in measuring the time delays, we simulated light curves (50 iterations)
within the measurement uncertainties of the RHESSI data during the calculation
process. For the GBM/Fermi data, the standard deviation within the
investigated time window was considered the measurement error of the initial
data. The resulting TD spectrum error for each energy channel was determined
as the standard deviation for an ensemble of 50 delay values obtained for the
channel under investigation. The TD spectral curve was computed as the mean of
the resulting spectral ensemble. The cross-correlation function of the
radiation curves in various energy ranges was approximated by using a
quadratic function at three points close to its peak (Aschwanden & Schwartz,
1995).

As an example, the bottom panel of Figure 4 displays the radiation curves in
two energy channels: 26.6–34.6 keV and 34.6–45.2 keV, retrieved from RHESSI,
and 31.9–38.7 keV and 38.7–47.9 keV, obtained from GBM/Fermi. The colored
rectangles designate the time domains used to calculate the TD spectra, as
illustrated in the top panel of Figure 4. The solid lines in the upper panel
represent the TD spectra derived from the GBM/Fermi data, and the dashed lines
correspond to those obtained from the RHESSI data. The colors of the curves
depicted in the top panel correspond to those of the rectangles in the bottom
panel. In this study, we focused on analyzing the HXR emission from the onset
of the flare to the emission peak, which occurred at 02:10:22 UT. When
nonthermal electrons propagate within flare loops owing to magnetic mirroring,
Coulomb, or turbulence scattering, a portion of these electrons may be
captured in a magnetic trap for durations ranging from seconds to tens of
seconds. The structure of the time profile of the radiation curve may comprise
individual pulses superimposed on one another, representing instances of
electron acceleration. To minimize the impact of these effects on the TD
spectra, we selected the shortest segments of the time profile for analysis.
Furthermore, we attempted to analyze the phases of the radiation rise and
decay separately. The bottom panel of Figure 4 displays the intervals in which
we obtained TD spectra with acceptable error values. These represent two
phases of a local increase in the radiation flux and one phase of decay
(windows 1, 2, and 3 in Figure 4, bottom). A qualitative agreement can be
observed when comparing the GBM/Fermi and RHESSI data (solid versus dashed
curves), notwithstanding the temporal resolution and channel count
differences. During the rising phases of the HXR flux (windows 1 and 2,
represented by the black and green curves in Figure 4, respectively), the TD
spectra decreased in the approximate range of 26–70 keV. Above the inflection
point in the (50–70) keV region, the TD spectra become flat. During the decay
phase (window 3, red curves), the TD spectra exhibit a pronounced V-shaped
profile, decreasing in the 26–44 keV range and increasing beyond $\sim$44 keV.

Figure 5 shows the TD spectra obtained for the HXR of the SOL2013-05-13T15:51
event. We calculated the TD spectra for regions of interest (ROIs) in the HXR
images. The images were obtained via the Clean imaging algorithm applied to
the RHESSI data within the 12.0–18.3, 18.3–28.0, 28.0–42.8, 42.8–65.4, and
65.4–100 keV channels. The HXR sources from the ROIs at the looptop and two
footpoints of the flare arcade were analyzed. The colored rectangles in Figure
5 mark the phases of the rise and decay of the flare HXR, for which the TD
spectra were calculated. The shaded white time interval (15:59:04–15:59:44)
was excluded from the analysis because of potentially inaccurate flux values.
Windows 1 and 3 correspond to the rising phases of the HXR flux. During these
time intervals, the TD spectra at the loop footpoints exhibit an increasing
trend in the spectral profiles. At the looptop, the TD spectrum exhibited a
growth shape (window 1, black curve) and a complex shape (window 3, black
curve). During the decay phases (windows 2 and 4), the TD spectral profiles
changed to a decreasing shape at the footpoints and looptop. A discussion of
the theoretical justification for the TD spectral data is provided in Section
7.

##  5 The collapsing trap model. The statement of the problem.

We assume that the initial acceleration within the current sheet defines the
injected electrons parameters. In the multitude of relaxing magnetic loops, we
modeled one characterized by parameters representing ensemble-averaged loops
that exhibit similarity. Consequently, we modeled magnetic loop relaxation
accompanied by the injection of primary accelerated electrons at the looptop
region.

To model the properties of the HXR mentioned above, we numerically solved the
time-dependent relativistic kinetic equation for nonthermal electrons, which
has the following form: (Hamilton et al., 1990; Zharkova et al., 2010; Filatov
et al., 2013; Minoshima et al., 2010)

$\begin{split}\frac{\partial f}{\partial t}&=-\beta c\mu\frac{\partial
f}{\partial s}+\\\ &\beta c\frac{\partial\ln(B)}{\partial
s}\frac{\partial}{\partial\mu}\left[\frac{\left(1-\mu^{2}\right)}{2}f\right]+C_{1}\frac{c}{\lambda_{0}}\frac{\partial}{\partial
E}\left(\frac{f}{\beta}\right)+\\\
&C_{2}\frac{c}{\lambda_{0}\beta^{3}\gamma^{2}}\frac{\partial}{\partial\mu}\left[\left(1-\mu^{2}\right)\frac{\partial
f}{\partial\mu}\right]+\\\ &\frac{e\mathcal{E}\beta\mu}{m_{e}c}\frac{\partial
f}{\partial E}+\frac{e\mathcal{E}}{m_{e}\beta
c}\frac{\partial}{\partial\mu}\left[\left(1-\mu^{2}\right)f\right]+\\\
&\frac{\partial(\dot{E}f)}{\partial
E}+\frac{\partial(\dot{\mu}f)}{\partial\mu}+S\left(E,\mu,s,t\right)\end{split}$
(1)

where $f(E,\mu,s,t)$ is the non-thermal electron distribution function, $s$ is
the distance along the magnetic field line ($s=0$ corresponds to the loop
top), $t$ is the time, $\mu=\cos(\alpha)$ is the cosine of the pitch angle,
$\lambda_{0}(s)=\frac{10^{24}}{n(s)\ln\Lambda}$, $n(s)$ – plasma density,
$\Lambda=\frac{3k_{B}T_{e}}{2e^{2}}\left(\frac{k_{B}T_{e}}{8\pi
e^{2}n}\right)^{0.5}$ (Ginzburg, 1981), $k_{B}$ – Boltzmann constant, $T_{e}$
– electron temperature, $e$ – electron charge, $\beta=vc^{-1}$, $v$ – non-
thermal electrons speed, $c$ – speed of light, $m_{e}$ – electron mass,
$\gamma=E+1$ – Lorentz factor of the electron, $E$ is the kinetic energy of an
electron, expressed in units of the electron rest mass energy,
$C_{1}=x+\frac{1-x}{2}\frac{\ln\beta^{2}g^{2}E/\alpha_{F}^{4}}{\ln\Lambda}$,
$C_{2}=\frac{1}{2}+\frac{1+g}{4}C_{1}$ (Leach & Petrosian, 1981; Emslie, 1978)
– the coefficients take into account the contribution of partially ionized
plasma to the energy loss and angular scattering of fast electrons,
$\alpha_{F}$ – fine structure constant, $x$ – fraction of ionized hydrogen
atoms. Initial plasma density at the looptop is $n^{LT}_{(t=0)}=7\times
10^{8}$ cm-3. In the chromosphere it corresponds to hydrostatic equilibrium at
$10^{4}$K (Mariska et al., 1989) and is exponential between the looptop and
chromosphere. For comparison, we will also use a model with stationary
magnetic field, time-independent plasma distribution and density at the
looptop $n^{LT}=10^{10}\,$ cm-3. For a more detailed description of the terms
in equation (1) and the initial conditions, see Shabalin et al. (2022).

If the compensation of the beam current $\textbf{{j}}_{b}$ by the return
current $\textbf{{j}}_{rc}$ in the flaring plasma is fulfilled, that is,
$\textbf{{j}}_{rc}=-\textbf{{j}}_{b}$, the induced electric field must satisfy
the following relation:

$\begin{split}\mathcal{E}(s,t)=\frac{j_{b}(s,t)}{\sigma(s)}=\\\
\frac{e}{\sigma(s)}\int_{E_{min}}^{E_{max}}v(E)&\,dE\int_{-1}^{1}f(E,\mu,s,t)\mu\,d\mu\end{split}$
(2)

where $\sigma(s)$ is the classical Spitzer conductivity of flare plasma
determined by pair collisions of charged particles. In the case of partial
plasma ionization, the conductivity $\sigma(s)$ was defined in Shabalin et al.
(2022).

The source of injected electrons was modeled as the product of four
independent factors representing the distributions on the energy, pitch angle,
space location, and time, that is,
$S(E,\mu,s,t)=KS_{1}(E)S_{2}(\mu)S_{3}(s)S_{4}(t)$, where $K$ is a
normalization constant scaled to match the chosen energy flux. The peak
electron energy flux was $\sim 10^{10}$ erg cm-2s-1, typical for M-X-class
solar flares. We consider the cases of stationary isotropic $S_{2}(\alpha)=1$
and strong anisotropic $S_{2}(\alpha)=\cos^{12}(\alpha)$ pitch-angle
distributions.

The time profile of the electron injection function $S_{4}(t)$ represent a
pulse followed a Gaussian function
$S_{4}(t)=\exp{-\frac{(t-t_{1})^{2}}{t_{0}^{2}}}$ centered at time $t_{1}=2.6$
s with a width of $t_{0}=1.4$ s. The electron injection occurred spatially at
the looptop, and was modeled as a Gaussian
$S_{3}(s)=\exp{-\frac{(s-s_{1})^{2}}{s_{0}^{2}}}$ centered at location
$s_{1}=0$ cm.

The energy dependence $S_{1}(E)$ is represented as a broken power law with
$E_{br}=200$ keV, $\delta_{1}=3$ or $7$ below $E_{br}$ depending on the model
that corresponds to the typical flare’s hard and soft spectra and
$\delta_{2}=10$ above $E_{br}$.

The betatron and Fermi accelerations of electrons in the collTr model were
calculated using the equation (Filatov et al., 2013; Minoshima et al., 2010):

$\dot{E}=\gamma\beta^{2}\frac{1-\mu^{2}}{2}\frac{B_{t}}{B}-\gamma\beta^{2}\mu^{2}\frac{\dot{l}}{l}$
(3)

where $B_{t}$ is the time derivative of the magnetic field. From equation (3),
it follows that electrons with pitch angles close to $\frac{\pi}{2}$ are
mainly accelerated by the betatron mechanism, which boosts the transverse
velocity component, while the longitudinal component of the electron velocity
increases due to Fermi acceleration when the magnetic mirrors converge
($\dot{l}<0$). The betatron and Fermi pitch-angle terms are set as:

$\dot{\mu}=-\mu\frac{1-\mu^{2}}{2}\frac{B_{t}}{B}-2\mu(1-\mu^{2})\frac{\dot{l}}{l}$
(4)

The magnetic field in the collTr model with a vertical current sheet cannot be
strictly determined. We consider a model of the magnetic field evolution, with
an initial rapid change in the magnetic field strength and length of the
magnetic field lines mainly in the cusp, in the following form:

$B(s,t)=B_{min}+\left[\frac{s-b_{1}}{l}\right]^{2}(B_{max}-B_{min}),$ (5)

$B_{min}=B_{0}+\cos\left(\frac{2\pi
t}{T_{cusp}}\right)e^{-D_{b}t}(B_{cusp}-B_{0}),$ (6)

where the magnetic field at the footpoints is fixed $B_{max}=mB_{0}$, the
period of magnetic field oscillations $T_{cusp}=16\,$s, the initial magnetic
field at the top of the loop $B_{cusp}=20$ G, $b_{1}=0\,$cm defines the
magnetic field minimum location, the damping factor $D_{b}=0.14\,$ s-1, and
the average magnetic field at the looptop of the relaxed magnetic loop
$B_{0}=200\,$G. Thus, in the considered model, the collapse time $t_{col}$ is
half of $T_{cusp}$, that is, 8 s. The coefficient $D_{b}$ is selected such
that the possible following oscillations of the magnetic field dampen
sufficiently quickly. The values of the magnetic field $B_{cusp}$ and $B_{0}$
were chosen as typical for the cusp region and the tops of the arcade of
closed loops (Gary et al., 2018; Kuridze et al., 2019). The mirror ratio $m$
for closed flare loops is determined by the magnetic field gradient and varies
in the interval 2–20 (Shabalin et al., 2019). We considered the results of the
calculations for a symmetric magnetic loop with $m=8$. The mirror ratio is
constant for the model without collapsing traps and is the average for the
model with collapsing traps. Figure 2b in Shabalin et al. (2022) shows the
magnetic field variations at the looptop, according to Equation 5.

The length of the collapsing loops during their relaxation decreases from
$1.5l_{0}$ with a looptop in the cusp to $l_{0}$ for low closed loops:

$l(t)=l_{0}-\cos\left(\frac{2\pi t}{T_{cusp}}\right)e^{-D_{b}t}(l_{1}-l_{0})$
(7)

where $l(t)$ is the semidistance between trap ends, $l_{0}=3\times 10^{9}$ cm,
$l_{1}=\frac{1}{2}l_{0}$.

The X-ray flux was calculated according to the formulas of relativistic
bremsstrahlung (Bai & Ramaty, 1978; Gluckstern & Hull, 1953; Koch & Motz,
1959) for accelerated electrons (see equation (9) in Shabalin et al. (2022)).

##  6 Results of modeling the kinetics of accelerated electrons

###  6.1 Energy spectra

Magnetic field reconnection and particle acceleration can occur sporadically
in time and with varying intensities as well as with different localizations
within the flare region. To identify the main trends in the changes in the
energy spectrum of the HXR radiation, we considered a simplified model
employing successive collapse of several magnetic loops. The total injection
function of the accelerated electrons for such a model appears as two
Gaussian-shaped elementary bursts with different peak fluxes, which follow
each other for approximately 10 s (gray dashed curve in Figure 6, top panel).
The interval between the peaks and the full width at half maximum (FWHM) of
the electron source was selected to minimize the influence of one collapsing
loop (previously referred to as the CollTr model) on the subsequent loop.
Additionally, the top panel displays the integral (over several collapsing
loops) HXR flux in the 29–58 keV range from the looptop (solid black curve)
and footpoint (dashed black curve). The red curves (solid and dashed lines)
represent variations in the power-law spectrum index within the 29–58 keV
energy range. The spectrum index values are shown on the right vertical axis.
The labels ”soft” and ”hard” on the top panel indicate the formation of a
soft-hard-soft pattern. During the initial increase in HXR flux, the spectrum
of HXRs was soft. At the HXR peak and the onset of decay, the spectrum of the
HXR radiation became harder. Furthermore, the hardness of the subsequent peaks
in the spectrum index curve increased. Thus, on a time scale corresponding to
the collapse of several magnetic loops, the dynamics of the spectral indices
form the SHH pattern. Hence, consecutive collapses of two magnetic loops form
the SHS pattern, whereas a combination of three or more collapsing loops gives
rise to the SHH pattern.

Figure 6: Top panel: Model HXR fluxes and spectrum index (right axis) in the
29–58 keV range for the model of sequentially collapsing traps. Middle panel:
TD spectra for the CollTr model with $S(\alpha)=1$ and $\delta=3$. Bottom
panel: On the left, TD spectra for the DMTI model with parameters
$S(\alpha)=1$, $\delta=3$, $b_{1}=0$ cm; on the right, TD spectra for the DMTI
model with parameters $S(\alpha)=1$, $\delta=3$, $b_{1}=-2\times 10^{9}$ cm,
representing an asymmetric magnetic field in the flare loop. The plasma
density at the looptop in the DMTI models was $10^{10}$ cm-3.

###  6.2 Time Delay Spectra

The TD spectra for the CollTr model are shown on the left side of the middle
panel in Figure 6. The model parameters are $S(\alpha)=1$, $\delta=3$, and
$b_{1}=0$ cm. Following the same approach as the analysis of flares in Section
4, we calculated the model TD spectra by examining the phases of increasing
HXR flux up to and including the peak, as well as the decay phases
encompassing the peak. The middle panel on the right displays examples of
calculated HXR fluxes at two energies, 15.2 keV and 74.1 keV, for the looptop
and footpoint. It is important to note the increasing TD spectra in the CollTr
model at both the footpoint and the looptop. The delays ranged from 18 to 1110
ms, depending on the analyzed spatial region of the flare loop. Alternative
models were also calculated, including one with a softer electron source
$S(\alpha)=1$ and $\delta=7$ and another with an anisotropic source
$S(\alpha)=\cos^{12}(\alpha)$ and $\delta$ values of 3 and 7. Graphs for these
models are not presented here. However, it is important to note that the same
increasing trend in the TD spectra was observed for all calculated cases.

To reveal the features of the CollTr model, we also calculated the TD spectra
for the other models, in which the distributions of plasma density and
magnetic field along the loop were time-independent (hereafter referred to as
the DMTI models). In the bottom left panel of Figure 6, the TD spectra are
presented for the DMTI model with the parameters $S(\alpha)=1$, $\delta=3$,
and $b_{1}=0$ cm. It is important to note the decreasing TD spectra at the
footpoints of the magnetic loop (black and light-gray curves). Conversely, at
the looptop, the TD spectra exhibit an increasing trend. We also examined
another DMTI model with the parameter $b_{1}=-2\times 10^{9}$ cm. In this
model, the distribution of the magnetic field along the loop was asymmetric,
featuring a $B_{max}/B_{min}$ ratio of 8 at one footpoint and 1.28 at the
second one. Consequently, the geometric apex of the loop, where the majority
of accelerated electrons are injected, is no longer a magnetic trap for
charged nonthermal particles. The TD spectra in this model, with the
parameters $S(\alpha)=1$, $\delta=3$, and $b_{1}=-2\times 10^{9}$ cm, are
displayed in the bottom right panel of Figure 6. In this case, the looptop TD
spectra showed a decreasing pattern. The delays span from 4 to 100 ms.

##  7 Summary and discussion

###  7.1 Hard X-ray source motion

We found that the height of the coronal HXR source decreased during the early
phase of flux rise in the considered flares. According to Table 1, in the
selected flares SOL2013-05-13T01:50 and SOL2013-05-13T15:51, the velocities
absolute values of the downward and upward motions of the HXR sources varied
widely and were (0.6 – 23.5) and (0.6 – 22.1) km s-1, respectively. Note that
for both flares, the HXR sources with higher energy had higher velocities for
downward and upward motions (Table 1, SOL2013-05-13T01:50, intervals 1, 2;
SOL2013-05-13T15:51, intervals 1, 3). A decreasing tendency in the height of
the X-ray source in solar flares at the initial stage of the HXR flux rise was
observed by other authors. Thus, Sui et al. (2004) recorded a decrease in the
source height for three flares, and the downward motion velocities were 8–20
km s-1, after the height decrease phase, the HXR sources moved upward at
velocities of 5–40 km s-1. A similar trend was observed in the events studied
by Veronig et al. (2006), where the average velocities of the source within
the 10–30 keV energy range increased from 14 to 45 km s-1 with increasing
energy. One possible explanation for the downward motion of the HXR source may
be the projection effect. That is, the heights of the arcade magnetic loops
decrease along the line of sight, and sequential ignition of loops is
realized. However, this interpretation contradicts the stereoscopic
observations of flares in the EUV range by the Extreme Ultraviolet Imager
(Wuelser et al., 2004) onboard the Solar TErrestrial RElations Observatory
(STEREO) spacecraft (Howard et al., 2008). These observations indicate that
heating of the flare loops occurs unevenly and inconsistently throughout the
arcade. Additionally, it remains unclear why the velocity of the height
decrease is dependent on the energy of the HXR source, and why the downward
motion of the source is observed during the initial phase of the flare only.

From another perspective, the downward movement of the HXR source at the
initial flare stage was considered to be a result of the sequential relaxation
of the magnetic field (Sui et al., 2004; Veronig et al., 2006). It should be
noted, that during the relaxation of magnetic field, a decrease in the height
of the coronal HXR source may not necessarily be observed. Certain conditions
are necessary to observe a bright coronal HXR source: a high plasma density
$n^{LT}>10^{10}$ cm-3 and the ability of the trap to accumulate high-energy
electrons, which implies an isotropic or quasi-transverse pitch-angle
distribution of electrons after their primary acceleration in the current
sheet (Charikov & Shabalin, 2021; Shabalin et al., 2022). Moreover, the
spatial distribution of the magnetic field in such traps should be nearly
symmetrical with the same magnetic field values at the footpoints. If we
assume a symmetric magnetic field, a sufficient density of plasma and
accelerated particles in the region of the magnetic field minimum can be
formed shortly before the vertical motion of the relaxing magnetic loop stops.
Under these conditions, a minimal decrease in the height of the HXR source may
be observed. Therefore, the absence of an increase in the height of the
coronal HXR source at an early stage does not contradict the CollTr model.

###  7.2 Spatial separation of X-ray source centroids at different energy
ranges

In our opinion, one of the most important results of observations indicating
the collapse of magnetic loops in a flare is the spatial separation of X-ray
sources in different energy ranges: an increase in HXR source height from
lower to higher energies. Sequential involvement in the relaxation process of
newly formed magnetic structures results in a configuration of magnetic loops
with different heights at different energies. The nonthermal electrons in the
formed earlier lower loops lose energy within seconds to tens of seconds. The
upper bound of captured accelerated electrons lifetime with energy
$\lesssim$100 keV can be estimated from the kinetic equation 1 as follows
$\tau_{coll}=\frac{E}{\dot{E}_{coll}}\varpropto n^{-1}E^{3/2}$ where
$\dot{E}_{coll}=\frac{dE}{dt}$ determines the rate of energy loss by an
electron during scattering by plasma ions. It can be seen that Coulomb loss
time depends on the energy as $E^{3/2}$. Therefore, for example, $\tau_{coll}$
for a 30 keV electron in a homogeneous plasma with a density of $10^{10}$ cm-3
will be 3.21 s, and for a 98 keV electron 17.4 s. In a flare loop plasma, as
follows from equation 1, accelerated electrons, when moving along the magnetic
field, pass through the loss cone into the chromosphere, scatter, reflect from
magnetic mirrors, and are additionally accelerated by Fermi and betatron
mechanisms. In addition, in the collTr model, at the beginning of the collapse
(in a higher loop), the efficiency of the capture of accelerated electrons is
high due to the large values of the magnetic ratio $m\gg 8$, and the Coulomb
losses are small, because the density is low $n<10^{10}$ cm-3. The scenario is
different for relaxed low loops. The magnetic ratio in these loops is not as
high, with $m\sim 5-8$, and the Coulomb losses are significantly greater
because $n>10^{10}$ cm-3. Therefore, the lower loops radiate in a softer
energy range because of hot plasma only. In magnetic loops at higher
altitudes, nonthermal electrons are yet to thermalize. These electrons
accumulate at the looptop due to trapping. Therefore, electron propagation in
magnetic traps, combined with traps formation at different times, led to
height separation of the centroids of the HXR source in various energy ranges
in the considered events SOL2013-05-13T01:50 and SOL2013-05-13T15:51. Source
centroid separation in various energy channels has also been observed by
Veronig et al. (2006) and simulated by Kong et al. (2019). The values of the
magnetic field minima are different for relaxing magnetic loops of different
heights. For higher loops, these values are lower than those for the lower
loops. Therefore, the magnetic loop velocities depend on the height owing to
the conservation of the magnetic flux $B\cdot v=Const$ (Veronig et al., 2006).
Thus, we observed the effect of increasing the velocities with source energy
for the considered flares.

###  7.3 Hard X-ray flux—area correlation

In addition to the height separation of HXR sources at different energies, we
examined the temporal variation in coronal HXR source size. Owing to the
conservation of magnetic flux, an increase in the magnetic field during
collapse is expected to result in a reduction in the cross-sectional area of
the magnetic loop. Although this value cannot be measured, it is possible to
estimate the size of the HXR source associated with the loop cross-section.
During the collapse, the magnetic field in the coronal part of the loop
increases several times, which leads to a decrease in the cross section of the
loop by the same factor. We are not expecting to observe such a significant
change in the HXR source area. Nonetheless, a negative correlation may be
anticipated between the cross-sectional area of the HXR sources during
collapse and HXR flux. We found a negative correlation between the area of the
coronal source and the HXR flux in phases with increasing flux during downward
and upward motion of the sources for both considered flares (Table 1). For
SOL2013-05-13T15:51, aside from the negative correlation, a positive
correlation between the area and flux was observed during the phase without an
intensive increase in flux within the time interval of 15:55-15:59 UT.

###  7.4 Hard X-ray energy spectra variations

In the considered flares, we also analyzed the temporal variations in the
energy spectrum. To date, numerous hypotheses have been proposed to explain
the evolution of the HXR energy spectrum during solar flares. These hypotheses
can be divided into four categories. First, changes in the HXR spectrum may be
associated with accelerator properties or various acceleration mechanisms
during several-step acceleration. Second, electron distribution functions are
transformed during transport along magnetic loops via Coulomb collisions and
magnetic mirroring. This transformation influences the HXR spectra. Third, the
propagation of an accelerated electron beam results in the emergence of
physical processes that influence the electrons responsible for initiating
these processes. For example, the return current or the generation of various
turbulent modes significantly modifies the distribution function of radiating
electrons. Fourth, the electron propagation environment can contain MHD waves,
shock fronts, and magnetic field fluctuations generated during energy release
or by an external trigger.

The types of energy spectrum evolution that correlate with HXR flux are of
particular interest for research. We identified the sub-minute and minute SHS
patterns in the flares under discussion. SHS patterns are challenging to
explain based solely on kinetic effects during electron beam propagation.
Magnetic mirroring of electrons, Coulomb losses, and return current leads to a
change in the power law spectrum index in the range of 20–100 keV by fractions
of units when the integral electron flux (erg cm-2 s-1) varies several times
according to our numerical simulations. However, in the observations, the
spectrum index changes by values comparable to one or more (see, for example,
Figure 3). In this study, we interpreted SHS patterns as a superposition of
several magnetic loops within the flare region. In the CollTr model, it can be
assumed that each HXR pulse, ranging from seconds to tens of seconds,
corresponds to a different relaxing magnetic structure. In each of these
magnetic structures, accelerated electrons undergo propagation, accumulation,
and emission.

Numerical simulations of electron propagation along the loop show that at the
looptop, the HXR spectrum hardens over time because low-energy electrons lose
energy in Coulomb collisions more efficiently. The rate of spectrum hardening
depends on the plasma density in the coronal part of the loop. The plasma
density at the looptop can increase in each loop because of magnetic loop
relaxation and chromospheric evaporation. In addition, the response in the HXR
range exhibited a delay of several seconds relative to the electron injection
time in the CollTr model.

Considering the aspects mentioned above, we modeled electron propagation in
the CollTr model, assuming that the collapse of different loops occurs
sequentially. In a flare environment, the formation of a separate relaxing
magnetic structure strongly depends on the local parameters of the plasma and
magnetic field. Therefore, the duration of the injection into a separate
collapsing structure and its physical parameters can vary. A simplified model
of identical successive collapsing traps was suitable to demonstrate the
fundamental possibility of SHS pattern formation in the HXR spectrum. The
results of this simulation are shown in the top panel of Figure 6. As
discussed in Section 6.1, the successive relaxation of several magnetic loops
results in the formation of an SHS pattern in the spectrum index dynamics.
Furthermore, a global (over several HXR peaks) SHH pattern in HXR radiation is
formed owing to the gradual hardening of the distribution function of the
accelerated electrons, which is integrated over several magnetic loops.
Hardening is caused by Coulomb losses, which transform the distribution
function of the accelerated electrons in the low-energy region and due to the
effective trapping of electrons in the high-energy region.

###  7.5 Hard X-ray time delays spectra

The HXR TD spectrum is an additional tool for diagnosing accelerated electrons
during flares. The dynamic of the acceleration and propagation of high-energy
electrons in the flare-loop plasma determines the HXR TD spectra and their
evolution. In the simplest models of electron propagation, the TD spectra are
determined by a power-law dependence of approximately $E^{-1/2}$ in the free-
streaming model and $\sim E^{3/2}$ in the trap-plus-precipitation model
(Aschwanden & Schwartz, 1995). Due to the speed difference between electrons,
high-energy electrons reach the chromosphere before lower-energy electrons.
The TD spectra exhibit a descending profile in the absence of particle
scattering and trapping. The processes of electron scattering and mirroring in
a magnetic field lead to the trapping of electrons. These electrons spend some
time in a magnetic trap before falling into the loss cone and radiating. If
there is an increase in the plasma density at the loop top and/or the
generation of turbulence that scatters longitudinally distributed electrons,
the trapped particles will radiate more intensely in the coronal region of the
loop. This results in a complex emission pattern, leading to the formation of
different types of TD spectra. Numerical calculations of the propagation of
accelerated electrons and bremsstrahlung HXR radiation enable a thorough
analysis and identification of the factors responsible for the formation of a
particular time delay spectrum.

In Section 4, for flare SOL2013-05-13T01:50, decaying TD spectra were observed
in the coronal HXR source during two bursts (windows 1 and 2), according to
data from both GBM/Fermi and RHESSI instruments (Figure 4). During the decay
phase (window 3), the TD spectrum also exhibited a decrease in the
$\sim$(26–45) keV range. The observation of decreasing TD spectra at the
looptop is unusual when considering the accelerated electron propagation and
emission simulation.

For the second flare, SOL2013-05-13T15:51 (Figure 5), a decrease in the TD
spectra at the looptop was observed in time windows 2 and 4, corresponding to
the local decay phases of the HXR flux. In this event, at the footpoints
during the rising phases of the HXR flux (windows 1 and 3, Figure 5),
increasing TD spectra, which are unusual for this location, were also
observed. Based on the accelerated electron propagation DMTI model, decreasing
spectra are expected at the footpoints of the flare loop, as illustrated in
Figure 6 (bottom left panel). Nonetheless, in the CollTr model, the TD spectra
at the footpoints exhibited growth. This is attributed to a combination of
factors: a high initial $B_{max}/B_{min}$ ratio during the onset of magnetic
loop relaxation, additional electron acceleration, and an increase in plasma
density at the looptop. The modeling revealed that betatron acceleration has a
dominant influence on the formation of growing TD spectra at footpoints.
Because betatron acceleration is significant only during the growth phase of
the HXR flux, the transition from increasing TD spectra at the footpoints to
decreasing patterns in windows 2 and 4 in Figure 5 can also be explained. That
is, during the increase of the HXR flux, growing TD spectra are formed at the
footpoints owing to the betatron acceleration of the electrons. Subsequently,
during the decay of the HXR flux, when the betatron acceleration was no longer
effective and the $B_{max}/B_{min}$ ratio was not substantial (approximately
8), the TD spectra transformed to a decreasing pattern, as anticipated by the
DMTI model of electron propagation with a stationary magnetic field.

Justifying the decreasing TD spectra at the looptop is more challenging. As
demonstrated on the right-hand side of the bottom panel of Figure 6, for the
model $S(\alpha)=1$, $\delta=3$, $b_{1}=-2\times 10^{9}$ cm, with an
asymmetric magnetic field, the TD spectra decrease at the looptop. This occurs
because, in an asymmetric magnetic loop, the geometrical top of the loop does
not serve as a trap for accelerated electrons. The transition from increasing
TD spectra at the looptop to decreasing ones in the SOL2013-05-13T15:51 event
may be due to a reduction in the $B_{max}/B_{min}$ ratio over time during
magnetic field relaxation. Assuming an initial asymmetric distribution of the
magnetic field along the flare loops before flare onset, a decrease in the
$B_{max}/B_{min}$ ratio from large values of approximately 80 at the beginning
of the relaxation (collapse) process to $B_{max}/B_{min}\simeq 8$ would lead
to a more pronounced manifestation of the magnetic field asymmetry during the
decay phases of the HXR flux. The results of magnetic field approximations
using the nonlinear force-free field (NLFFF) method for solar disk flares
suggest that asymmetry in the magnetic field distribution along magnetic loops
in active regions is a relatively common occurrence. Hence, the hypothesis
regarding the asymmetric magnetic field in the examined limb flares appears
entirely plausible. This was further supported by the diverse brightness of
the footpoints in the SOL2013-05-13T15:51 event, indirectly indicating
different values for the loss cone.

###  7.6 Conclusions

In summary, we outlined the key conclusions. Numerical simulations of
accelerated electron kinetics and the associated HXR emission in the CollTr
model are consistent with the observed HXR data. First, the separation of the
HXR sources by height and energy occurs naturally during magnetic field
relaxation in a model with a vertical current sheet in the corona. Second,
there was a negative correlation between the area of the HXR sources in the
corona and HXR flux for both flares examined in this study. Third, there were
particular characteristics of the TD spectra in the investigated flares,
specifically the decreasing TD spectra in the coronal source and the
increasing TD spectra in the footpoints of the SOL2013-05-13T15:51 event. We
explain these characteristics using numerical simulations conducted within the
CollTr model framework. Fourth, our modeling of the HXR spectrum index
evolution suggests that the SHS and SHH patterns may not appear due to
specific processes in the accelerator or the region of suprathermal electron
propagation but rather as a consequence of the superposition of HXR fluxes
from various magnetic structures, which are at different stages in terms of
accelerated electron accumulation and radiation. A potential extension of this
research is to calculate the gyrosynchrotron radiation in similar events using
the CollTr model and compare the results with observational data obtained from
radio telescopes. Various turbulence modes (such as Langmuir, ion-acoustic,
and whistler modes) and magnetic fluctuations may arise in secondary
acceleration and electron radiation regions. Incorporating these processes
into the CollTr model is another avenue of future research.

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#  On Newton’s identities in positive characteristic

Sjoerd de Vries

###### Abstract

Newton’s identities provide a way to express elementary symmetric polynomials
in terms of power polynomials over fields of characteristic zero. In this
article we study symmetric polynomials in positive characteristic. Our main
result shows that in this setting, one can recover the elementary symmetric
polynomials as rational functions in the power polynomials.

##  1 Introduction

The elementary symmetric polynomials, resp. power polynomials, in $n$
variables are defined as

$\displaystyle e_{k}(x_{1},\ldots,x_{n})$ $\displaystyle=\sum_{1\leq
i_{1}<\ldots<i_{k}\leq n}x_{i_{1}}x_{i_{2}}\ldots x_{i_{k}};$ (1)
$\displaystyle p_{k}(x_{1},\ldots,x_{n})$
$\displaystyle=\sum_{i=1}^{n}x_{i}^{k}.$ (2)

In the following, we write $e_{k}$ (resp. $p_{k}$) for the elementary
symmetric (resp. power) polynomials in $n$ variables, where $n$ is some fixed
number which should be clear from the context. Equation (1) holds for $k\geq
1$, and $e_{0}$ is defined to be $1$. Equation (2) holds for $k\geq 0$, so
that $p_{0}(x_{1},\ldots,x_{n})=n$. In particular, $e_{k}=0$ for $k>n$, but
$p_{k}\neq 0$ for all $k$. For a partition $\lambda=(k_{1},\ldots,k_{l})$, we
write $e_{\lambda}:=e_{k_{1}}\ldots e_{k_{l}}$ and
$p_{\lambda}:=p_{k_{1}}\ldots p_{k_{l}}$.

The importance of the elementary symmetric polynomials is made evident by the
following theorem (cf. Theorem 2.6):

######  Theorem 1.1.

Let $K$ be any field. Then the $K$-algebra of symmetric polynomials in $n$
variables is equal to $K[e_{1},\ldots,e_{n}]$.

The power polynomials are related to the elementary symmetric polynomials via
Newton’s identities:

$ke_{k}=\sum_{i=1}^{k}(-1)^{i-1}e_{k-i}p_{i}.$ (3)

It follows that for every $1\leq k\leq n$, one has (see e.g. [Mac95, p.28])

$e_{k}=\frac{1}{k!}\det\begin{pmatrix}p_{1}&1&0&\cdots&0\\\
p_{2}&p_{1}&2&\cdots&0\\\ \vdots&\vdots&\ddots&\ddots&\vdots\\\
p_{k-1}&p_{k-2}&\cdots&p_{1}&k-1\\\
p_{k}&p_{k-1}&\cdots&p_{2}&p_{1}\end{pmatrix}.$ (4)

An immediate corollary of Theorem 1.1 and the above equation is the following:

######  Theorem 1.2.

Let $K$ be a field of characteristic zero. Then
$K[e_{1},\ldots,e_{n}]=K[p_{1},\ldots,p_{n}]$.

On the other hand, suppose $K$ has characteristic $r>0$ (we avoid using $p$ to
avoid confusion with the power polynomials). Then we have algebraic relations

$p_{kr}=x_{1}^{kr}+\ldots+x_{n}^{kr}=(x_{1}^{k}+\ldots+x_{n}^{k})^{r}=p_{k}^{r}$
(5)

for any integer $k\geq 1$. Thus, contrary to Theorem 1.2, it is not true that
$K[e_{1},\ldots,e_{n}]=K[p_{1},\ldots,p_{n}]$ when $n\geq r$, because the
left-hand side has transcendence degree $n$ while the right-hand side has
transcendence degree at most $n-\lfloor n/r\rfloor$. This is related to the
fact that the identity (4) is not valid when $k$ is not invertible. However,
one may wonder if the set of _all_ power polynomials still generates all
symmetric polynomials over fields of positive characteristic.

This question has a negative answer, as the following example shows. Suppose
we are working with symmetric polynomials in two variables over
$\mathbb{F}_{2}$. Then $p_{k}(1,1)=1+1=0$ for all values of $k$. If
$e_{2}=x_{1}x_{2}$ could be written as an algebraic combination of
$\\{p_{k}\\}_{k\in\mathbb{N}}$, i.e. as a finite sum over integer partitions

$e_{2}=\sum_{m\in\mathbb{N}}\sum_{\lambda\vdash m}c_{\lambda}p_{\lambda}$ (6)

with $c_{\lambda}\in\mathbb{F}_{2}$, then evaluating both sides at $(1,1)$
(resp. $(0,0)$) gives $c_{0}=1$ (resp. $c_{0}=0$). Hence no such equation (6)
can exist.

However, notice that in the above case, one can write $e_{2}$ in two variables
in terms of power polynomials as

$e_{2}=x_{1}x_{2}=\frac{x_{1}^{2}x_{2}+x_{1}x_{2}^{2}}{x_{1}+x_{2}}=\frac{(x_{1}^{3}+x_{1}^{2}x_{2}+x_{1}x_{2}^{2}+x_{2}^{3})+x_{1}^{3}+x_{2}^{3}}{x_{1}+x_{2}}=\frac{p_{1}^{3}+p_{3}}{p_{1}}.$
(7)

A natural question is then whether it is true that the elementary symmetric
polynomials can always be obtained as rational functions in the power
polynomials. In this note, we answer this question in the affirmative:

######  Theorem 1.3 (cf. Theorem 2.8).

Let $K$ be any field. Then $K(e_{1},\ldots,e_{n})=K(p_{1},p_{2},\ldots)$.

Importantly, the proof of Theorem 2.8 provides an algorithm to explicitly
determine any elementary symmetric polynomial as a rational function of power
polynomials.

The outline of this paper is as follows. In §2, we derive a formula for the
elementary symmetric polynomials as rational functions in the power
polynomials, and give examples and applications. In §3, we study the
subalgebra generated by the power polynomials in positive characteristic.

###  1.1 Acknowledgements

I am grateful to Per Alexandersson for helpful discussions and providing a
combinatorist’s perspective on the contents of this paper. I would also like
to thank my PhD-advisor Jonas Bergström.

##  2 Elementary symmetric polynomials in terms of power polynomials

We first prove some preliminary results needed in the proof of the main
theorem.

######  Lemma 2.1.

Let $M=(m_{i,j})$ be a $d\times d$ matrix with entries given by power
polynomials: $m_{i,j}=p_{r_{i,j}}(x_{1},\ldots,x_{n_{i,j}})$ for some integers
$r_{i,j},n_{i,j}\geq 0$. Write $m_{i,j}$ for
$m_{i,j}(x_{1},\ldots,x_{n_{i,j}})$ and $m_{i,j}(\hat{x}_{k})$ for
$m_{i,j}(x_{1},\ldots,\hat{x}_{k},\ldots,x_{n_{i,j}})$, i.e. the power
polynomial evaluated at all variables except $x_{k}$. Note that
$m_{i,j}(\hat{x}_{k})=m_{i,j}$ if and only if $k>n_{i,j}$. Define

$\epsilon_{i,j,k}:=\begin{cases}1&k\leq n_{i,j};\\\
0&\text{otherwise}.\end{cases}$

Then for any $1\leq i\leq d$ and any $k\geq 1$, we have

$\det M=\det\begin{pmatrix}m_{1,1}&m_{1,2}&\cdots&m_{1,d}\\\ \vdots&\vdots&\
&\vdots\\\ m_{{i-1},1}&m_{{i-1},2}&\cdots&m_{{i-1},d}\\\
\epsilon_{i,1,k}x_{k}^{r_{i,1}}&\epsilon_{i,2,k}x_{k}^{r_{i,2}}&\cdots&\epsilon_{i,d,k}x_{k}^{r_{i,d}}\\\
m_{{i+1},1}&m_{{i+1},2}&\cdots&m_{{i+1},d}\\\ \vdots&\vdots&\ &\vdots\\\
m_{d,1}&m_{d,2}&\cdots&m_{d,d}\end{pmatrix}+\det\begin{pmatrix}m_{1,1}&m_{1,2}&\cdots&m_{1,d}\\\
\vdots&\vdots&\ &\vdots\\\ m_{{i-1},1}&m_{{i-1},2}&\cdots&m_{{i-1},d}\\\
m_{i,1}(\hat{x}_{k})&m_{i,2}(\hat{x}_{k})&\cdots&m_{i,d}(\hat{x}_{k})\\\
m_{{i+1},1}&m_{{i+1},2}&\cdots&m_{{i+1},d}\\\ \vdots&\vdots&\ &\vdots\\\
m_{d,1}&m_{d,2}&\cdots&m_{d,d}\end{pmatrix}$

###### Proof.

This follows from the $i$-th row expansion of $\det M$, since
$m_{i,j}=\epsilon_{i,j,k}x_{k}^{r_{i,j}}+m_{i,j}(\hat{x}_{k})$. ∎

######  Proposition 2.2.

Let $d,n\in\mathbb{Z}_{\geq 1}$ and consider the Hankel matrix

$P_{d,n}(x_{1},\ldots,x_{n})=\begin{pmatrix}p_{1}&p_{2}&p_{3}&\cdots&p_{d}\\\
p_{2}&p_{3}&p_{4}&\cdots&p_{d+1}\\\ p_{3}&p_{4}&p_{5}&\cdots&p_{d+2}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
p_{d}&p_{d+1}&p_{d+2}&\cdots&p_{2d-1}\end{pmatrix}$ (8)

Then the following hold:

  1. 1.

For any $n$, we have

$\det P_{n,n}=e_{n}\prod_{1\leq i<j\leq n}(x_{i}-x_{j})^{2}.$

  2. 2.

For any $d$ and $n$, we have

$\det P_{d,n}=\sum_{1\leq i_{1}<i_{2}<\ldots<i_{d}\leq n}\det
P_{d,d}(x_{i_{1}},x_{i_{2}},\ldots,x_{i_{d}}).$

In particular, $\det P_{d,n}$ is non-zero if and only if $d\leq n$.

###### Proof.

Proof of 1. If we write

$P_{n,n}=\begin{pmatrix}1&1&\cdots&1\\\ x_{1}&x_{2}&\cdots&x_{n}\\\
x_{1}^{2}&x_{2}^{2}&\cdots&x_{n}^{2}\\\ \vdots&\vdots&\ddots&\vdots\\\
x_{1}^{n-1}&x_{2}^{n-1}&\cdots&x_{n}^{n-1}\end{pmatrix}\begin{pmatrix}x_{1}&x_{1}^{2}&\cdots&x_{1}^{n}\\\
x_{2}&x_{2}^{2}&\cdots&x_{2}^{n}\\\ x_{3}&x_{3}^{2}&\cdots&x_{3}^{n}\\\
\vdots&\vdots&\ddots&\vdots\\\ x_{n}&x_{n}^{2}&\cdots&x_{n}^{n}\end{pmatrix}$

then the first matrix is a standard Vandermonde matrix, which has determinant

$\prod_{1\leq i<j\leq n}(x_{j}-x_{i}).$

The second matrix is of the same form after dividing row $i$ by $x_{i}$ and
taking the transpose; hence it has determinant $e_{n}\prod_{1\leq i<j\leq
n}(x_{j}-x_{i})$. The result follows.

Proof of 2. Fix $d$. The idea is to decompose $\det P_{d,n}$ completely using
Lemma 2.1. First consider $P_{d,d}$. By applying Lemma 2.1 with $i=k=1$, we
obtain

$\det P_{d,d}=\det\begin{pmatrix}x_{1}&x_{1}^{2}&\cdots&x_{1}^{d}\\\
p_{2}&p_{3}&\cdots&p_{d+1}\\\ \vdots&\vdots&\ddots&\vdots\\\
p_{d}&p_{d+1}&\cdots&p_{2d-1}\end{pmatrix}+\det\begin{pmatrix}p_{1}(\hat{x}_{1})&p_{2}(\hat{x}_{1})&\cdots&p_{d}(\hat{x}_{1})\\\
p_{2}&p_{3}&\cdots&p_{d+1}\\\ \vdots&\vdots&\ddots&\vdots\\\
p_{d}&p_{d+1}&\cdots&p_{2d-1}\end{pmatrix}$

In the first matrix, one can replace each power polynomial $p_{j}$ with
$p_{j}(\hat{x}_{1})$, by replacing row $i$ with row $i-(x_{1}^{i-1}$ times row
1) for all $i=2,\ldots,d$. The second determinant we can reduce again by
applying Lemma 2.1 with $i=1$ and $k=2$. Continuing this process gives the
expression

$\det
P_{d,d}=\sum_{k=1}^{d}\det\begin{pmatrix}x_{k}&x_{k}^{2}&\cdots&x_{k}^{d}\\\
p_{2}(\hat{x}_{k})&p_{3}(\hat{x}_{k})&\cdots&p_{d+1}(\hat{x}_{k})\\\
\vdots&\vdots&\ddots&\vdots\\\
p_{d}(\hat{x}_{k})&p_{d+1}(\hat{x}_{k})&\cdots&p_{2d-1}(\hat{x}_{k})\end{pmatrix}$

By applying Lemma 2.1 inductively to the other rows, one obtains the following
expression for $\det P_{d,d}$:

$\det P_{d,d}=\sum_{\sigma\in
S_{d}}\det\begin{pmatrix}x_{\sigma(1)}&x_{\sigma(1)}^{2}&\cdots&x_{\sigma(1)}^{d}\\\
x_{\sigma(2)}^{2}&x_{\sigma(2)}^{3}&\cdots&x_{\sigma(2)}^{d+1}\\\
\vdots&\vdots&\ddots&\vdots\\\
x_{\sigma(d)}^{d}&x_{\sigma(d)}^{d+1}&\cdots&x_{\sigma(d)}^{2d-1}\end{pmatrix}$
(9)

where $S_{d}$ denotes the symmetric group on $d$ letters.

For arbitrary $n$, one can use the same method to express $\det P_{d,n}$ as

$\det P_{d,n}=\sum_{1\leq i_{1},\ldots,i_{d}\leq
n}\det\begin{pmatrix}x_{i_{1}}&x_{i_{1}}^{2}&\cdots&x_{i_{1}}^{d}\\\
x_{i_{2}}^{2}&x_{i_{2}}^{3}&\cdots&x_{i_{2}}^{d+1}\\\
\vdots&\vdots&\ddots&\vdots\\\
x_{i_{d}}^{d}&x_{i_{d}}^{d+1}&\cdots&x_{i_{d}}^{2d-1}\end{pmatrix}$

where such a determinant is automatically zero if there is a repetition in the
indices (since one row will be a multiple of another). Thanks to (9), we may
write this determinant as

$\det P_{d,n}=\sum_{1\leq i_{1}<i_{2}<\ldots<i_{d}\leq n}\det
P_{d,d}(x_{i_{1}},x_{i_{2}},\ldots,x_{i_{d}}).$

When $d\leq n$, the above expression is non-zero because it contains the
monomial $x_{1}x_{2}^{3}\ldots x_{d}^{2d-1}$. This finishes the proof. ∎

######  Remark 2.3.

The argument in the proof of part 2 of Proposition 2.2 works for any matrix
$M=(m_{i,j})$ consisting of power polynomials in $n$ variables such that $\deg
m_{i,j}-\deg m_{i+1,j}$ is independent of $j$ for all $i$.

We can give our main result in more general form than Theorem 1.3 and so we
fix the following notation.

######  Definition 2.4.

Let $R$ be a commutative ring and let $\\{f_{i}\ |\ i\in I\\}$ be a set of
polynomials in $R[x_{1},\ldots,x_{n}]$. Let $S$ denote the multiplicative
system of non-zero divisors in $R[f_{i}\ |\ i\in I]\subset
R[x_{1},\ldots,x_{n}]$. We denote by

$R(f_{i}\ |\ i\in I):=S^{-1}R[f_{i}\ |\ i\in I]=Q(R[f_{i}\ |\ i\in I])$

the total ring of fractions of $R[f_{i}\ |\ i\in I]$.

######  Lemma 2.5.

[AM69, Exercise 1.2] Let $R$ be a commutative ring, and let $f\in
R[x_{1},\ldots,x_{n}]$ be a zero divisor. Then there exists a zero divisor
$r\in R$ such that $rm=0$ for any monomial $m$ in $f$.

######  Theorem 2.6 (Fundamental Theorem on Symmetric Polynomials).

Let $R$ be a commutative ring. Then $R[e_{1},\ldots,e_{n}]\subset
R[x_{1},\ldots,x_{n}]$ is equal to the ring of symmetric polynomials over $R$,
and the set $\\{e_{1},\ldots,e_{n}\\}$ is algebraically independent.

###### Proof.

Let $S_{n}$ denote the symmetric group on $n$ letters, which acts on the
variables $x_{1},\ldots,x_{n}$ by permutation. By [Mac95, I.2.4], the theorem
holds with $R=\mathbb{Z}$. In particular, for the trivial action of $S_{n}$ on
$R$, we have

$R[e_{1},\ldots,e_{n}]=R\otimes\mathbb{Z}[e_{1},\ldots,e_{n}]=R\otimes\mathbb{Z}[x_{1},\ldots,x_{n}]^{S_{n}}=R[x_{1},\ldots,x_{n}]^{S_{n}}.$

This implies the first part of the theorem. The algebraic independence follows
from [Mac95, I.2.3], which is stated for $\mathbb{Z}$ but whose proof works
over any ring $R$. ∎

######  Remark 2.7.

The fact that $\\{e_{1},\ldots,e_{n}\\}$ is algebraically independent over any
ring $R$ does not follow directly from the statement for $R=\mathbb{Z}$
without using properties of the elementary symmetric polynomials. For
instance, as shown in [Ham86, Example 1.4], there are rings $R$ and
polynomials $f_{1},\ldots,f_{m}\in\mathbb{Z}[x_{1},\ldots,x_{n}]$ such that

$\text{trdeg}_{R}\\{f_{1},\ldots,f_{m}\\}<\text{trdeg}_{\mathbb{Z}}\\{f_{1},\ldots,f_{m}\\},$

even when none of the coefficients of the $f_{i}$ are zero divisors in $R$.
Another example of this phenomenon is that for a prime number $r$,

$\text{trdeg}_{\mathbb{F}_{r}}\\{p_{1},p_{r}\\}=1$

for any number of variables $n$, since $p_{r}\equiv p_{1}^{r}\pmod{r}$.

######  Theorem 2.8.

Let $R$ be any commutative ring not containing $\mathbb{Q}$, and let
$r_{0}\in\mathbb{Z}_{\geq 1}$ be the smallest integer which is not invertible
in $R$. Fix the number of variables $n\geq r_{0}$. Then
$R(e_{1},\ldots,e_{n})=R(p_{1},\ldots,p_{2n+1-r_{0}})$.

###### Proof.

The power polynomials are symmetric polynomials, so $p_{i}\in
R(e_{1},\ldots,e_{n})$ for all $i$ by Theorem 2.6.

Conversely, fix $k\in\mathbb{N}$. We argue by induction that if $e_{i}\in
R(p_{1},p_{2},\ldots)$ for all $i\leq k-1$, then also $e_{k}\in
R(p_{1},p_{2},\ldots)$. Note that $e_{1}=p_{1}$, so the base case holds.

If $k$ is invertible in $R$, we obtain $e_{k}$ from the usual Newton identity

$ke_{k}=\sum_{i=1}^{k}(-1)^{i-1}e_{k-i}p_{i}.$

So suppose $k$ is not invertible. Since $e_{N}=0$ for $N>n$, the Newton
identities imply that

$\sum_{i=N-n}^{N}(-1)^{i-1}e_{N-i}p_{i}=0$ (10)

for all $N>n$. Applying (10) with $N=n+1,\ldots,2n-k+1$, we obtain the system
of equations

$\begin{pmatrix}p_{1}&-p_{2}&\cdots&(-1)^{n}p_{n+1}\\\
-p_{2}&p_{3}&\cdots&(-1)^{n+1}p_{n+2}\\\ \vdots&\vdots&\ddots&\vdots\\\
(-1)^{n-k}p_{n-k+1}&\cdots&\cdots&(-1)^{k}p_{2n-k+1}\end{pmatrix}\begin{pmatrix}e_{n}\\\
e_{n-1}\\\ \vdots\\\ e_{1}\\\ 1\end{pmatrix}=\begin{pmatrix}0\\\ 0\\\
\vdots\\\ 0\\\ 0\end{pmatrix}$ (11)

The $(n-k+1)\times(n+1)$ matrix above is of the form $(\tilde{P}|*)$ for a
square matrix $\tilde{P}$. Multiplying each second row of $\tilde{P}$ by $-1$
and then multiplying each second column by $-1$, we see that

$\det\tilde{P}=\det P_{n-k+1,n},$

with notation from (8). This determinant is not a zero-divisor in
$R[x_{1},\ldots,x_{n}]$ by Proposition 2.2 and Lemma 2.5, since $\det P_{d,n}$
for $d\leq n$ always contains the monomial $x_{1}x_{2}^{3}\ldots x_{d}^{2d-1}$
with coefficient $1\in R^{\times}$. Hence the matrix in (11) can be reduced to
the form $(I_{n-k+1}|*)$ by performing elementary row operations over
$R(p_{1},p_{2},\ldots)$. Considering the final equation obtained from the
system

$\begin{pmatrix}1&c_{1,2}&\cdots&c_{1,n-k+1}&c_{1,n-k+2}&\cdots&c_{1,n+1}\\\
0&1&\cdots&c_{2,n-k+1}&c_{2,n-k+2}&\cdots&c_{2,n+1}\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&1&c_{n,n-k+2}&\cdots&c_{n,n+1}\end{pmatrix}\begin{pmatrix}e_{n}\\\
e_{n-1}\\\ \vdots\\\ e_{1}\\\ 1\end{pmatrix}=\begin{pmatrix}0\\\ 0\\\
\vdots\\\ 0\\\ 0\end{pmatrix}$ (12)

we see that $e_{k}=\sum_{i=0}^{k-1}-c_{n,n+1-i}e_{i}$ with each $c_{n,j}\in
R(p_{1},p_{2},\ldots)$. This finishes the proof. ∎

######  Remark 2.9.

The reduction of the system (11) to the system (12) only requires division by
$\det P_{n-k+1,n}$, which we know explicitly by Proposition 2.2. Thus, one
only needs to invert algebraic expressions in the power polynomials $p_{i}$
for $1\leq i\leq 2n-2r_{0}+1$ (not $2n-r_{0}+1$) to obtain all elementary
symmetric polynomials.

Moreover, this observation gives a test as to whether one can evaluate
$e_{k}(\alpha_{1},\ldots,\alpha_{n})$ for $\alpha_{i}\in R$ given the power
polynomials and the $e_{i}(\alpha_{1},\ldots,\alpha_{n})$ for $i<k$: namely,
it suffices that

$\det P_{n-k+1,n}(\alpha_{1},\ldots,\alpha_{n})\neq 0.$

Indeed, this is the denominator in the expression for
$e_{k}(\alpha_{1},\ldots,\alpha_{n})$, which is a polynomial; hence the
denominator divides the numerator whenever the denominator is non-zero.

######  Example 2.10.

In this example, we write $e_{k}^{(n)}:=e_{k}(x_{1},\ldots,x_{n})$.
Implementing the algorithm from the proof, we obtain for $k=2$:

$\displaystyle\qquad\qquad\qquad\qquad
e_{2}^{(2)}=e_{2}(x_{1},x_{2})=\frac{p_{1}p_{2}-p_{3}}{p_{1}};$
$\displaystyle\qquad\qquad\qquad\qquad
e_{2}^{(3)}=e_{2}(x_{1},x_{2},x_{3})=\frac{p_{1}p_{2}p_{3}-p_{1}^{2}p_{4}-p_{2}p_{4}+p_{1}p_{5}}{p_{2}^{2}-p_{1}p_{3}};$
$\displaystyle e_{2}^{(4)}=$
$\displaystyle\frac{p_{1334}-p_{1244}-p_{1235}+p_{1145}-p_{335}+p_{245}+p_{1226}-p_{1136}-p_{146}+p_{137}+p_{236}-p_{227}+p_{245}-p_{335}}{p_{3}^{3}-2p_{2}p_{3}p_{4}+p_{1}p_{4}^{2}+p_{2}^{2}p_{5}-p_{1}p_{3}p_{5}},$

where the shorthand $p_{1334}$ for example means $p_{1}p_{3}^{2}p_{4}$. Note
that the expression for $e_{2}^{(2)}$ reduces to Equation (7) modulo $2$,
since $p_{2}=p_{1}^{2}$ in characteristic 2. For $k=3$, we obtain

$\displaystyle\begin{split}e_{3}^{(3)}&=\frac{-p_{1}p_{3}+p_{2}e_{2}^{(3)}+p_{4}}{p_{1}};\\\
e_{3}^{(4)}&=\frac{p_{1}^{2}p_{5}-p_{1}p_{2}p_{4}-p_{1}e_{2}^{(4)}p_{4}-p_{1}p_{6}+p_{2}e_{2}^{(4)}p_{3}+p_{2}p_{5}}{p_{2}^{2}-p_{1}p_{3}}.\end{split}$
(13)

######  Corollary 2.11.

Let $K$ be a field of characteristic $r>0$, and fix the number of variables to
be $n\geq r$. Then $\text{trdeg}\hskip 1.99997ptK[p_{1},\ldots,p_{2n+1-r}]=n$.

###### Proof.

By Theorem 2.6, the set $\\{e_{1},\ldots,e_{n}\\}$ is algebraically
independent over $K$. By Theorem 2.8, the fraction fields
$K(e_{1},\ldots,e_{n})$ and $K(p_{1},\ldots,p_{2n+1-r})$ are equal. Since the
transcendence degree of a polynomial ring over a field can be defined either
as the cardinality of a maximal algebraically independent subset or as the
transcendence degree of its field of fractions, we are done. ∎

######  Remark 2.12.

Our methods do not show that $2n+1-r$ is the minimal $m$ such that
$K[p_{1},\ldots,p_{m}]$ has transcendence degree $n$ (except when $r=2$). The
fact that $p_{kr}=p_{k}^{r}$ for all $k\geq 1$ means that $m$ must be at least
$\lfloor nr/(r-1)\rfloor$ (the smallest integer $m$ satisfying $m-\lfloor
m/r\rfloor\geq n$). It would be interesting to know if one can do better than
$2n+1-r$.

######  Corollary 2.13.

Let $K$ be a field of characteristic $r>0$, $V$ an $n$-dimensional $K$-vector
space, and $T\in\operatorname{End}_{K}(V)$ a linear operator on $V$. Suppose
one knows $\operatorname{Tr}(T^{d})$ for all $1\leq d\leq 2n+1-r$. If for
every $1<k\leq n$ with $r\mid k$, the determinant

$\det\begin{pmatrix}\operatorname{Tr}(T)&\operatorname{Tr}(T^{2})&\cdots&\operatorname{Tr}(T^{n-k+1})\\\
\operatorname{Tr}(T^{2})&\operatorname{Tr}(T^{3})&\cdots&\operatorname{Tr}(T^{n-k+2})\\\
\vdots&\vdots&\ddots&\vdots\\\
\operatorname{Tr}(T^{n-k+1})&\operatorname{Tr}(T^{n-k+2})&\cdots&\operatorname{Tr}(T^{2n-2k+1})\end{pmatrix}$
(14)

is non-zero, then one can compute the characteristic polynomial of $T$.

###### Proof.

If $v$ is an eigenvector for $T$ with eigenvalue $\lambda$, then $v$ is an
eigenvector for $T^{d}$ with eigenvalue $\lambda^{d}$. Hence

$\operatorname{Tr}(T^{d})=\sum_{i=1}^{n}\lambda_{i}^{d}=p_{d}(\lambda_{1},\ldots,\lambda_{n}),$

where $\lambda_{1},\ldots,\lambda_{n}\in\overline{K}$ are the eigenvalues of
$T$.
On the other hand, the characteristic polynomial of $T$ is given by

$\text{ch}_{T}(X)=\prod_{i=1}^{n}(X-\lambda_{i})=\sum_{k=0}^{n}(-1)^{k}e_{k}(\lambda_{1},\ldots,\lambda_{n})X^{n-k}.$

We argue by induction that one can compute all
$e_{k}(\lambda_{1},\ldots,\lambda_{n})$. The base case holds as $e_{1}=p_{1}$.
Next, if $r\nmid k$, one can use Newton’s identities (3) to obtain $e_{k}$
from $\operatorname{Tr}(T),\ldots,\operatorname{Tr}(T^{k})$ and
$e_{1},\ldots,e_{k-1}$. If $r\mid k$, one can use Theorem 2.8 to obtain
$e_{k}$ as a rational function in $p_{1},p_{2},\ldots,p_{2n-k+1}$ and
$e_{1},\ldots,e_{k-1}$. This rational function can be evaluated at
$(\lambda_{1},\ldots,\lambda_{n})$ if the denominator of the rational function
is non-zero at this point, which happens if and only if the determinant (14)
is non-zero (cf. Remark 2.9). ∎

######  Example 2.14.

Due to removable poles, it may be possible to compute the characteristic
polynomial of $T$ even if the determinant (14) is zero, as the following
example shows. Let $K=\mathbb{F}_{3}$ and suppose $T$ acts on the
3-dimensional vector space $V$ with the traces
$\operatorname{Tr}(T)=0,\operatorname{Tr}(T^{2})=-1,\operatorname{Tr}(T^{3})=0,\operatorname{Tr}(T^{4})=-1$.
Since $2\in K^{\times}$, we can use Newton’s identities (3) to obtain

$e_{2}(\lambda_{1},\lambda_{2},\lambda_{3})=2^{-1}(\operatorname{Tr}(T)^{2}-\operatorname{Tr}(T^{2}))=-1.$

By Equation (13), this gives

$e_{3}(\lambda_{1},\lambda_{2},\lambda_{3})=\frac{-\operatorname{Tr}(T)\operatorname{Tr}(T^{3})+\operatorname{Tr}(T^{2})e_{2}+\operatorname{Tr}(T^{4})}{\operatorname{Tr}(T)}=\frac{-\operatorname{Tr}(T)\operatorname{Tr}(T^{3})}{\operatorname{Tr}(T)}=-\operatorname{Tr}(T^{3})=0.$

Hence the characteristic polynomial of $T$ is $\text{ch}_{T}(X)=X^{3}-X$.

######  Remark 2.15.

The complete homogeneous symmetric polynomials are defined by

$h_{k}(x_{1},\ldots,x_{n})=\sum_{1\leq i_{1}\leq i_{2}\leq\ldots\leq i_{k}\leq
n}x_{i_{1}}x_{i_{2}}\ldots x_{i_{k}}.$

One can express the power polynomials in terms of the complete homogeneous
symmetric polynomials without problems, but going the other way, there are
again denominators. Fortunately, the fact that
$\mathbb{Z}[e_{1},e_{2},\ldots,e_{n}]=\mathbb{Z}[h_{1},h_{2},\ldots,h_{n}]$
[Mac95, I.2.8] in conjunction with Theorem 2.8 allows one to compute the
$h_{k}$ in terms of the power polynomials over any ring.

##  3 The subalgebra generated by power polynomials

For the remainder of this article, we work over a field $K$ of characteristic
$r>0$. We assume that $n\geq r$, as otherwise the situation is identical to
the characteristic 0 case. A natural question is whether one can describe the
subalgebra $K[p_{1},p_{2},\ldots]\subseteq K[e_{1},\ldots,e_{n}]$ in terms of
elementary symmetric polynomials. For this purpose, we work in the $K$-basis
$\\{e_{\lambda}\ |\ \lambda\text{ is an integer partition}\\}$ of
$K[e_{1},\ldots,e_{n}]$, and when we write “monomial”, we mean a monomial
expressed in terms of elementary symmetric polynomials; that is, a constant
multiple of some $e_{\lambda}$.

The question posed above in some sense has a direct answer. Namely, by
applying the Newton identities inductively, one obtains the following
expression for the power polynomials in terms of the elementary symmetric
polynomials:

$p_{m}=(-1)^{m}\sum_{t_{1}+2t_{2}+\ldots+mt_{m}=m}c_{t_{1},\ldots,t_{m}}\prod_{i=1}^{m}(-e_{i})^{t_{i}}$
(15)

where

$c_{t_{1},\ldots,t_{m}}=\frac{m\cdot(t_{1}+t_{2}+\ldots+t_{m}-1)!}{t_{1}!t_{2}!\cdots
t_{m}!}.$

Since the coefficients $c_{t_{1},\ldots,t_{m}}$ are integers, this is a
formula that holds in $K$. However, it is not clear from the formula (15) how
to judge whether a given symmetric polynomial lies in $K[p_{1},p_{2},\ldots]$
or not.

Our first result shows that the containment $K[p_{1},p_{2},\ldots]\subset
K[e_{1},\ldots,e_{n}]$ is always proper when $n\geq r$ by exhibiting a proper
subalgebra of symmetric polynomials which is not generated by power
polynomials. Our second result shows that no finite number of power
polynomials generates $K[p_{1},p_{2},\ldots]$.

######  Proposition 3.1.

Denote by $E$ the $K$-algebra generated by monomials $e_{\lambda}$ such that
at least one part of $\lambda$ is coprime to $r$. Then
$K[p_{1},p_{2},\ldots]\subseteq E$.

###### Proof.

Since $p_{kr}=p_{k}^{r}$ for all $k$, we see that $K[p_{1},p_{2},\ldots]$ is
generated by power polynomials $p_{k}$ with $(k,r)=1$. Any such power
polynomial lies in $E$. Indeed, any monomial $e_{\lambda}$ corresponding to a
partition $\lambda$ with all parts divisible by $r$ has degree a multiple of
$r$, while $p_{k}$ is homogeneous of degree $k$, which is not divisible by $r$
by assumption. The result follows because $E$ is a subalgebra of
$K[e_{1},\ldots,e_{n}]$. ∎

######  Example 3.2.

One can check by hand that for $K=\mathbb{F}_{2}$ and $n=3$, one cannot write
$e_{3}=x_{1}x_{2}x_{3}$ as an algebraic combination of power polynomials. This
shows that Proposition 3.1 is not sharp in general.

######  Proposition 3.3.

Let $K$ be a field of characteristic $r>0$. Fix an integer $k$ not divisible
by $r$. Let $n\geq r$ be the number of variables. Then

$K[p_{1},\ldots,p_{k-1}]\subsetneq K[p_{1},\ldots,p_{k}].$

###### Proof.

Write $k=ar+b$ for integers $a\geq 0$ and $0<b<r$. Applying the identity (15)
with $m=k$, $t_{b}=1$, and $t_{r}=a$ tells us that $p_{k}$ contains the
monomial $(-1)^{a+1}e_{r}^{a}e_{b}$.

We claim that there are no symmetric polynomials in $K[p_{1},\ldots,p_{k-1}]$
which, when written in terms of the elementary symmetric polynomials, contain
the monomial $e_{r}^{a}e_{b}$; this will finish the proof. Suppose for
contradiction that $f$ is a counterexample. Consider an expression

$f=\sum_{v\in\mathbb{N}^{k-1}}c_{v}p_{1}^{v_{1}}\ldots p_{k-1}^{v_{k-1}},$

with $v_{i}=0$ for all $r\mid i$; we can achieve this because
$p_{jr}=p_{j}^{r}$. By the algebraic independence of the elementary symmetric
polynomials, the monomial $e_{r}^{a}e_{b}$ in $f$ must be a product of
monomials of the form $e_{r}^{\alpha}e_{b}^{\beta}$, with $\alpha\leq a$ and
$\beta\leq 1$, occurring in $p_{1}^{v_{1}},\ldots,p_{k-1}^{v-1}$. Since there
are no monomials of this form with $\beta=0$ by Proposition 3.1, we see that
the entire monomial $e_{r}^{a}e_{b}$ must occur in some $p_{m}$. This is a
contradiction for degree reasons: we obtain $m=ra+b=k>m$. This finishes the
proof. ∎

######  Corollary 3.4.

Whenever $n\geq r$, the subalgebra

$K[p_{1},p_{2},\ldots]\subset K[x_{1},\ldots,x_{n}]$

is not finitely generated.

###### Proof.

Given any finite set $S=\\{f_{1},\ldots,f_{m}\\}$ of products of power
polynomials in $n$ variables, consider $p_{N}$ for any $N$ satisfying
$N>\max\\{\deg f_{i}\\}$ and $r\nmid N$. Then by the same argument as in the
proof of Proposition 3.3, $p_{N}$ (when written in terms of the elementary
symmetric polynomials) contains a monomial of the form $e_{r}^{a}e_{b}$ for
$N=ra+b$, but no polynomial generated by $S$ contains a monomial of this form.
∎

######  Example 3.5.

Let $K=\mathbb{F}_{2}$ and $n=2$. Then we have for any $N=2m+1$:

$\mathbb{F}_{2}[p_{1},p_{3},\ldots,p_{N}]=\mathbb{F}_{2}[e_{1},e_{1}e_{2},e_{1}e_{2}^{2},\ldots,e_{1}e_{2}^{m}].$

Note that this example shows that $K[e_{1},\ldots,e_{n}]$ is not the integral
closure of $K[p_{1},p_{2},\ldots]$.

###  3.1 Concluding remarks

The subalgebra generated by power polynomials over $K$ is the whole algebra of
symmetric polynomials when $n<r$, but becomes much more difficult to describe
when $n\geq r$. As there is no finite generating set, it is not clear whether
there is a more satisfactory description of $K[p_{1},p_{2},\ldots]$ in terms
of elementary symmetric polynomials than the expressions given by the identity
(15), which may be slightly simplified by the fact that
$e_{1},\ldots,e_{r-1}\in K[p_{1},\ldots,p_{r-1}]$.

The discrepancy between the cases $n<r$ and $n\geq r$ is reminiscent of the
modular representation theory of the symmetric group $S_{n}$. It would be
interesting to know whether this theory can be applied to offer a more
conceptual interpretation of the results in this paper.

## References

  * [AM69] M.. Atiyah and I.. Macdonald  “Introduction to commutative algebra”  Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969
  * [Ham86] Eloise Hamann  “Transcendence degree over an arbitrary commutative ring”  In _Journal of Algebra_ 101.1, 1986, pp. 110–119
  * [Mac95] I.. Macdonald  “Symmetric functions and Hall polynomials”, Oxford Mathematical Monographs  The Clarendon Press, Oxford University Press, New York, 1995

Matematiska institutionen, Stockholms universitet, 106 91 Stockholm, Sweden
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