# The nonuniformity of poleward flux transport on the solar surface: a
statistical method applied to solar cycles 21-24

Zi-Fan Wang  Key Laboratory of Solar Activity, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, China  School of
Astronomy and Space Science, University of Chinese Academy of Sciences,
Beijing, China  Jie Jiang School of Space and Environment, Beihang University,
Beijing, China  Key Laboratory of Space Environment Monitoring and Information
Processing of MIIT, Beijing, China  Jing-Xiu Wang  School of Astronomy and
Space Science, University of Chinese Academy of Sciences, Beijing, China  Key
Laboratory of Solar Activity, National Astronomical Observatories, Chinese
Academy of Sciences, Beijing 100101, China

###### Abstract

The poleward migration of the active regions’ magnetic flux on the solar
surface plays an important role in the development of the large-scale field
development, especially the polar field reversal, which is a key process in
the Babcock-Leighton-type solar dynamos. The poleward flux transport is
nonuniform, centered around poleward surges as suggested by previous
observations. The strong, long-lasting surges are related to activity
complexes, and often result in violent polar field reversal. However, the
nonuniformity of poleward flux transport has not been evaluated
quantitatively. We propose a statistical method to analyze the poleward flux
transport during solar cycles 21-24 by considering the frequency distributions
of the magnetic field at latitudes of poleward surges occurrence during solar
cycles. The nonuniformity is quantified as the kurtosis statistics
representing the tailedness of the distributions. We test the method on
results of surface flux transport simulations, and apply to WSO, NSO, MWO, and
HMI data. We confirm that the poleward surges are of significance during solar
cycles 21-24 in general. The kurtosis within a solar cycle is affected by
different latitudes of the magnetic field and different data sources. The
southern hemisphere of cycle 24 exhibits the largest kurtosis, agreeing the
super surge concept from previous work. The significant nonuniformity of
poleward flux transport originates from the nonrandomness of active regions,
which favors the activity complexes origin of poleward surges.

##  1 Introduction

The emergence of active regions (ARs) and the transport of AR flux on the
solar surface are key processes in Babcock-Leighton-type solar dynamos that
aim to explain the 11-year sunspot cycle (Babcock, 1961; Leighton, 1969).
Typically, an AR emerges with two opposite polarities that exhibit a tilt
angle with respect to the east-west direction. Since the tilt angles follow
Joy’s law (Hale et al., 1919), the following polarity of ARs is usually closer
to the pole of the corresponding hemisphere, while the leading polarity of ARs
is usually closer to the equator. As the AR flux evolves on the solar surface,
more flux of the following polarity is transported to the pole, contributing
to the development of the polar field. The polar field at cycle minimum is an
important precursor of the strength of the next cycle, as shown by analysis of
observations and dynamo models (Schatten et al., 1978; Svalgaard et al., 2005;
Jiang et al., 2007).

Besides the individual AR’s characteristics such as their emerging latitudes
and tilt angles, the clustering tendency of ARs emergence is a notable
characteristic during solar cycles. The continuous emergence of closely
located ARs during several Carrington rotations (CRs) is defined as a complex
of activity (Bumba & Howard, 1965; Gaizauskas et al., 1983), or an activity
complex. It is similar to the concept of sunspot nests proposed by
Castenmiller et al. (1986), as a sequence of AR emergence in a compact solar
surface area lasting for months. Subsequent quantitative analysis of sunspot
nests by Brouwer & Zwaan (1990) suggests at least a third of sunspots are
nested. Activity complexes represent the important non-axisymmetric property
of the large-scale field, as reviewed by van Driel-Gesztelyi & Green (2015).

As the emergence of ARs tends to be concentrated in activity complexes, the
poleward migration of AR flux is also not uniform during solar cycles, but in
the form of poleward surges (streamers, plumes) seen on the magnetic butterfly
diagrams (longitudinally averaged surface magnetic field), as recognized by
Howard & Labonte (1981). The poleward surges’ strength, temporal width and
influence to the polar field is determined by the originating ARs and the
surface flux transport parameters (Sun et al., 2015). Observationally, long-
lasting activity complexes can be related to prominent poleward surges,
accompanied by intense polar field reversal (Petrie & Ettinger, 2017;
Mordvinov & Kitchatinov, 2019). Especially, the polar field development of
cycle 24 is majorly determined by a “super surge” at the southern hemisphere
originating from activity complexes around the year 2014, as shown by Wang et
al. (2020). Considering the nesting nature of ARs, super surges are likely to
occur. Such surges are able to greatly influence the short term development of
the polar field, introducing violent variations, affecting the solar open
field (Jiang et al., 2010), fast solar wind (Tu et al., 2005), and
interplanetary field (Balogh et al., 1995). Since the polar field observations
have been suspicious in the open flux models (Linker et al., 2017), it is
important to examine the generation and development of these surges
contributing to the polar field. The surges provide us a direct perspective to
understand and reconstruct the polar field development, according to the
surface flux transport (SFT) models (Jiang et al., 2014). These surges are
also highly important in the long term development of solar cycles as they can
contribute a major part to the polar field at cycle minimum, and in turn to
the development of following generation of toroidal field, especially the next
solar cycle considered.

Previous studies on poleward surges involve the observations and simulations
of specific surges, or the qualitative analysis of their relationship with the
general nesting tendency of activity complexes. SFT simulations of Yeates et
al. (2015) reveal the origin and contribution to the polar field of a poleward
surge. The surge they analyze originates from higher latitudes and has low
influence to the polar field. The super surge with large polar field
contribution studied by Wang et al. (2020) is also examined with SFT
simulations. Properties of the super surge’s configuration is described by the
magnetic field profiles at middle to high latitudes, that is, latitudes higher
than AR emergence and lower than polar regions. The peaks (or dips) correspond
to poleward surges in these profiles. The magnetic field at middle latitudes
is also used by Sun et al. (2015) to represent the surges as they seek
correlations between surge properties and AR parameters, and by Baranov et al.
(2021) to study the occurrence and periodicity of surges. The causal
relationship between poleward surges and activity complexes has been evaluated
qualitatively, by combining the magnetic butterfly diagrams, AR emergence
data, and polar field evolutions (Mordvinov & Kitchatinov, 2019).

The poleward surges originate from ARs, which are discrete in time and
location during solar cycles. That is, the magnetic field on activity
latitudes cannot remain constant during one solar cycle, so the magnetic field
on middle to high latitudes cannot be constant as well. Hence, qualitatively
speaking, the poleward flux transport is always nonuniform to a certain
degree. However, from the quantitative perspective, it is expected that the
nonuniformity of poleward transport from ARs with and without nesting tendency
will show difference, if there is a strong causal relationship between
activity complexes and poleward surges. It is required to know how nonuniform
the poleward transport is quantitatively, and its possible implications on the
originating ARs. This leads us to propose a statistical method to analyze the
properties of the magnetic field at middle to high latitudes.

In this article we present a statistical method to evaluate the quantitative
nonuniformity of poleward flux transport during solar cycles 21-24 by
analyzing the non-Gaussianity of the frequency distribution of magnetic field
at middle to high latitudes, mainly expressed by the kurtosis statistic. We
establish the basis of the statistical method on the central limit theorem,
and construct the relationship between poleward flux transport nonuniformity
and AR emergence nonrandomness. We validate the method with a series of SFT
simulations. We apply the method to solar cycles 21-24 from different sources
of data, evaluating the kurtosis of different cycles, and the values of the
data combined by the 4 cycles together. We are able to show that generally the
cycles exhibit positive kurtosis, implying significant occurrence of poleward
surges. Especially, the southern hemisphere of cycle 24 has the largest
nonuniformity of poleward flux transport, indicating strong poleward surges.

This article is organized as follows. In Section 2, we introduce the
statistical analysis method, and evaluate its effectiveness with SFT
simulations. We then apply the method to observational data and present the
results in Section 3. We discuss and conclude in Section 4.

##  2 Analysis method

###  2.1 The statistical method to quantify the nonuniformity of poleward flux
transport

The basis of our statistical method is the central limit theorem. The theorem
states that, for a number of independent random variables satisfying a certain
distribution, no matter what the distribution is, their addition will follow a
Gaussian distribution as the number of the random variables becomes
sufficiently large. It is also usually expressed as, for a certain
distribution, the average of a number of independent sampling will follow a
Gaussian distribution, as the number of the sampling increases large enough.
The most widely applied form of the theorem has all the random variables
following the same distribution. However, more generalized forms of the
central limit theorem do not require the distributions of the random variables
to be exactly the same, but require that the sum is primely determined by a
number of them, instead of some extreme few (see e.g. Lindeburg, 1922). This
inspires us to consider the large-scale field development of the Sun. If the
emergence of ARs during a period of time is random and uncorrelated, which
refers to the case where activity complexes are completely absent, the
emergence of ARs will meet the prerequisites of the central limit theorem.

We consider the poleward flux transport generated by such random and
uncorrelated ARs. The magnetic field at a certain time $t$ at middle to high
latitudes is an linear addition of the contribution from many ARs emerged
before, as we assume that the surface flux transport process is described by a
linear, kinematic magnetic induction equation on the 2 dimensional solar
surface, which is often used in previous SFT studies (e.g. DeVore et al.,
1985; Wang et al., 1989; van Ballegooijen et al., 1998; Mackay et al., 2002;
Jiang et al., 2014). This means that the nonlinear processes such as inflows
toward activity belts (Haber et al., 2002; Zhao & Kosovichev, 2004; Gizon,
2004; Jiang et al., 2010; Cameron & Schüssler, 2012) in the surface flux
transport process are neglected. The _i_ th AR with emergence time $t_{i}$
generates a contribution to the magnetic field denoted as
$B_{i}\left(t-t_{i}\right)$ at time $t$, and the magnetic field can be
expressed as $B\left(t\right)=\sum_{i}B_{i}\left(t_{i}-t\right)$. If the
“$B_{i}$”s have the same distributions, the magnetic field can be approximated
as $B\left(t\right)=\sum_{i}B_{0}\left(t_{i}-t\right)$. If the emergence time
“$t_{i}$”s are randomly and independently distributed during the cycle,
$B\left(t\right)$ can be regarded as an addition of repeatedly independently
sampling $B_{0}$. Hence, according to the central limit theorem,
$B\left(t\right)$ should follow a Gaussian distribution.

The ARs in reality are not identical in terms of their characteristics such as
emerging latitudes, size, and shape, so the magnetic fields at middle to high
latitudes they generate are not exactly the same. Generally, this does not
affect the application of the central limit theorem under the condition that
the difference between the magnetic field distributions is not too large, and
that the variance of the sum is not dominated by a few particular
distributions. However, some ARs have extreme characteristics, for example, an
AR with large size may emerge at higher latitudes than normal, close to the
latitudes we evaluate. In such cases, the magnetic field at the middle to high
latitudes is primarily dominated by the particular AR, instead of the linear
addition of many ARs. Therefore, such magnetic fields can no longer be
regarded as determined by a large number of ARs, which results in deviations
from Gaussian distributions. In observations, such ARs are possible to occur,
for example, the high emerging latitudes ARs that generate the prominent surge
studied by Yeates et al. (2015).

The Gaussianity, or non-Gaussianity of the distribution that $B\left(t\right)$
follows shows the statistical properties of the poleward flux transport, and
the originating ARs. If the distribution exhibits significant non-Gaussianity,
it shows that the ARs generating the poleward flux transport are either too
few to satisfy the central limit theorem, or are not randomly distributed and
uncorrelated. The activity complexes are a probable source of the introduction
of nonrandom AR emergence. The poleward surges, as the nonuniform component,
can be referred to as the the source for the distribution deviating from the
Gaussian distribution. As the activity complexes generate more prominent
poleward surges, the distribution will exhibit heavier than Gaussian tails on
the sign of the polarity of the surges. Therefore, if we get heavier than
Gaussian tails of the magnetic field distributions from observational data, we
can tell that the surges are not generated by random ARs. In such cases, the
nonuniformity of poleward flux transport can be considered statistically
significant, as it represents important nonrandom AR characteristics that
favor the production of poleward surges. This can be an important statistical
evidence to the aforementioned activity complexes – poleward surges
relationship.

Hence, our statistical method analyzes the frequency distribution of the
magnetic fields at middle to high latitudes, focusing on the non-Gaussianity
of its tailedness. The tailedness of a frequency distribution is often
measured by the kurtosis statistical parameter, derived from the 4th central
moment. The kurtosis of a series of values $x_{1},...,x_{n}$ is defined as
follows,

$kurtosis=\frac{1}{n\sigma^{4}}\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{4}-3,$
(1)

where $n$ is the total number of the values, $\bar{x}$ is the mean of the
values, and $\sigma$ is the standard deviation. The kurtosis of a Gaussian
distribution is 0, while the value of a distribution with heavier tail than
Gaussian is larger than 0. Hence, we use the kurtosis statistics to quantify
the nonuniformity. We get the middle to high latitude magnetic field on a
certain hemisphere during a solar cycle from magnetic butterfly diagrams, and
get the frequency distribution. We then calculate and evaluate the kurtosis
values. In practice, the standard error of kurtosis (SEK) should be
considered. Approximately, the standard error is $\sqrt{24/n}$, where $n$
represents the sample size (see e.g. Jarque & Bera, 1987; Ðorić et al., 2009).
We use $2\times SEK$ to judge the statistical significance of nonuniformity. A
kurtosis value large than $2\times SEK$ can be regarded as significantly
nonuniform, and the originating ARs exhibit nonrandom characteristics.

###  2.2 Evaluation of the statistical method with a surface flux transport
model

In order to evaluate the ability of the statistical method to quantify the
nonuniformity of poleward flux transport, and to justify whether the method
can reveal the probable nonrandom ARs’ characteristics that favor the
formation of surges, we employ a series of SFT simulations with artificially
designed source terms, generate the corresponding magnetic synoptic diagrams,
and examine the magnetic field strength frequency distributions. SFT models
solve the magnetic induction equation on the solar surface for the radial
component of the magnetic field, with a set of transport parameters including
surface meridional flow, differential rotation and supergranular diffusion, as
well as newly emerged ARs as source terms. The SFT model that we utilize is
from Baumann et al. (2004), with differential rotation profile adopted from
Snodgrass (1983) and meridional flow from van Ballegooijen et al. (1998). The
supergranular diffusivity that we use is 500 km2s-1.

We design two cases of AR emergence, namely the unnested ARs case and nested
ARs case. Both cases are simulated without an initial magnetic field. For the
unnested case, we consider a series of AR emergence during a time period with
a constant average number of ARs per unit time. We consider an average of 1500
ARs during 10 years, hence the average number of AR emergence per day is
0.082. The exact number of ARs emerged on a specific day during a simulation
is determined by a Poisson distribution, so the emergence of ARs is random and
uncorrelated. The ARs have a size of 200 millionth of solar hemisphere, and
are placed on $10^{\circ}$ latitude on either hemisphere, with a Gaussian
scatter of latitude with $\sigma=2^{\circ}$. The tilt angles of ARs are
determined by a typical form of the Joy’s law, which is
$1.3\times\sqrt{\left|\lambda\right|}$ with $\lambda$ being the latitude. The
scatter of tilt angle follows a Gaussian profile, the width of which is
determined by the AR area – tilt scatter relationship given by Jiao et al.
(2021). We conduct 25 simulation runs for 10 years each.

In the nested case, the random and uncorrelated emergence across the entire 10
years is reduced by half, that is, averagely 750 ARs during 10 years.
Meanwhile, 5 concentrated nests with a length of 5 Carrington rotations are
added to the simulations on 5 separate days randomly selected in the 10 years.
Each nest contains an average of 150 ARs, hence the total expectation of AR
emergence during each simulation is also 1500. Then we get the modified
average AR emergence per day constructed by a low value during most of the
simulations, and a high value during the relatively short nests. The exact
number of ARs emerged on each day during a simulation is also determined by a
Poisson distribution based on the average AR emergence per day. The parameters
such as latitudes and tilt angles of ARs are determined by identical means as
the unnested case.

The AR emergence in the SFT simulations are designed in order to generate a
series of generally even poleward flux transport, and a series of nonuniform
poleward flux transport mainly in the form of poleward surges. In reality, the
AR emergence during a solar cycle follows much more empirical rules from
observations (see e.g. Jiang et al., 2011), which are possible to influence
the nonuniformity of poleward flux transport. For example, the latitudes of AR
emergence during a solar cycle is different. When ARs emerge at higher
latitudes, the magnetic fluxes from them are less diffusive as they reach the
middle to high latitudes we evaluates. In this case the magnetic field is
primarily contributed by fewer ARs, which is not consistent with the
perquisites of the central limit theorem. So the effects of these varied
situations are not considered in the simulations for a clearer example. In
reality, the difference of emergence latitudes of ARs influences the analysis
of the nonuniformity in the form of the latitudinal differences of kurtosis
values, as shown in the following Section.

Figure 1: Results of SFT simulations evaluating the statistical method. Panels
(a) and (b) are examples of magnetic butterfly diagrams from the unnested case
and the nested case, respectively. Panels (c), (d), (e), and (f) are magnetic
field frequency distributions of the combined data of 25 runs for the two
cases. The solid curves show the distributions of the simulated data. The
dashed curves represent Gaussian profiles with identical mean and standard
deviation of the distributions. The solid vertical lines mark the means, while
the dashed vertical lines represent the standard deviations. Panels (c) and
(e) are results for the northern and southern hemisphere for the unnested
case, while panels (d) and (f) are results for the northern and southern
hemisphere for the nested case.

The simulation results are analyzed using the method described in the previous
subsection. Each simulation contains 137 Carrington rotations, but we omit the
first 14 rotations when we construct the magnetic field distributions, in
order to remove the effect of the start field on the analysis. The data of 25
runs are combined together by merging the data into one frequency
distribution, as the AR emergence of each run is generated by identical means.
Hence, there are 3075 data points for each case in total. For $n=3075$ we have
$SEK=0.09$, so $2\times SEK$ is 0.18, which is used to judge the statistical
significance of the non-Gaussianity of kurtosis values. Two typical magnetic
butterfly diagrams drawn from the two cases are shown in Figures 1 (a) and
(b). As shown, the nested case shows clear generation of prominent poleward
surges. The unnested case shows much smoother poleward flux transport, which
is close to uniform spanning across the simulation, but surges still occur
occasionally if the emerged active regions are in proximity by chance. Hence,
the statistical method is needed to quantitatively distinguish the difference
between the two cases. Considering the latitudes the surges located, we choose
the average of latitudes $35^{\circ}$ to $40^{\circ}$ to construct the
magnetic field frequency distribution, by using 20 equal size bins across the
magnetic field values. The distributions are shown in Figures 1 (c)$\sim$(f).
As shown, the distributions of the unnested case are mostly Gaussian. The
nested case, on the contrary, shows prominent non-Gaussianity in its
distributions, which are not symmetric, and have heavier tails on the side of
the following polarity. Then we calculate the central moments of the magnetic
field frequency distribution. Table 1 shows the central moments of the
distributions. As shown, both cases have the same mean value, hence the total
amount of poleward flux is the same for the two cases. The nested case has a
larger standard deviation, and is more skewed against the sign of the mean,
while the unnested case is generally symmetric and has close to 0 skewness,
satisfying the properties of a Gaussian distribution. The kurtosis of the
unnested case is less than $2\times SEK$, while the kurtosis of the nested
case is significantly greater than $2\times SEK$. The central moments are
consistent with the characteristics of the distributions shown in Figures 1
(c)$\sim$(f), and the kurtosis well describes the distributions’ tailedness
that represents the poleward surges. The results show that the statistical
analysis based on the kurtosis effectively distinguishes the difference
between the nonuniform poleward flux transport generated by nested ARs, and
generally uniform poleward flux transport generated by unnested ARs.

Table 1: Central moments of the simulated poleward magnetic field frequency distribution  Sunspot nests | Latitudes | Mean | Standard deviation | Skewness | Kurtosis |
---|---|---|---|---|---|---
Unnested | $-35^{\circ}\sim-40^{\circ}$ | -0.80 | 0.38 | -0.06 | 0.00 |
$35^{\circ}\sim 40^{\circ}$ | 0.84 | 0.40 | -0.02 | -0.12 |
Nested | $-35^{\circ}\sim-40^{\circ}$ | -0.80 | 0.61 | -1.19 | 1.76 |
$35^{\circ}\sim 40^{\circ}$ | 0.84 | 0.59 | 1.20 | 2.70 |

##  3 Analysis of observational data

In this section we quantify the nonuniformity of solar cycles 21-24 using the
statistical method established in the previous section. We employ the surface
line-of-sight magnetic field synoptic maps from Wilcox Solar Observatory
(WSO), and surface radial magnetic field synoptic maps from National Solar
Observatory (NSO) and Mount Wilson Observatory (MWO). The WSO data and NSO
data cover solar cycles 21-24, while the MWO data covers solar cycles 21-23.
We also include the more recent results from the synoptic maps of Helioseismic
and Magnetic Imager of Solar Dynamical Observatories (SDO/HMI) for solar cycle
24 for better quality. The magnetic butterfly diagrams generated from WSO,
NSO, and MWO data are displayed in Figure 2. The poleward surges can be
clearly seen on all diagrams. However, different sources of data have
different resolutions and hence show different levels of details, which can be
directly expressed in the magnetic butterfly diagrams. Generally, the WSO data
shows the least details of the surges, while the MWO data shows the most. This
may lead to differences in the results of different sources in the following
analysis.

Figure 2: Magnetic butterfly diagrams generated from longitudinally averaged
synoptic maps. Panels (a), (b), and (c) are generated from WSO, NSO, and MWO
data, respectively. The diagrams are equally spaced in latitude.

We analyze the magnetic field strength frequency distribution at middle to
high latitudes. The WSO data is composed of 30 data points equally spaced in
sine latitude between $-75^{\circ}$ and $75^{\circ}$, so we choose the
latitudes $34^{\circ}$, $39^{\circ}$, and $44^{\circ}$ for both hemispheres
from the original data points. The NSO/KPVT data and the NSO/SOLIS data have
180 and 900 data points equally spaced in sine latitudes, respectively. The
MWO data has 512 points equally spaced in latitudes. The HMI data has 1440
points equally spaced in sine latitudes. In order to compare the results of
NSO, MWO, and HMI with the results of WSO, we choose to use the average of
several close latitudes when we deal with NSO, MWO, and HMI data. The
latitudes $35^{\circ}$-$40^{\circ}$ and $40^{\circ}$-$45^{\circ}$ are used for
NSO, MWO, and HMI. Then we construct the frequency distribution and calculate
the kurtosis values at these latitudes.

The kurtosis values for each cycle, hemisphere, and latitude considered from
the 4 data sources are shown in Tables 2, 3, 4, and 5. The values are positive
for many cycles and latitudes, and are subject to changes for different
latitudes, cycles, and data sources. Averagely, each solar cycle lasts for 11
years, that is, approximately 149 Carrington rotations. For $n=149$,
$SEK=0.4$. Hence the $2\times SEK$ criterion should be 0.8, which will be used
in the analysis to judge statistical significance.

The kurtosis values show latitudinal variations within a certain hemisphere
and cycle, which shows statistical significance in general. We consider the
kurtosis value variations statistically significant if the variations are
larger than $2\times SEK=0.8$. Among the 22 cases considering all hemispheres,
cycles, and data sources, about 68% show a statistically significant decrease
in kurtosis values as the latitude increases. The general decrease of kurtosis
is consistent with the considerations of our statistical method, as poleward
surges tend to be more diffusive as they migrate to higher latitudes, which
makes the overall poleward transport smoother. In this case the distribution
has less values in the tails, so the kurtosis values would be smaller.
Meanwhile, since the magnetic field at higher latitudes can be associated with
more ARs as a result of AR flux being more diffusive on reaching higher
latitudes, they are more consistent with the requirement of the central limit
theorem. Hence, they tend to follow a distribution closer to a Gaussian
distribution. However, we also note that there are exceptions to the general
trend, as there are about 14% of the cases show a statistically significant
increase in kurtosis values, such as the southern hemisphere of cycle 23 for
NSO and MWO. The originating latitudes of the surges are different, which is a
possible cause of kurtosis latitudinal variations, especially the variations
opposite to the general trend.

The kurtosis values also show differences for different data sources.
Considering the $2\times SEK$ criterion for judging non-Gaussianity, some
solar cycles have an agreement in terms of non-Gaussianity of poleward flux
transport distribution, while some cycles do not have an agreement. For solar
cycles 21 and 23, the WSO data source does not show significant non-
Gaussianity, while both the NSO and MWO data sources show significant non-
Gaussianity at some hemispheres and latitudes. This can be possibly explained
by the fact that the different data sources have different resolutions, and
show different levels of details of the surges in the magnetic butterfly
diagrams. As described above in Figure 2, the details of the surges are less
displayed in the diagram of WSO than in NSO and MWO, which is possible to
affect the results of the kurtosis analysis. More details of the surge
generally mean more uneven structures in the poleward flux transport,
increasing overall nonuniformity. This interpretation agrees with the result
that the largest kurtosis value 6.33 is obtained from HMI data on the southern
hemisphere of cycle 24, as the HMI data shows the most detailed structure of
cycle 24. The difference between the HMI results and other results of the
southern hemisphere is larger than $2\times SEK=0.8$, indicating a significant
data quality difference.

The kurtosis values represent the characteristics of poleward flux transport
in different hemispheres and solar cycles. From the results with better
agreement among the data sources, we can see that the northern hemisphere of
solar cycle 22 and the southern hemisphere of cycle 24 have kurtosis values
that are larger than the value of a Gaussian distribution of statistical
significance. This indicates that the poleward flux transport during these
cycles on the certain hemisphere is more uneven than what randomly generated
ARs could produce, so the associated ARs emergence should not be random. The
results of the southern hemisphere of cycle 24 have the largest kurtosis
values, and are highly in agreement among the results of different latitudes
and data sources, which agrees with the result of Wang et al. (2020) that the
super surge considered has a profound and long-lasting effect on the
development of the solar cycle. The northern hemisphere, on the contrary, does
not show non-Gaussianity of statistical significance. The results of the HMI
data are consistent with the interpretation of cycle 24, as the kurtosis on
the southern hemisphere is essentially large. The northern hemisphere has
significantly lower kurtosis values. However, since the HMI data source tends
to produce larger kurtosis values than other data sources in general, the
kurtosis on $35^{\circ}\sim 40^{\circ}$ still exceeds $2\times SEK=0.8$.
Still, the super surge occurrence on the southern hemisphere is strongly
favored by HMI results.

Table 2: Kurtosis of the frequency distribution of poleward magnetic field based on the WSO data for cycles 21-24  Latitudes | Cycle 21 | Cycle 22 | Cycle 23 | Cycle 24 |
---|---|---|---|---|---
$-34^{\circ}$ | 0.98 | 1.56 | 1.15 | 3.70 |
$-39^{\circ}$ | -0.32 | 0.48 | 0.19 | 3.26 |
$-44^{\circ}$ | -0.28 | -0.13 | 0.50 | 3.60 |
$34^{\circ}$ | -0.43 | 2.80 | 0.39 | 0.17 |
$39^{\circ}$ | 0.44 | 2.44 | -0.38 | -0.87 |
$44^{\circ}$ | 0.79 | 1.06 | -0.44 | -0.82 |
Table 3: Kurtosis of the frequency distribution of poleward magnetic field based on the NSO data for cycles 21-24  Latitudes | Cycle 21 | Cycle 22 | Cycle 23 | Cycle 24 |
---|---|---|---|---|---
$-35^{\circ}\sim-40^{\circ}$ | -0.13 | 1.52 | 0.07 | 3.55 |
$-40^{\circ}\sim-45^{\circ}$ | -0.11 | -0.37 | 1.23 | 2.67 |
$35^{\circ}\sim 40^{\circ}$ | 2.27 | 2.13 | 1.77 | 0.46 |
$40^{\circ}\sim 45^{\circ}$ | 1.09 | 0.73 | 1.06 | -0.62 |
Table 4: Kurtosis of the frequency distribution of poleward magnetic field based on the MWO data for cycles 21-23  Latitudes | Cycle 21 | Cycle 22 | Cycle 23 |
---|---|---|---|---
$-35^{\circ}\sim-40^{\circ}$ | 1.19 | 2.86 | 0.30 |
$-40^{\circ}\sim-45^{\circ}$ | -0.04 | 0.45 | 1.15 |
$35^{\circ}\sim 40^{\circ}$ | 3.88 | 1.60 | 1.28 |
$40^{\circ}\sim 45^{\circ}$ | 1.31 | 1.26 | 0.43 |
Table 5: Kurtosis of the frequency distribution of poleward magnetic field based on the HMI data for cycle 24  Latitudes | Cycle 24 |
---|---|---
$-35^{\circ}\sim-40^{\circ}$ | 6.33 |
$-40^{\circ}\sim-45^{\circ}$ | 4.22 |
$35^{\circ}\sim 40^{\circ}$ | 1.32 |
$40^{\circ}\sim 45^{\circ}$ | -0.30 |

We further apply the analysis method to the data sets combining the data from
different solar cycles. As shown above, the results of an individual cycle
show nonuniformity at some cycles, while some other cycles do not have
nonuniformity with statistical significance. In order to answer whether the
poleward flux transport is nonuniform in general terms, combining the data
from different cycles is needed. Combining the data also decreases the
standard error of kurtosis as the sample size is increased. The data of an
individual cycle suffers from a relatively high standard error, which limits
the credibility. The general results from the combined data can be more
credible when judging the statistical significance of kurtosis. We consider
that the total amount of poleward transported flux is different for different
cycles, hence the absolute values of “tails” of the frequency distributions
should be different. Therefore, we normalize the data from different cycles by
dividing the data by the mean of the corresponding cycle and hemisphere. Then
the frequency distribution is constructed for each data source and the
different orders of central moments are obtained.

We present the frequency distributions of the poleward magnetic field based on
the WSO data as a typical example. The distributions for different latitudes
of the WSO data are displayed in Figure 3. A Gaussian profile with identical
mean and standard deviation is also plotted in each panel, to show the
difference between the distributions and a Gaussian distribution. The
distributions show significant non-Gaussianity, as they are much different
from the Gaussian profiles. They are constructed by a narrower peak, and a
heavier tail at the following polarity side, much similar to the frequency
distribution of the nested case described in Subsection 2.2. Note that since
we divide the data of an individual cycle by the mean of the corresponding
cycle and hemisphere, the mean of the data set for cycles 21-24 is 1. The
heavy tails imply that the contribution of poleward surges to the poleward
flux transport during cycles 21-24 is higher than the poleward magnetic field
satisfying a Gaussian distribution in general, indicating a non-random AR
emergence during the cycles.

The central moments for the combined data of all three data sources are shown
in Table 6. As shown, the skewness and kurtosis are all positive, which is
consistent with Figure 3 that shows heavier tails especially on the positive
side. For the data combining cycles 21-24 (21-23 for MWO), the $SEK$ value is
0.2 (0.20 for 4 cycles, and 0.23 for 3 cycles). Compared to $2\times SEK=0.4$,
the kurtosis values for all cases listed show the significant non-Gaussian
property. Hence, the data sources agree that the poleward surges during the
cycles considered have statistical significance in terms of kurtosis.
Meanwhile, there exist latitudinal and hemispheric differences with
statistical significance compared to $2\times SEK$, for all data sources
considered. The results suggest that the poleward surges, as the nonuniform
component, play an important role in the poleward transport of AR flux. The
surges are not generated by some random ARs that happen to emerge in proximity
by chance, but generated by the AR emergence that is not random and
uncorrelated during the solar cycles.

Figure 3: Magnetic field strength distributions for the WSO data during cycles
21-24, on different latitudes. The solid curves show the distributions of the
observed data. The dashed curves represent Gaussian profiles with identical
means and standard deviations of the corresponding distributions. The solid
vertical lines mark the means, while the dashed vertical lines represent the
standard deviations. Panels (a) to (f) refer to latitudes $35^{\circ}$,
$-35^{\circ}$, $40^{\circ}$, $-40^{\circ}$, $45^{\circ}$, and $-45^{\circ}$,
respectively.  Table 6: Central moments of the frequency distributions of
poleward magnetic field for WSO, NSO, and MWO

Observation source | Latitudes | Standard deviation | Skewness | Kurtosis |
---|---|---|---|---|---
WSO | $-34^{\circ}$ | 1.86 | 0.83 | 3.14 |
$-39^{\circ}$ | 1.56 | 1.22 | 2.42 |
$-44^{\circ}$ | 1.32 | 1.42 | 2.88 |
$34^{\circ}$ | 2.39 | 1.33 | 3.94 |
$39^{\circ}$ | 2.04 | 0.96 | 2.48 |
$44^{\circ}$ | 1.84 | 0.35 | 2.47 |
NSO | $-35^{\circ}\sim-40^{\circ}$ | 2.31 | 0.62 | 1.47 |
$-40^{\circ}\sim-45^{\circ}$ | 1.78 | 0.89 | 1.30 |
$35^{\circ}\sim 40^{\circ}$ | 2.24 | 0.85 | 4.39 |
$40^{\circ}\sim 45^{\circ}$ | 1.96 | 0.64 | 2.35 |
MWO | $-35^{\circ}\sim-40^{\circ}$ | 2.02 | 0.36 | 1.82 |
$-40^{\circ}\sim-45^{\circ}$ | 1.50 | 0.75 | 1.03 |
$35^{\circ}\sim 40^{\circ}$ | 2.40 | 1.41 | 3.85 |
$40^{\circ}\sim 45^{\circ}$ | 1.94 | 1.31 | 1.65 |

Note. — As the combination of the data for different cycles involving dividing
the data by the mean of the cycle, the 1st order central moments, i.e., the
means for the combined data should be 1 for all cases, hence they are not
listed here.

##  4 Discussion and conclusion

In the article, we quantitatively describe and analyze the nonuniformity of
poleward flux transport, which manifests as poleward surges on magnetic
butterfly diagrams, by a statistical approach we propose. We investigate the
magnetic field frequency distribution at middle to high latitudes. The 4th
central moment kurtosis describing the tailedness of the distribution is a
representation of the magnetic fields with large, extreme values of the
poleward flux transport, hence implying the occurrence of strong poleward
surges. For randomly and independently emerging ARs, the resulting poleward
flux transport should follow a Gaussian distribution as a result of the
central limit theorem. Therefore, a distribution with a larger kurtosis than
that of the Gaussian distribution indicates that the poleward flux transport
is significantly nonuniform, characterized by prominent surges probably
generated by ARs with nesting tendencies. The statistical method is examined
by a series of SFT simulations, confirming its ability to quantify the
nonuniformity of poleward flux transport, and to distinguish poleward flux
transport generated by ARs with and without nesting tendency.

We apply the statistical method to the surface magnetic field synoptic map
data of WSO, NSO, MWO, and HMI. Generally, the poleward flux transport during
the solar cycles 21-24 exhibits considerable nonuniformity. The kurtosis for
the data sets combining all cycles is higher than $2\times$ standard error.
For the kurtosis of an individual cycle, latitudinal, hemispheric, and data
source differences are observed. The kurtosis tends to decrease at higher
latitudes, as the poleward surges become more diffusive at higher latitudes.
The kurtosis values are subject to differences in different data sources, as
they tend to show different levels of details of the surges in their magnetic
butterfly diagrams. The characteristic of the cycles and hemispheres can be
shown from the results. Concerning the agreement of data sources, the northern
hemisphere of cycle 22 has significant poleward surges from the statistical
view. The southern hemisphere of cycle 24 has the highest kurtosis values
among the cycles, while the northern hemisphere does not have non-Gaussian
kurtosis of statistical significance. This can be attributed to the super
surge on the southern hemisphere during cycle 24 studied by Wang et al.
(2020).

The significant nonuniformity of poleward flux transport during solar cycles
21-24 indicates that the generation of the poleward surges is not a totally
random process. The ARs generating the poleward surges deviate from the
requirements of the central limit theorem. They may not be randomly,
independently generated, which agrees with the concept that the long-lasting
activity complexes introduce poleward surges (Petrie & Ettinger, 2017;
Mordvinov & Kitchatinov, 2019). This means that the poleward surges during
solar cycles we consider are not generated by chance. Instead, they are likely
to be generated as a result of the nesting tendency of ARs.

As we apply the method to obtain the results of an individual cycle, we meet
the limitations of the relatively large standard error of the kurtosis as a
result of limited data points during one solar cycle. Meanwhile, the
latitudinal variations and different data sources complicate the method and
the analysis of the results. Further utilization of the method needs more
suited latitudes to express the characteristics of surges, and more consistent
synoptic magnetic field data with good reliability. Especially, the
latitudinal variations of magnetic field distributions are related to the
diffusion of magnetic fluxes as they migrate poleward, as well as the
different emerging latitudes of the originating ARs of poleward flux
transport. The possible AR emergence at relatively higher latitudes is also
able to introduce non-Gaussianity to the distribution. These factors are
complicated components of the analysis that should be considered as we improve
the method.

The statistical view in this work is important in the analysis of poleward
surges, and we expect it to be further utilized in analyzing the solar large-
scale field development. The statistical view paves the way of identifying
properties of the large-scale field. This removes the ambiguity when we
discuss the relatively strong or weak magnetic phenomena of the large-scale
field. Meanwhile, there are other cases in the development of the large-scale
field where the magnetic field is contributed by a number of sources. For
example, the polar field at a certain time is contributed by a large number of
ARs emerged before. Hence, it is possible to further utilize the central limit
theorem in the analysis of the polar field evolution, and other perspectives
of the large scale field. However, we state that the current method utilizes 1
dimensional data that only focus on the nonuniformity, while other details of
the poleward flux transport are neglected. This is because that our focus in
this article concerns primarily the features on the magnetic butterfly
diagram, which does not include longitudinal information. It is expected that
more advanced statistical method will be utilized in analyzing the solar
large-scale field, especially the 2 dimensional methods, in order to obtain
more results. For example, the mixture of Kent distributions on 2 dimensional
spherical surface (Kent, 1982) can represent the distribution and evolution of
the surface magnetic features with more detailed results, and is expected to
be utilized in the future.

Wilcox Solar Observatory data was obtained via the web site
http://wso.stanford.edu. The Wilcox Solar Observatory is currently supported
by NASA. KPVT and SOLIS data are obtained by the NSO Integrated Synoptic
Program, managed by the National Solar Observatory, which is operated by the
Association of Universities for Research in Astronomy (AURA), Inc. under a
cooperative agreement with the National Science Foundation. The MWO data are
from the synoptic program at the 150-Foot Solar Tower of the Mt. Wilson
Observatory. The Mt. Wilson 150-Foot Solar Tower is operated by UCLA, with
funding from NASA, ONR and NSF, under agreement with the Mt. Wilson Institute.
The SDO/HMI data are courtesy of NASA and the SDO/HMI team. This research was
supported by the Strategic Priority Program of Chinese Academy of Sciences,
Grant No. XDB41000000, the National Natural Science Foundation of China
through grant Nos. 11873023, 11873059, and 12173005, Key Research Program of
Frontier Sciences of CAS through grant No. ZDBS-LY-SLH013, and Yunnan
Academician Workstation of Wang Jingxiu (No. 202005AF150025).

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# Roots of Gårding hyperbolic polynomials

Armin Rainer  Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-
Platz 1, A-1090 Wien, Austria<EMAIL_ADDRESS>
###### Abstract.

We explore the regularity of the roots of Gårding hyperbolic polynomials and
real stable polynomials. As an application we obtain new regularity results of
Sobolev type for the eigenvalues of Hermitian matrices and for the singular
values of arbitrary matrices. These results are optimal among all Sobolev
spaces.

###### Key words and phrases:

Hyperbolic polynomials, real stable polynomials, regularity of roots,
Hermitian matrices, regularity of eigenvalues and singular values

###### 2010 Mathematics Subject Classification:

26C05, 15A18, 32A08, 46E35,

The author was supported by the Austrian Science Fund (FWF), START Programme
Y963 and P 32905-N

##  1\. Introduction

A celebrated theorem of Bronshtein [9] (see also [26]) states that the roots
of a monic univariate _real-rooted_ polynomial of degree $d$ with coefficients
$C^{d-1,1}$-functions defined in some open set $U\subseteq\mathbb{R}^{m}$ can
be represented by functions that are locally Lipschitz in $U$. (A
$C^{k,\alpha}$-function is by definition a function that is $k$-times
continuously differentiable and whose partial derivatives of order $k$ satisfy
a local $\alpha$-Hölder condition.) That means, if $a_{j}\in C^{d-1,1}(U)$,
$j=1,\ldots,d$, and all roots of the monic polynomial

$P_{a(x)}(Z)=Z^{d}+\sum_{j=1}^{d}a_{j}(x)Z^{d-j}$

are real for all $x\in U$, then there exist functions $\lambda_{j}\in
C^{0,1}(U)$, $j=1,\ldots,d$, such that

$P_{a(x)}(Z)=\prod_{j=1}^{d}(Z-\lambda_{j}(x)),\quad x\in U.$ (1.1)

Let us call any $d$-tuple of functions
$\lambda=(\lambda_{1},\ldots,\lambda_{d})$ that satisfy (1.1) a _system of the
roots_ of $P_{a}$. One can show that _any_ continuous system
$\lambda=(\lambda_{1},\ldots,\lambda_{d})$ of the roots of $P_{a}$ with
$a_{j}\in C^{d-1,1}(U)$, $j=1,\ldots,d$, is locally Lipschitz. And there are
explicit bounds for the $C^{0,1}$-norm of the $\lambda_{j}$ in terms of the
coefficients $a_{j}$; see [26]. We shall refer to these results as
_Bronshtein’s theorem_.

That Bronshtein’s theorem is sharp is manifest already in dimension $m=1$. If
the coefficients $a=(a_{1},\ldots,a_{d})$ are of class $C^{d-1,1}$, then
generally there are no differentiable systems of the roots of $P_{a}$; e.g. no
function $\lambda$ with $\lambda^{2}=f$, where $f$ is the $C^{1,1}$-function
$f(t)=t^{2}\sin^{2}(\log|t|)$ for $t\neq 0$ and $f(0)=0$, is differentiable at
$0$. On the other hand, if $a$ is of class $C^{d}$ (resp. $C^{2d}$) and $m=1$,
then there exists a $C^{1}$ (resp. twice differentiable) system of the roots
of $P_{a}$; see [10] (and also [26]). However, there are non-negative
$C^{\infty}$-functions $f$ on $\mathbb{R}$ such that no function $\lambda$
with $\lambda^{2}=f$ is of class $C^{1,\alpha}$ for any $\alpha>0$; cf. [6].

There are various ways to generalize univariate real-rooted polynomials to
multivariate polynomials. In this paper we will consider _Gårding hyperbolic
polynomials_ and _real stable polynomials_. These classes of polynomials are
intimately related and have important applications in many different fields
including PDEs, combinatorics, optimization, functional analysis, probability
etc.

We will focus on the regularity of the roots of Gårding hyperbolic polynomials
and real stable polynomials. By combining Bronshtein’s theorem with the
observation that the roots of Gårding hyperbolic polynomials are _difference-
convex_ functions we obtain uniform regularity results for the roots of the
considered classes of real-rooted polynomials which in general are optimal
among Sobolev spaces, see Theorem 4.6.

As an application we obtain new regularity results for the eigenvalues of
Hermitian matrices and for the singular values of arbitrary matrices. In
particular, we will prove in Theorem 6.2 that any $C^{1,1}$-curve of Hermitian
$d\times d$ matrices admits a system of its eigenvalues that is locally of
Sobolev class $W^{2,1}$. This result is uniform, in the sense explained in
Theorem 6.2(3), and optimal among all Sobolev spaces $W^{k,p}$, see Remark
6.3. (A $W^{k,p}$-function is by definition a $p$-integrable function with
$p$-integrable weak partial derivatives up to order $k$.)

As a byproduct we obtain a new simple proof of the fact that a $C^{1}$-curve
of Hermitian matrices admits a $C^{1}$-system of its eigenvalues. This was
proved for symmetric matrices by [30] and for normal matrices by [28] by
different methods.

### Acknowledgment

This note arose from a discussion with Alexander Lytchak who introduced me to
difference-convex functions.

##  2\. Hyperbolic and real stable polynomials

We call a univariate polynomial $f(Z)\in\mathbb{C}[Z]$ _real-rooted_ if all
roots of $f$ are real; then also all coefficients are real and
$f(Z)\in\mathbb{R}[Z]$. There is the following multivariate generalization.

######  Definition 2.1.

A polynomial $f(Z_{1},\ldots,Z_{n})\in\mathbb{C}[Z_{1},\ldots,Z_{n}]$ is said
to be _stable_ if $f(z_{1},\ldots,z_{n})\neq 0$ for all
$(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}$ with $\mathrm{Im}(z_{j})>0$ for
$j=1,\ldots,n$. A stable polynomial with real coefficients is called a _real
stable_ polynomial.

A univariate polynomial with real coefficients is real stable if and only if
it is real-rooted. A polynomial
$f(Z_{1},\ldots,Z_{n})\in\mathbb{C}[Z_{1},\ldots,Z_{n}]$ is stable (resp. real
stable) if and only if for all $x\in\mathbb{R}^{n}$ and all
$v\in\mathbb{R}^{n}_{>0}$ the univariate polynomial $f(x+Tv)\in\mathbb{C}[T]$
is stable (resp. real stable). Real stable polynomials played a crucial role
in the recent proof of the Kadison–Singer conjecture [24].

A different but related multivariate generalization of real-rootedness is
Gårding hyperbolicity.

######  Definition 2.2.

A homogeneous polynomial
$f(Z_{1},\ldots,Z_{n})\in\mathbb{R}[Z_{1},\ldots,Z_{n}]$ is said to be
_Gårding hyperbolic_ with respect to a given vector $v\in\mathbb{R}^{n}$ if
$f(v)\neq 0$ and for all $x\in\mathbb{R}^{n}$ the univariate polynomial
$f(x-Tv)\in\mathbb{R}[T]$ is real-rooted.

This notion was introduced by Gårding [12] in the 1950s. He showed that $f$
being hyperbolic with respect to a direction $v$ is a necessary and sufficient
condition for local well-posedness of the Cauchy problem with principal symbol
$f$ and initial data on a hyperplane with normal vector $v$. Hyperbolic
polynomials have found many applications ever since, for instance, in Gurvits’
proof [14] of the van der Waerden conjecture.

######  Example 2.3.

The determinant on the real vector space of $d\times d$ Hermitian matrices is
Gårding hyperbolic with respect to the identity matrix $\mathbb{I}$.

The relation between real stable and Gårding hyperbolic polynomials is
captured in the following

######  Proposition 2.4 ([7, Proposition 1.1]).

Let $f\in\mathbb{R}[Z_{1},\ldots,Z_{n}]$ be of degree $d$ and let
$f_{H}\in\mathbb{R}[Z_{1},\ldots,Z_{n},W]$ be the unique homogeneous
polynomial of degree $d$ such that

$f_{H}(Z_{1},\ldots,Z_{n},1)=f(Z_{1},\ldots,Z_{n}).$

Then $f$ is real stable if and only if $f_{H}$ is Gårding hyperbolic with
respect to every vector $v\in\mathbb{R}^{n}_{>0}\times\\{0\\}$.

Interesting examples of real stable polynomials (and hence of hyperbolic
polynomials) are generated as follows (see [7, Proposition 1.12]): Let
$A_{1},\ldots,A_{n}$ be positive semidefinite $m\times m$ matrices and $B$ a
(complex) Hermitian $m\times m$ matrix. Then the polynomial

$f(Z_{1},\ldots,Z_{n})=\det\Big{(}\sum_{j=1}^{n}Z_{j}A_{j}+B\Big{)}$

is either real stable or identically zero. Conversely, any real stable
polynomial in two variables $Z_{1}$ and $Z_{2}$ can be written as
$\pm\det(Z_{1}A_{1}+Z_{2}A_{2}+B)$, where $A_{j}$ are positive semidefinite
and $B$ is symmetric [7, Theorem 1.13]. The latter result is based on (and
corresponds to) the Lax conjecture [21] for Gårding hyperbolic polynomials
which was recently established by [23] relying on work by [17], [34]; see also
Remark 6.4.

We refer to the survey article [5] for background on Gårding hyperbolic
polynomials and more ways to generate examples.

##  3\. Difference-convex functions

We shall work with the the class of difference-convex functions. This is a
subclass of the class of locally Lipschitz functions which arises as the
smallest vector space that contains all continuous convex functions on a given
set. We follow the survey article [4]; see also [18].

######  Definition 3.1.

Let $U\subseteq\mathbb{R}^{n}$ be a convex set. A function $f:U\to\mathbb{R}$
is said to be _difference-convex_ ($DC$) if it can be written as the
difference of two continuous convex functions on $U$. A function $f$ is called
_locally difference-convex_ if each point in the domain of $f$ has a convex
neighborhood on which $f$ is difference-convex.

Let us denote by $DC(U)$ the space of all difference-convex functions on $U$
and use the obvious notation $DC_{\operatorname{loc}}(U)$ for the local class.
We have $DC_{\operatorname{loc}}(U)\subseteq C^{0,1}(U)$, by the local
Lipschitz properties of locally bounded convex functions.

Next we collect some basic properties of difference-convex functions.

######  Lemma 3.2 ([4]).

Let $U\subseteq\mathbb{R}^{n}$ be an open convex set. Then:

  1. (1)

$DC(U)$ is an algebra.

  2. (2)

Each $DC$-function on $U$ is the difference of two non-negative convex
continuous functions on $U$.

  3. (3)

If $f\in DC(U)$ then positive and negative parts $f^{\pm}$ of $f$ as well as
$|f|$ belong to $DC(U)$.

For us the following properties will be crucial.

######  Lemma 3.3 ([15]).

We have:

  1. (1)

$f\in DC(\mathbb{R}^{n})$ if and only if $f\in
DC_{\operatorname{loc}}(\mathbb{R}^{n})$.

  2. (2)

Let $A\subseteq\mathbb{R}^{m}$ be convex and either open or closed. Let
$B\subseteq\mathbb{R}^{n}$ be convex and open. If $f:A\to B$ is $DC$ (that is
each component is $DC$) and $g:B\to\mathbb{R}$ is $DC$, then $g\circ f$ is
locally $DC$.

  3. (3)

Let $A\subseteq\mathbb{R}^{m}$ be convex and either open or closed. If a
$DC$-function $f$ is nowhere vanishing on $A$, then $1/f$ is $DC$ on $A$.

We shall also need

######  Lemma 3.4 ([33, Lemma 4.8]).

Let $U\subseteq\mathbb{R}^{n}$ be an open convex set and let $f_{j}\in DC(U)$,
$j=1,\ldots,d$. If $f:U\to\mathbb{R}$ is continuous and
$f(x)\in\\{f_{1}(x),\ldots,f_{d}(x)\\}$ for all $x\in U$, then $f\in DC(U)$.

Note that, if $U\subseteq\mathbb{R}^{n}$ is open, then $C^{1,1}(U)\subseteq
DC_{\operatorname{loc}}(U)$, cf. [32, Theorem 11] and [18, Section II]. In
particular, all real polynomials on $\mathbb{R}^{n}$ belong to
$DC(\mathbb{R}^{n})$; see also [4, Section 3.3] for a direct argument.

On the other hand the first order partial derivatives of a difference-convex
function $f:U\to\mathbb{R}$ have bounded variation, i.e., the weak second
order partial derivatives of $f$ are signed Radon measures. This follows from
Dudley’s result [11] that a Schwartz distribution is a convex function if and
only if its second derivative is a non-negative matrix-valued Radon measure.
In one dimension, a real-valued function $f$ on a compact interval is
difference-convex if and only if $f$ is absolutely continuous and $f^{\prime}$
has bounded variation.

##  4\. Roots of Gårding hyperbolic polynomials

Let $f(Z_{1},\ldots,Z_{n})\in\mathbb{R}[Z_{1},\ldots,Z_{n}]$ be a homogeneous
polynomial of degree $d$ which is Gårding hyperbolic with respect to a
direction $v\in\mathbb{R}^{n}$. We may factorize

$f(x+Tv)=f(v)\prod_{j=1}^{d}(T+\lambda_{j}^{\downarrow}(x)),$

where

$\lambda^{\downarrow}_{1}(x)\geq\ldots\geq\lambda^{\downarrow}_{d}(x)$

are the decreasingly ordered roots of $f(x-Tv)\in\mathbb{R}[T]$. We call

$\lambda^{\downarrow}=(\lambda^{\downarrow}_{1},\ldots,\lambda^{\downarrow}_{d}):\mathbb{R}^{n}\to\mathbb{R}^{d}$

the _characteristic map_ of $f$ with respect to $v$ and
$\lambda^{\downarrow}_{1},\ldots,\lambda^{\downarrow}_{d}$ (in no particular
order) the _characteristic roots_. The polynomial $f$ is recovered by
$f(x)=f(v)\prod_{j=1}^{d}\lambda_{j}^{\downarrow}(x)$. We see that being
Gårding hyperbolic with respect to a direction $v$ geometrically means that
any affine line with direction $v$ meets the real hypersurface
$\\{x\in\mathbb{R}^{n}:f(x)=0\\}$ in $d$ points (with multiplicities).

Note that for all $j=1,\ldots,d$, $r\in\mathbb{R}_{\geq 0}$, and
$s\in\mathbb{R}$ we have

$\displaystyle\begin{split}\lambda^{\downarrow}_{j}(rx+sv)&=r\lambda^{\downarrow}_{j}(x)+s,\quad\text{
and }\\\
\lambda^{\downarrow}_{j}(-x)&=-\lambda^{\downarrow}_{d+1-j}(x).\end{split}$
(4.1)

Indeed,
$f(rx+sv+Tv)=r^{d}f(x+r^{-1}(s+T)v)=r^{d}f(v)\prod_{j=1}^{d}(r^{-1}(s+T)+\lambda_{j}^{\downarrow}(x))=f(v)\prod_{j=1}^{d}(T+(s+r\lambda_{j}^{\downarrow}(x)))$,
by homogeneity. The case $r=0$ is even simpler: $f(Tv)=T^{d}f(v)$. The second
assertion can be seen analogously; the negative factor reverses the order of
the roots.

Gårding [13, Theorem 2] proved that $\lambda^{\downarrow}_{d}$ is concave
which, by (4.1), is equivalent to $\lambda^{\downarrow}_{1}$ being convex. In
view of (4.1) the largest root $\lambda^{\downarrow}_{1}$ is also positively
homogeneous and thus sublinear. Recall that a function $\varphi$ on
$\mathbb{R}^{n}$ is _sublinear_ if $\varphi(rx+sy)\leq
r\varphi(x)+s\varphi(y)$ for all $x,y\in\mathbb{R}^{n}$ and $r,s\geq 0$; it is
easy to see that a positively homogeneous function $\varphi$ is convex if and
only if it is sublinear. The connected component $C_{f}$ of the set
$\mathbb{R}^{n}\setminus\\{f=0\\}$ which contains $v$ is an open convex cone,
one has $C_{f}=\\{x\in\mathbb{R}^{n}:\lambda^{\downarrow}_{d}(x)>0\\}$, and
$f$ is Gårding hyperbolic with respect to each $w\in C_{f}$.

In [5, Theorem 3.1] a method for generating Gårding hyperbolic polynomials
from a given one was presented. Suppose that $f$ is a homogeneous polynomial
of degree $d$ which is Gårding hyperbolic with respect to $v$ and has
characteristic map $\lambda^{\downarrow}$. If $g$ is a homogeneous symmetric
polynomial of degree $e$ on $\mathbb{R}^{d}$ which is Gårding hyperbolic with
respect to $e_{1}+e_{2}+\cdots+e_{d}$ and has characteristic map
$\mu^{\downarrow}$, then $g\circ\lambda^{\downarrow}$ is Gårding hyperbolic of
degree $e$ with respect to $v$ and its characteristic map is
$\mu^{\downarrow}\circ\lambda^{\downarrow}$. (Here and below $e_{i}$ denotes
the $i$-th standard unit vector in $\mathbb{R}^{d}$.)

For instance, the homogeneous symmetric polynomial

$g_{k}(Y_{1},\ldots,Y_{d}):=\prod_{\begin{subarray}{c}I\subseteq\\{1,\ldots,d\\}\\\
|I|=k\end{subarray}}\sum_{i\in I}Y_{i}.$

of degree $\binom{d}{k}$ is Gårding hyperbolic with respect to
$e_{1}+e_{2}+\cdots+e_{d}$ and has the characteristic roots
$\mu_{I}(Y)=\frac{1}{k}\sum_{i\in I}Y_{i}$, where $I$ ranges over the subsets
of $\\{1,\ldots,d\\}$ with $k$ elements. On the set $\\{Y_{1}\geq
Y_{2}\geq\cdots\geq Y_{d}\\}$ we have $\mu_{I}(Y)\geq\mu_{J}(Y)$ if and only
if $\sum_{i\in I}i\leq\sum_{j\in J}j$, in particular, $\mu_{\\{1,\ldots,k\\}}$
is the largest characteristic root. Then $g_{k}\circ\lambda^{\downarrow}$ is
Gårding hyperbolic with respect to $v$ and has the largest characteristic root
$\mu_{\\{1,\ldots,k\\}}\circ\lambda^{\downarrow}=\frac{1}{k}\sum_{i=1}^{k}\lambda_{i}^{\downarrow}$.
From Gårding’s result we may conclude that, for all $k=1,\ldots,d$, the sum of
the $k$ largest roots

$\sigma_{k}:=\sum_{i=1}^{k}\lambda_{i}^{\downarrow}$

is a sublinear function on $\mathbb{R}^{n}$. We immediately obtain

######  Proposition 4.1.

Let $f(Z_{1},\ldots,Z_{n})\in\mathbb{R}[Z_{1},\ldots,Z_{n}]$ be a homogeneous
polynomial of degree $d$ which is Gårding hyperbolic with respect to a
direction $v\in\mathbb{R}^{n}$. The characteristic map
$\lambda^{\downarrow}:\mathbb{R}^{n}\to\mathbb{R}^{d}$ is globally Lipschitz
and difference-convex on $\mathbb{R}^{n}$.

###### Proof.

Any finite sublinear function on $\mathbb{R}^{n}$ is globally Lipschitz; this
follows from [31, Corollary 10.5.1]. By the above discussion, the functions
$\sigma_{k}$, $k=1,\ldots,d$, are convex and globally Lipschitz. Thus each
$\lambda^{\downarrow}_{k}=\sigma_{k}-\sigma_{k-1}$ is difference-convex and
globally Lipschitz. ∎

Combined with Lemma 3.4 this leads to

######  Corollary 4.2.

In the setting of Proposition 4.1 every continuous function
$\lambda:\mathbb{R}^{n}\to\mathbb{R}$ satisfying $f(x-\lambda(x)v)=0$ for all
$x\in\mathbb{R}^{n}$ is globally Lipschitz and difference-convex on
$\mathbb{R}^{n}$.

###### Proof.

That $\lambda\in DC(\mathbb{R}^{n})$ is an immediate consequence of Lemma 3.4
and Proposition 4.1. For arbitrary $x,y\in\mathbb{R}^{n}$ consider the set
$[x,y]:=\\{(1-t)x+ty:t\in[0,1]\\}$ equipped with the linear order induced by
$[0,1]$. By assumption, there is a finite sequence of points
$x=:x_{0}<x_{1}<\cdots<x_{k}:=y$ with $k\leq d$ such that for each
$i=1,\ldots,k$ there exists $j_{i}$ such that
$\lambda(x_{i-1})=\lambda_{j_{i}}^{\downarrow}(x_{i-1})$ and
$\lambda(x_{i})=\lambda_{j_{i}}^{\downarrow}(x_{i})$. Then

$\frac{|\lambda(y)-\lambda(x)|}{|y-x|}\leq\sum_{i=1}^{k}\frac{|\lambda(x_{i})-\lambda(x_{i-1})|}{|x_{i}-x_{i-1}|}\cdot\frac{|x_{i}-x_{i-1}|}{|y-x|}\leq\sum_{i=1}^{k}\frac{|\lambda_{j_{i}}^{\downarrow}(x_{i})-\lambda_{j_{i}}^{\downarrow}(x_{i-1})|}{|x_{i}-x_{i-1}|}$

which implies that $\lambda$ is globally Lipschitz, by Proposition 4.1. ∎

Let $\operatorname{Hyp}_{n}^{d}(v)$ denote the space of all homogeneous
polynomials $f(Z_{1},\ldots,Z_{n})\in\mathbb{R}[Z_{1},\ldots,Z_{n}]$ of degree
$d$ which are Gårding hyperbolic with respect to a fixed vector
$v\in\mathbb{R}^{n}$. By identifying $f\in\operatorname{Hyp}_{n}^{d}(v)$ with
the sequence of its coefficients (with respect to a fixed order of the
monomials) we view $\operatorname{Hyp}_{n}^{d}(v)$ as a subset of
$\mathbb{R}^{N}$ where $N=\binom{n+d-1}{d}$. Then the subset
$\mathring{\operatorname{Hyp}}_{n}^{d}(v)$, consisting of all those
$f\in\operatorname{Hyp}_{n}^{d}(v)$ for which the roots of $f(x-Tv)$ are
simple for every $x$ not proportional to $v$, is open. And
$\operatorname{Hyp}_{n}^{d}(v)$ is the part of the closure of
$\mathring{\operatorname{Hyp}}_{n}^{d}(v)$ where $f(v)\neq 0$. See [25] in
which also the connectivity properties of these spaces are discussed.

By a _$C^{k,\alpha}$ -mapping_
$f:\mathbb{R}^{m}\to\operatorname{Hyp}_{n}^{d}(v)$ we mean a mapping with
values in $\operatorname{Hyp}_{n}^{d}(v)$ which is of class $C^{k,\alpha}$
when considered as mapping to $\mathbb{R}^{N}$. For each $y\in\mathbb{R}^{m}$
we have the characteristic map $\lambda^{\downarrow}(y)(\cdot)$ of $f(y)$. By
slight abuse of notation we write

$\lambda^{\downarrow}:\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}^{d}$

and call it the _characteristic map_ of $f$ with respect to $v$. Its
regularity properties are stated in the following

######  Proposition 4.3.

If $f:\mathbb{R}^{m}\to\operatorname{Hyp}_{n}^{d}(v)$ is a mapping of class
$C^{d-1,1}$, then its characteristic map
$\lambda^{\downarrow}:\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}^{d}$ is
locally Lipschitz.

###### Proof.

Bronshtein’s theorem [9] (see also [26]) implies that the roots
$\lambda^{\downarrow}_{j}(y,x)$, $j=1,\ldots,d$ of the univariate real-rooted
polynomial

$\mathbb{R}^{m}\times\mathbb{R}^{n}\ni(y,x)\mapsto f(y)(x-Tv)\in\mathbb{R}[T]$
(4.2)

are locally Lipschitz in $(y,x)$. ∎

Note that in Proposition 4.3 we need the assumption that the coefficients of
$f$ are of class $C^{d-1,1}$, since the assumptions of Bronshtein’s theorem
are sharp.

By Proposition 4.1 the behavior of the characteristic map
$\lambda^{\downarrow}:\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}^{d}$,
$(y,x)\mapsto\lambda^{\downarrow}(y,x)$ is essentially better in the $x$
variables. This will be further investigated in the next theorem, that is
Theorem 4.6. To this end we need the following

######  Lemma 4.4.

Let $\lambda=(\lambda_{1},\ldots,\lambda_{d}):\mathbb{R}\to\mathbb{R}^{d}$ be
of class $C^{1}$ and assume that the coefficients of the polynomial
$P(t)(Z)=\prod_{j=1}^{d}(Z-\lambda_{j}(t))$ are of class
$W^{2,1}_{\operatorname{loc}}$. Then each derivative $\lambda_{j}^{\prime}$,
$j=1,\ldots,d$, has the Luzin (N) property (i.e. sets of Lebesgue measure zero
are mapped to sets of Lebesgue measure zero).

###### Proof.

Note that it is enough to show the Luzin (N) property on bounded intervals,
since $\mathbb{R}$ can be covered by countably many of them. We proceed by
induction on $d$. The case $d=1$ is clear, since in this case the derivative
of the only root is locally absolutely continuous, by assumption. Let $d>1$.

Let $A$ denote the closed set of all $t\in\mathbb{R}$ such that
$\lambda_{1}(t)=\cdots=\lambda_{d}(t)$. On an open neighborhood $I$ of each
point $t_{0}\in A^{c}:=\mathbb{R}\setminus A$ we have a splitting
$P(t)=P_{1}(t)P_{2}(t)$ into polynomials with positive degree and
$W^{2,1}_{\operatorname{loc}}$ coefficients, by the inverse function theorem
(cf. [26, Lemma 3.2]); here we use that the superposition with a real analytic
function preserves the class $W^{2,1}_{\operatorname{loc}}$, cf. [8, Theorem
2]. On $I$ we have a partition of $\\{\lambda_{1},\ldots,\lambda_{d}\\}$ into
the roots of $P_{1}$ and the roots of $P_{2}$. By induction hypothesis, for
each $j$ the restriction $\lambda_{j}^{\prime}|_{I}$ has the Luzin (N)
property. Since $A^{c}$ can be covered by countably many open neighborhoods of
type $I$, we see that each $\lambda_{j}^{\prime}|_{A^{c}}$ has the Luzin (N)
property.

Let $A^{\prime}$ denote the set of accumulation points of $A$. Let $a_{1}(t)$
denote the coefficient of $Z^{d-1}$ in $P(t)$. For $t\in A$ we have

$a_{1}(t)=-\lambda_{1}(t)-\cdots-\lambda_{d}(t)=-d\cdot\lambda_{j}(t),\quad\text{
for all }j.$

And hence for $t\in A^{\prime}$ and all $j$ we have

$\lambda_{j}^{\prime}(t)=\lim_{A\ni t_{n}\to
t}\frac{\lambda_{j}(t_{n})-\lambda_{j}(t)}{t_{n}-t}=-\frac{1}{d}\lim_{A\ni
t_{n}\to
t}\frac{a_{1}(t_{n})-a_{1}(t)}{t_{n}-t}=-\frac{a_{1}^{\prime}(t)}{d}.$

By assumption $a_{1}^{\prime}$ is absolutely continuous and hence has the
Luzin (N) property. If $E\subseteq\mathbb{R}$ is a set of Lebesgue measure
zero, then also

$\displaystyle\lambda_{j}^{\prime}(E)$
$\displaystyle=\lambda_{j}^{\prime}(E\cap A^{c})\cup\lambda_{j}^{\prime}(E\cap
A^{\prime})\cup\lambda_{j}^{\prime}(E\cap A\setminus A^{\prime})$
$\displaystyle=\lambda_{j}^{\prime}(E\cap
A^{c})\cup(-\tfrac{1}{d})a_{1}^{\prime}(E\cap
A^{\prime})\cup\lambda_{j}^{\prime}(E\cap A\setminus A^{\prime})$

has Lebesgue measure zero, noting that $A\setminus A^{\prime}$ is discrete and
thus countable. ∎

Let $f\in\operatorname{Hyp}_{n}^{d}(v)$ have characteristic map
$\lambda^{\downarrow}:\mathbb{R}^{n}\to\mathbb{R}^{d}$. Let
$x:\mathbb{R}\to\mathbb{R}^{n}$ be a curve. A mapping
$\lambda=(\lambda_{1},\ldots,\lambda_{d}):\mathbb{R}\to\mathbb{R}^{d}$ is
called a _system of the roots of $f$ along $x$_ if $\lambda(t)$ and
$\lambda^{\downarrow}(x(t))$ coincide as unordered $d$-tuples for all $t$.

######  Lemma 4.5.

Let $f\in\operatorname{Hyp}^{d}_{n}(v)$ and let
$x:\mathbb{R}\to\mathbb{R}^{n}$ be a $C^{1}$-curve. Then there exists a
differentiable system $\lambda=(\lambda_{1},\ldots,\lambda_{d})$ of the roots
of $f$ along $x$.

###### Proof.

Let $\lambda_{0}$ be a $p$-fold root of $f(x(t_{0})-Tv)$. We claim that the
polynomial $f(x(t)-Tv)$ has $p$ roots of the form
$\lambda_{0}+(t-t_{0})\mu_{j}(t)$, where the $\mu_{j}$ are functions which are
defined near $t_{0}$ and continuous at $t_{0}$.

We may assume without loss of generality that $t_{0}=0$. In view of (4.1) we
may also assume that $\lambda_{0}=0$. We consider the localization $f_{x(0)}$
of $f$ at $x(0)$ defined by

$t^{d}f(t^{-1}x(0)+\xi)=f(x(0)+t\xi)=t^{p}f_{x(0)}(\xi)+O(t^{p+1}),$ (4.3)

where $f_{x(0)}(\xi)$ is the first coefficient in the above expansion which
does not vanish identically. Since

$f(x(0)+tv)=f(v)\prod_{j=1}^{d}(t+\lambda^{\downarrow}_{j}(x(0)))=t^{p}f(v)\prod_{\lambda^{\downarrow}_{j}(x(0))\neq\lambda_{0}}(t+\lambda^{\downarrow}_{j}(x(0))),$

the exponent $p$ in (4.3) is precisely the multiplicity of $\lambda_{0}$. The
homogeneous polynomial $f_{x(0)}$ has degree $p$ and is Gårding hyperbolic
with respect to $v$; see [3, Lemma 3.42].

Since $x$ is of class $C^{1}$, we have $x(t)=x(0)+t\tilde{x}(t)$ for a
continuous function $\tilde{x}$. We are interested in the solutions of the
equation

$f(x(0)+t\tilde{x}(t)-tYv)=0$

By (4.3) we may equivalently study the equation

$f_{x(0)}(\tilde{x}(t)-Yv)+O(t)=0.$

This is a polynomial equation in the unknown $Y$ with coefficients which
depend continuously on $t$ in a neighborhood of $0$. It follows that there
exist $p$ solutions $\mu_{1}(t),\ldots,\mu_{p}(t)$ (since $p$ is the degree of
$f_{x(0)}$) for $t$ near $0$ which are continuous at $t=0$. This implies the
claim.

Let us indicate how to construct a global differentiable system
$\lambda=(\lambda_{1},\ldots,\lambda_{d})$ of the roots of $f$ along $x$ from
the local data provided by the claim; we follow the argument in [1, 4.3]. We
start with the continuous system $\nu_{1}\geq\nu_{2}\geq\cdots\geq\nu_{d}$,
where $\nu_{j}:=\lambda^{\downarrow}_{j}\circ x$. Fix $i\neq j$ and
$t_{0}\in\mathbb{R}$. If $\nu_{i}(t_{0})=\nu_{j}(t_{0})$, then, by the claim,
$\nu_{i}(t)-\nu_{j}(t)=(t-t_{0})\nu_{ij}(t)$ near $t_{0}$, where $\nu_{ij}$ is
a function which is continuous at $t_{0}$. Let $E_{ij}$ be the closed set of
$t_{0}\in\mathbb{R}$ such that $\nu_{i}(t_{0})=\nu_{j}(t_{0})$ and
$\nu_{ij}(t_{0})=0$. At the points of $E_{ij}$ the roots $\nu_{i}$ and
$\nu_{j}$ are differentiable and the derivatives $\nu_{i}^{\prime}$ and
$\nu_{j}^{\prime}$ coincide. Let us define

$\lambda_{k}(t):=\nu_{\sigma_{t}(k)}(t),\quad k=1,\ldots,d,$

where $\sigma_{t}$ is the permutation

$\sigma_{t}=(1\;2)^{\epsilon_{12}(t)}\cdots(1\;d)^{\epsilon_{1d}(t)}(2\;3)^{\epsilon_{23}(t)}\cdots(2\;d)^{\epsilon_{2d}(t)}\cdots(d-1\;d)^{\epsilon_{d-1,d}(t)}$

and $\epsilon_{ij}(t)\in\\{0,1\\}$ are determined as follows. In each
connected component $I$ of $\mathbb{R}\setminus E_{ij}$ choose a point $t_{I}$
and set $\epsilon_{ij}(t_{I})=0$. Going left and right from $t_{I}$ change
$\epsilon_{ij}$ in each point $t_{0}\in I$, where
$\nu_{i}(t_{0})=\nu_{j}(t_{0})$ and $\nu_{ij}(t_{0})\neq 0$ (including $t_{I}$
if it is such a point). These points accumulate only in $E_{ij}$, where any
choice for $\epsilon_{ij}$ is good. It follows that
$\lambda=(\lambda_{1},\ldots,\lambda_{d})$ is a global differentiable system
of the roots of $f$ along $x$. ∎

We are ready to state and prove the theorem.

######  Theorem 4.6.

Let $f\in\operatorname{Hyp}_{n}^{d}(v)$. Let $x\in
DC(\mathbb{R},\mathbb{R}^{n})\cap
W^{2,1}_{\operatorname{loc}}(\mathbb{R},\mathbb{R}^{n})$; this is, for
instance, fulfilled if $x\in C^{1,1}(\mathbb{R},\mathbb{R}^{n})$.

  1. (1)

Any differentiable system $\lambda=(\lambda_{1},\ldots,\lambda_{d})$ of the
roots of $f$ along $x$ is actually of class

$\lambda\in C^{1}(\mathbb{R},\mathbb{R}^{d})\cap
DC(\mathbb{R},\mathbb{R}^{d})\cap
W^{2,1}_{\operatorname{loc}}(\mathbb{R},\mathbb{R}^{d}).$ (4.4)

  2. (2)

If $x:\mathbb{R}\to\mathbb{R}^{n}$ is of class $C^{1}$, then there exists a
differentiable (thus $C^{1}$) system
$\lambda=(\lambda_{1},\ldots,\lambda_{d})$ of the roots of $f$ along $x$.

  3. (3)

The result is uniform in the following sense: Let $I\subseteq\mathbb{R}$ be a
bounded open interval. Let $U$ be an open neighborhood of the closure of
$I^{1+k}$ in $\mathbb{R}^{1+k}$. Suppose that $x:U\to\mathbb{R}^{n}$ is such
that

     * •

$x$ is locally DC on $U$,

     * •

$x(\cdot,r)\in C^{1}(\overline{I},\mathbb{R}^{n})\cap
W^{2,1}(I,\mathbb{R}^{n})$ for all $r\in\overline{I}^{k}$.

Assume that, for each $r\in\overline{I}^{k}$, a $C^{1}$-system
$\lambda(\cdot,r)$ of the roots of $f$ along $x(\cdot,r)$ is fixed. Then the
family $\lambda(\cdot,r)$, for $r\in\overline{I}^{k}$, is bounded in
$C^{1}(\overline{I},\mathbb{R}^{d})$ and there is a non-negative
$L^{1}$-function $m:I^{k}\to\mathbb{R}_{\geq 0}$ such that

$\|\lambda(\cdot,r)\|_{W^{2,1}(I,\mathbb{R}^{d})}\leq m(r),\quad\text{ for
a.e.\ }r\in I^{k}.$ (4.5)

###### Proof.

(1) Let $\lambda=(\lambda_{1},\ldots,\lambda_{d})$ be a differentiable system
of the roots of $f$ along $x$. By Lemma 3.3, $\lambda^{\downarrow}\circ x$ is
$DC$ on $\mathbb{R}$. Then $\lambda$ is $DC$ on $\mathbb{R}$, by Corollary
4.2. A differentiable $DC$-function $\varphi$ on $\mathbb{R}$ is of class
$C^{1}$ and the difference of two convex $C^{1}$-functions (see [18, Theorem
2.1]); indeed, by Darboux’s theorem, $\varphi^{\prime}$ has the intermediate
value property and so $\varphi^{\prime}$ having (locally) bounded variation
implies that $\varphi^{\prime}$ is continuous. It follows that $\lambda$ is of
class $C^{1}$.

In order to conclude that $\lambda\in
W^{2,1}_{\operatorname{loc}}(\mathbb{R},\mathbb{R}^{n})$ it suffices to show
that each $\lambda_{j}^{\prime}$ has the Luzin (N) property which follows from
Lemma 4.4; in fact a continuous function of pointwise bounded variation with
the Luzin (N) property is absolutely continuous, cf. [22, Corollary 3.27].

(2) This is Lemma 4.5.

(3) The Lipschitz constant of $(\lambda^{\downarrow}\circ x)(\cdot,r)$ on
$\overline{I}$ is uniformly bounded in $r\in\overline{I}^{k}$ by the chain
rule, since $\lambda^{\downarrow}$ is Lipschitz, by Proposition 4.1. By an
argument similar to the one in the proof of Corollary 4.2 we may infer the
boundedness of the Lipschitz constant of $\lambda(\cdot,r)$ on $\overline{I}$.
It follows that the family $\lambda(\cdot,r)$, $r\in\overline{I}^{k}$, is
bounded in $C^{1}(\overline{I},\mathbb{R}^{d})$.

By Lemma 3.3, $\lambda^{\downarrow}\circ x$ is locally $DC$ on $U$. In
particular, the directional variation
$V_{e_{1}}(\partial_{1}(\lambda^{\downarrow}\circ x),I^{1+k})$ of
$\partial_{1}(\lambda^{\downarrow}\circ x)$ on $I^{1+k}$ is finite, where
$e_{1}$ denotes the first standard unit vector. We have, see e.g [2, (3.100)],

$V_{e_{1}}(\partial_{1}(\lambda^{\downarrow}\circ
x),I^{1+k})=\int_{I^{k}}V(\partial_{1}(\lambda^{\downarrow}\circ
x)(\cdot,r),I)\,dr,$

where $V(\partial_{1}(\lambda^{\downarrow}\circ x)(\cdot,r),I)$ is the
variation of $\partial_{1}(\lambda^{\downarrow}\circ x)(\cdot,r)$ on $I$. The
variation of $\partial_{1}(\lambda^{\downarrow}\circ x)(\cdot,r)$ on $I$
dominates the variation of $\partial_{1}\lambda(\cdot,r)$ on $I$ (up to a
constant which depends only on $d$), since for each $j=1,\ldots,d$ and a.e.
$t$ we have

$\partial_{1}\lambda_{j}(t,r)\in\\{\partial_{1}(\lambda_{1}^{\downarrow}\circ
x)(t,r),\ldots,\partial_{1}(\lambda_{d}^{\downarrow}\circ x)(t,r)\\},$

and $\partial_{1}\lambda_{j}(t,r)$ is continuous in $t$. Since
$\partial_{1}\lambda_{j}(\cdot,r)$ is absolutely continuous on $I$, by (1),
its $W^{1,1}$-norm coincides with its $BV$-norm on $I$. Together with the
boundedness in $C^{1}(\overline{I},\mathbb{R}^{d})$ the assertion (4.5)
follows easily. ∎

######  Remark 4.7.

If the system $\lambda=(\lambda_{1},\ldots,\lambda_{d})$ of the roots of $f$
along $x$ is not differentiable at $t_{0}$, then $\lambda$ is not of class
$W^{2,1}$ in any neighborhood of $t_{0}$.

In several variables the characteristic roots do not admit differentiable
rearrangements. But for real analytic families of Gårding hyperbolic
polynomials the characteristic roots can be chosen real analytically locally
after blowing up the parameter space.

######  Proposition 4.8.

If $f:\mathbb{R}^{m}\to\operatorname{Hyp}_{n}^{d}(v)$ is real analytic and
$x:\mathbb{R}^{\ell}\to\mathbb{R}^{n}$ is real analytic, then there exists a
locally finite composite of blowing-ups with smooth centers
$\tau=(\tau_{1},\tau_{2}):Y\to\mathbb{R}^{m}\times\mathbb{R}^{\ell}$ such that
for each $y_{0}\in Y$ there is a neighborhood $U$ and a real analytic system
of the roots of the univariate polynomial

$U\ni y\mapsto f(\tau_{1}(y))(x(\tau_{2}(y))-Tv)\in\mathbb{R}[T].$

If $f:\mathbb{R}\to\operatorname{Hyp}_{n}^{d}(v)$ is real analytic and
$x:\mathbb{R}\to\mathbb{R}^{n}$ is a real analytic curve, then there exists a
global real analytic system of the roots of

$\mathbb{R}\ni s\mapsto f(s)(x(s)-Tv)\in\mathbb{R}[T].$

###### Proof.

The statement follows from [20, Theorem 5.8] applied to the real-rooted
polynomial (4.2). The (one-dimensional) supplement follows from Rellich’s
theorem [29]; see also [1, Theorem 5.1]. ∎

######  Remark 4.9.

Analogous results follow for suitable quasianalytic classes thanks to a
corresponding theorem for real-rooted polynomials, see [27].

######  Remark 4.10.

It is shown in [16, Theorem 4.2] that for any
$f\in\operatorname{Hyp}_{n}^{d}(v)$, $x_{0}\in\mathbb{R}^{n}$, and $w\in
C_{f}$ there is a system
$\lambda=(\lambda_{1},\ldots,\lambda_{d}):\mathbb{R}\to\mathbb{R}^{d}$ of the
roots of $f$ along $t\mapsto x_{0}+tw$ which is real analytic and such that
each $\lambda_{j}:\mathbb{R}\to\mathbb{R}$ is strictly increasing and
surjective. The inverses of the $\lambda_{j}$ form a system of the
characteristic roots of $f\in\operatorname{Hyp}_{n}^{d}(w)$ along $t\mapsto
x_{0}+tv$.

##  5\. Roots of real stable polynomials

The regularity results for Gårding hyperbolic polynomials of the previous
section immediately give regularity results for real stable polynomials thanks
to the connection presented in Proposition 2.4. Let us denote by
$\operatorname{RStab}^{d}_{n}$ the set of real stable polynomials of degree
$d$ in $n$ variables. Then Proposition 2.4 defines a bijection

$\operatorname{RStab}^{d}_{n}\to\bigcap_{v\in\mathbb{R}^{n}_{>0}\times\\{0\\}}\operatorname{Hyp}^{d}_{n}(v),\quad
f\mapsto f_{H}.$

Suppose that $f\in\operatorname{RStab}^{d}_{n}$. For all
$v\in\mathbb{R}^{n}_{>0}\times\\{0\\}$ we may consider the characteristic map
$\lambda^{\downarrow}_{v}:\mathbb{R}^{n+1}\to\mathbb{R}^{d}$ of $f_{H}$ with
respect to $v$. Then the mapping $\lambda^{\downarrow}_{v}|_{\\{x_{n+1}=1\\}}$
is globally Lipschitz and difference-convex and its components represent the
roots of $f(x^{\prime}-Tv^{\prime})\in\mathbb{R}[T]$, where
$x^{\prime}=(x_{1},\ldots,x_{n})$ and $v^{\prime}=(v_{1},\ldots,v_{n})$. It is
then straightforward to transfer the regularity results for Gårding hyperbolic
polynomials of the previous section.

Finally, note that we can also vary the direction $v$: if
$f:\mathbb{R}^{m}\to\operatorname{RStab}^{d}_{n}$ _and_
$v:\mathbb{R}^{\ell}\to\mathbb{R}^{n}_{>0}$ are $C^{d-1,1}$-mappings, then the
roots of the univariate real-rooted polynomial

$\mathbb{R}^{m}\times\mathbb{R}^{\ell}\times\mathbb{R}^{n}\ni(s,t,x)\mapsto
f(s)(x-Tv(t))\in\mathbb{R}[T]$

are locally Lipschitz in $(s,t,x)$, by Bronshtein’s theorem [9] (see also
[26]).

##  6\. Eigenvalues of Hermitian matrices

Let $\operatorname{Herm}(d)$ denote the real vector space of complex $d\times
d$ Hermitian matrices. The determinant $\det$ is Gårding hyperbolic with
respect to the identity matrix $\mathbb{I}\in\operatorname{Herm}(d)$. The
characteristic map $\lambda^{\downarrow}$ of $\det$ with respect to
$\mathbb{I}$ assigns to a Hermitian matrix $A\in\operatorname{Herm}(d)$ its
eigenvalues in decreasing order

$\lambda^{\downarrow}_{1}(A)\geq\ldots\geq\lambda^{\downarrow}_{d}(A).$

As a direct consequence of Proposition 4.1 and the discussion preceding it we
get

######  Corollary 6.1.

The characteristic map
$\lambda^{\downarrow}:\operatorname{Herm}(d)\to\mathbb{R}^{d}$ is globally
Lipschitz and difference-convex on $\operatorname{Herm}(d)$. The sum
$\sum_{i=1}^{k}\lambda^{\downarrow}_{i}$, for $k=1,\ldots,d$, of the $k$
largest eigenvalues is sublinear.

The following result is a special case of Theorem 4.6.

######  Theorem 6.2.

Let $A:\mathbb{R}\to\operatorname{Herm}(d)$ be a curve of $d\times d$
Hermitian matrices of class $DC\cap W^{2,1}_{\operatorname{loc}}$ on
$\mathbb{R}$.

  1. (1)

Any differentiable system $\lambda=(\lambda_{1},\ldots,\lambda_{d})$ of the
eigenvalues of $A$ is actually of class

$\lambda\in C^{1}(\mathbb{R},\mathbb{R}^{d})\cap
DC(\mathbb{R},\mathbb{R}^{d})\cap
W^{2,1}_{\operatorname{loc}}(\mathbb{R},\mathbb{R}^{d}).$

  2. (2)

If $A$ additionally is of class $C^{1}$, then there exists a differentiable
(thus $C^{1}$) system $\lambda=(\lambda_{1},\ldots,\lambda_{d})$ of the
eigenvalues of $A$.

  3. (3)

The result is uniform in the following sense: Let $I\subseteq\mathbb{R}$ be a
bounded open interval. Let $U$ be an open neighborhood of the closure of
$I^{1+k}$ in $\mathbb{R}^{1+k}$. Suppose that $A:U\to\operatorname{Herm}(d)$
is such that

     * •

$A$ is locally DC on $U$,

     * •

$A(\cdot,r)$ is of class $C^{1}\cap W^{2,1}$ on $\overline{I}$ for all
$r\in\overline{I}^{k}$.

Assume that, for each $r\in\overline{I}^{k}$, a $C^{1}$-system
$\lambda(\cdot,r)$ of the eigenvalues of $A(\cdot,r)$ is fixed. Then the
family $\lambda(\cdot,r)$, for $r\in\overline{I}^{k}$, is bounded in
$C^{1}(\overline{I},\mathbb{R}^{d})$ and there is a non-negative
$L^{1}$-function $m:I^{k}\to\mathbb{R}_{\geq 0}$ such that

$\|\lambda(\cdot,r)\|_{W^{2,1}(I,\mathbb{R}^{d})}\leq m(r),\quad\text{ for
a.e.\ }r\in I^{k}.$

Note that the assumptions on $A$ are in particular satisfied if $A$ is of
class $C^{1,1}$.

######  Remark 6.3.

The conclusion of this theorem is best-possible among all Sobolev spaces
$W^{k,p}$. Indeed, by the Sobolev inequality, $W^{k,p}$-regularity with
$k+p>2$ would imply $C^{1,\alpha}$-regularity with some $\alpha>0$,
contradicting the counter-example of [19]: there is a $C^{\infty}$-curve of
symmetric $2\times 2$ matrices the eigenvalues of which do not admit a
$C^{1,\alpha}$-system for any $\alpha>0$.

######  Remark 6.4.

Statement (2) in Theorem 6.2 is due to Rellich [30] in the case of symmetric
matrices, and it was proved for normal matrices in [28]. Lemma 4.5 gives an
independent proof for Hermitian matrices.

Conversely, (2) in Theorem 6.2 implies Lemma 4.5 if $n\leq 3$: The Lax
conjecture (posed 1958 in [21]) which by now is a theorem (as [23] discovered
the conjecture follows from a theorem of [17], [34]) states that a homogeneous
polynomial $f$ on $\mathbb{R}^{3}$ is Gårding hyperbolic of degree $d$ with
respect to the direction $e_{1}$ with $f(e_{1})=1$ if and only if there exist
real symmetric $d\times d$ matrices $A$ and $B$ such that

$f(x,y,z)=\det(x\mathbb{I}+yA+zB).$ (6.1)

So, for $n\leq 3$, Theorem 4.6 follows from Theorem 6.2. A representation of
type (6.1) is in general not possible for Gårding hyperbolic polynomials on
$\mathbb{R}^{n}$ with $n\geq 4$. Indeed, the dimension of
$\operatorname{Hyp}^{d}_{n}(v)$ is $\binom{n+d-1}{d}$ while the set of
polynomials in $\mathbb{R}[X_{1},\ldots,X_{n}]$ of the form

$\det(X_{1}\mathbb{I}+X_{2}A_{2}+\cdots+X_{n}A_{n}),$ (6.2)

where $A_{i}$ are real symmetric $d\times d$ matrices, is at most
$(n-1)\cdot\binom{d+1}{2}$.

A particular homogeneous polynomial of degree $2$ which is Gårding hyperbolic
with respect to $e_{1}$ but cannot be represented in the form (6.2) is the
Lorentzian polynomial $f(X)=X_{1}^{2}-X_{2}^{2}-\cdots-X_{n}^{2}$ for $n\geq
4$. Indeed, if $x_{1}=0$ and $x_{j}\neq 0$ for some $j\in{2,\ldots,n}$ then
$f(x)<0$. But for any choice of real symmetric matrices $A_{i}$ we may find
$x\in\mathbb{R}^{n}\setminus\\{0\\}$ with $x_{1}=0$ such that the first row of
$x_{1}\mathbb{I}+x_{2}A_{2}+\cdots+x_{n}A_{n}$ is zero. Cf. [23, p.2498].

##  7\. Singular values

Let $A$ be any $m\times n$ matrix with complex entries and let

$\sigma^{\downarrow}_{1}(A)\geq\sigma^{\downarrow}_{2}(A)\geq\cdots\geq\sigma^{\downarrow}_{n}(A)\geq
0$

be the singular values of $A$ in decreasing order, i.e., the non-negative
square roots of the eigenvalues of $A^{*}A$. If $\operatorname{rank}A=\ell$,
then $\sigma^{\downarrow}_{\ell+1}(A)=\cdots=\sigma^{\downarrow}_{n}(A)=0$.
Thus we set $\ell:=\min\\{m,n\\}$ and consider only
$\sigma^{\downarrow}_{j}(A)$, for $j=1,\ldots,\ell$.

We may consider the $\sigma^{\downarrow}_{j}$ as functions on the vector space
$\mathbb{C}^{m\times n}$ of complex $m\times n$ matrices. Without loss of
generality assume that $m\leq n$ and let $\tilde{A}$ be the $n\times n$ matrix
resulting from $A$ by adding $n-m$ rows consisting of zeros. Then the
eigenvalues of the Hermitian matrix

$\begin{pmatrix}0&\tilde{A}\\\ \tilde{A}^{*}&0\end{pmatrix}$ (7.1)

are precisely

$\sigma^{\downarrow}_{1}(A)\geq\cdots\geq\sigma^{\downarrow}_{n}(A)\geq-\sigma^{\downarrow}_{n}(A)\geq\cdots\geq-\sigma^{\downarrow}_{1}(A).$

It follows from Corollary 6.1 that, for all $k=1,\ldots,\ell$, the sum

$\sum_{j=1}^{k}\sigma^{\downarrow}_{j}$ (7.2)

of the $k$ largest singular values is a sublinear function on
$\mathbb{C}^{m\times n}$ viewed as a real vector space. In fact, the sums
$A\mapsto\sum_{j=1}^{k}\sigma^{\downarrow}_{j}(A)$ are the so-called _Ky Fan
norms_. We obtain

######  Corollary 7.1.

The mapping
$\sigma^{\downarrow}=(\sigma^{\downarrow}_{1},\ldots,\sigma^{\downarrow}_{\ell}):\mathbb{C}^{m\times
n}\to\mathbb{R}^{\ell}$, where $\ell=\min\\{m,n\\}$, is globally Lipschitz and
difference-convex on $\mathbb{C}^{m\times n}$.

The following theorem is a consequence of Theorem 6.2.

######  Theorem 7.2.

Let $A:\mathbb{R}\to\mathbb{C}^{m\times n}$ be a curve of $m\times n$ complex
matrices of class $C^{1}\cap DC\cap W^{2,1}_{\operatorname{loc}}$. If
$\operatorname{rank}A(t)=\min\\{m,n\\}=:\ell$ for all $t$, then there exists a
system $\sigma=(\sigma_{1},\ldots,\sigma_{\ell})$ of the singular values of
$A$ such that $\sigma\in C^{1}(\mathbb{R},\mathbb{R}^{\ell})\cap
DC(\mathbb{R},\mathbb{R}^{\ell})\cap
W^{2,1}_{\operatorname{loc}}(\mathbb{R},\mathbb{R}^{\ell})$. This result is
uniform in the sense explained in Theorem 6.2.

###### Proof.

The assertions follow from Theorem 6.2 applied to the Hermitian matrix (7.1).
The condition on the rank of $A$ guarantees that the non-trivial singular
values of $A$ are always strictly positive and hence there exists a
$C^{1}$-system of them, since there exists a $C^{1}$-system of the eigenvalues
of (7.1). ∎

The rank condition is necessary; for instance, the singular value of the
symmetric $1\times 1$ matrix $A(t)=(t)$ is $|t|$ which does not admit a
$C^{1}$ parameterization.

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