# The unexpected uses of a bowling pin: anisotropic flow in fixed-target
208Pb+20Ne collisions as a probe of quark-gluon plasma

Giuliano Giacalone<EMAIL_ADDRESS>Institut für
Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120
Heidelberg, Germany  Wenbin Zhao<EMAIL_ADDRESS>Nuclear Science Division,
Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
Physics Department, University of California, Berkeley, California 94720, USA
Benjamin Bally  ESNT, IRFU, CEA, Université Paris-Saclay, 91191 Gif-sur-
Yvette, France  Shihang Shen  Institute for Advanced Simulation (IAS-4),
Forschungszentrum Jülich, D-52425 Jülich, Germany
Thomas Duguet  IRFU, CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette,
France  KU Leuven, Instituut voor Kern- en Stralingsfysica, 3001 Leuven,
Belgium  Jean-Paul Ebran  CEA, DAM, DIF, 91297 Arpajon, France  Université
Paris-Saclay, CEA, Laboratoire Matière en Conditions Extrêmes, 91680 Bruyères-
le-Châtel, France  Serdar Elhatisari  Faculty of Natural Sciences and
Engineering, Gaziantep Islam Science and Technology University, Gaziantep
27010, Turkey  Mikael Frosini  CEA, DES, IRESNE, DER, SPRC, LEPh, 13115 Saint-
Paul-lez-Durance, France
Timo A. Lähde  Institut für Kernphysik, Institute for Advanced Simulation and
Jülich Center for Hadron Physics, Forschungszentrum Julich, D-52425 Jülich,
Germany  Center for Advanced Simulation and Analytics (CASA),
Forschungszentrum Julich, D-52425 Jülich, Germany  Dean Lee  Facility for Rare
Isotope Beams and Department of Physics and Astronomy, Michigan State
University, MI 48824, USA  Bing-Nan Lu  Graduate School of China Academy of
Engineering Physics, Beijing 100193, China  Yuan-Zhuo Ma  Facility for Rare
Isotope Beams and Department of Physics and Astronomy, Michigan State
University, MI 48824, USA  Ulf-G. Meißner  Helmholtz-Institut für Strahlen-
und Kernphysik, Universität Bonn, D-53115 Bonn, Germany  Bethe Center for
Theoretical Physics, Universität Bonn, D-53115 Bonn, Germany  Institute for
Advanced Simulation (IAS-4), Forschungszentrum Jülich, D-52425 Jülich, Germany
Govert Nijs  Theoretical Physics Department, CERN, CH-1211 Genève 23,
Switzerland  Jacquelyn Noronha-Hostler  Illinois Center for Advanced Studies
of the Universe, Department of Physics, University of Illinois at Urbana-
Champaign, Urbana, IL 61801, USA  Christopher Plumberg  Natural Science
Division, Pepperdine University, Malibu, CA 90263, USA  Tomás R. Rodríguez
Departamento de Estructura de la Materia, Física Térmica y Electrónica and
IPARCOS, Universidad Complutense de Madrid, E-28040 Madrid, Spain
Robert Roth  Institut für Kernphysik, Technische Universität Darmstadt, 64289
Darmstadt, Germany  Helmholtz Forschungsakademie Hessen für FAIR, GSI
Helmholtzzentrum, 64289 Darmstadt, Germany  Wilke van der Schee  Theoretical
Physics Department, CERN, CH-1211 Genève 23, Switzerland  Institute for
Theoretical Physics, Utrecht University, 3584 CC Utrecht, The Netherlands
NIKHEF, Amsterdam, The Netherlands  Björn Schenke<EMAIL_ADDRESS>Physics
Department, Brookhaven National Laboratory, Upton, NY 11973, USA  Chun Shen
<EMAIL_ADDRESS>Department of Physics and Astronomy, Wayne State
University, Detroit, Michigan 48201, USA  RIKEN BNL Research Center,
Brookhaven National Laboratory, Upton, NY 11973, USA  Vittorio Somà  IRFU,
CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette, France

###### Abstract

The System for Measuring Overlap with Gas (SMOG2) at the LHCb detector enables
the study of fixed-target ion-ion collisions at relativistic energies
($\sqrt{s_{\rm NN}}\sim 100$ GeV in the centre-of-mass). With input from ab
initio calculations of the structure of 16O and 20Ne, we compute 3+1D
hydrodynamic predictions for the anisotropic flow of Pb+Ne and Pb+O
collisions, to be tested with upcoming LHCb data. This will allow the detailed
study of quark-gluon plasma (QGP) formation as well as experimental tests of
the predicted nuclear shapes. Elliptic flow ($v_{2}$) in Pb+Ne collisions is
greatly enhanced compared to the Pb+O baseline due to the shape of 20Ne, which
is deformed in a bowling-pin geometry. Owing to the large 208Pb radius, this
effect is seen in a broad centrality range, a unique feature of this collision
configuration. Larger elliptic flow further enhances the quadrangular flow
($v_{4}$) of Pb+Ne collisions via non-linear coupling, and impacts the sign of
the kurtosis of the elliptic flow vector distribution ($c_{2}\\{4\\}$).
Exploiting the shape of 20Ne proves thus an ideal method to investigate the
formation of QGP in fixed-target experiments at LHCb, and demonstrates the
power of SMOG2 as a tool to image nuclear ground states.

††preprint: CERN-TH-2024-074

Over the past decades, collider experiments with atomic nuclei have enabled
the exploration of the phase diagram of strong-interaction matter over a wide
range of temperature and densities. At ultra-relativistic energies,
experiments at the Relativistic Heavy Ion Collider (RHIC) and at the CERN
Large Hadron Collider (LHC) have established the formation of quark-gluon
plasma (QGP) Harris:2024aov , the hot phase of QCD behaving like a near-ideal
fluid of deconfined partons, via the observation of anisotropic flow
Ollitrault:2023wjk  in combination with the modification of hard probes due to
their interaction with the hot environment Apolinario:2022vzg . At these
energies, anisotropic flow signals persist down to small systems Nagle:2018nvi
; Shen:2020mgh ; Schenke:2021mxx ; Noronha:2024dtq , including $p$+$p$,
$\gamma+$Pb, and $e^{+}$+$e^{-}$ collisions CMS:2016fnw ; ATLAS:2018ngv ;
ALICE:2019zfl ; ATLAS:2021jhn ; Zhao:2022ayk ; Chen:2023njr , although without
any visible hard-probe modification, leaving the question of the formation of
QGP up to debate ATLAS:2014qaj ; ALICE:2016dei . Consequently, research on
small systems has shed light on the apparent hydrodynamization of out-of-
equilibrium QCD matter Berges:2020fwq ; Soloviev:2021lhs . In parallel, the
Beam Energy Scan (BES) at RHIC has elucidated the equation of state and the
transport properties of the QGP towards high net-baryon density PHOBOS:2006dbo
; STAR:2016vqt ; STAR:2017idk , while fixed-target experiments close to the
GeV scale have demonstrated the dominance of hadronic degrees of freedom and
opened connections with the physics of neutron stars HADES:2019auv ;
STAR:2021yiu ; HADES:2022osk . This wealth of experimental measurements in
combination with advanced Bayesian inference methods Annala:2019puf ;
OmanaKuttan:2022aml ; Giacalone:2023cet ; Paquet:2023rfd ; Shen:2023awv  has
helped shape a picture of nuclear matter gradually acquiring consistency
across experiments and theoretical results.

To add to this picture, exciting prospects lie ahead from fixed-target ion-ion
collisions at the LHC. The LHCb detector hosts in particular the System for
Measuring Overlap with Gas (SMOG2) LHCb:2021ysy ; LHCb:2022qvj ;
Mariani:2022klh , enabling us to perform beam-target collisions at
relativistic energy in the centre-of-mass frame ($\sqrt{s_{\rm NN}}=68.5$ GeV
for beam nucleons boosted to 2.5 TeV). This configuration is of unique
interest. At $\sqrt{s_{\rm NN}}=68.5$ GeV, Pb+O or Pb+Ne collisions yield
multiplicities close to those of collider-mode proton-nucleus or oxygen-oxygen
collisions at LHC energy, and should exhibit visible collective behavior. At
the same time, the beam energy is approximately that explored by the BES at
RHIC, implying a significantly baryon-rich system. However, the size of the
Pb+O and Pb+Ne droplets is somewhat intermediate between the $d$+Au and Au+Au
collisions explored at the BES. Fixed-target collisions at LHC probe, then, a
combination of size and baryon density which has not been explored (and will
not be explored) by any other experiment.

Characterizations of the matter produced in these experiments have only just
begun. The LHCb collaboration has recently published the measurement of
$J/\Psi$ and $D_{0}$ production in Pb+Ne collisions LHCb:2022qvj . The
resulting ratio between the two cross sections is consistent with the $p$+Ne
baseline, even in high-multiplicity events, suggesting that hot-medium
interactions may not be effective. Experimental studies of the soft sector and
the collective flow have not been performed yet. It is crucial to explore if
sizable anisotropic flow is observed in these collisions: in the absence of
hard-probe modification, this would effectively open a small system puzzle in
a new type of system. Surprisingly, theoretical studies of anisotropic flow in
fixed-target mode are missing. To motivate and underscore the potential of the
experimental effort, in this Letter we aim to fill this important gap.

Owing to recent advances in the understanding of the ground-state structure of
16O and 20Ne, as well as of the longitudinal structure of heavy-ion
collisions, we work within an end-to-end hydrodynamic framework based on
PGCM/NLEFT+3D-Glauber+MUSIC+UrQMD simulations of Pb+Ne and Pb+O collisions.

The simulations start with configurations of nucleons for 20Ne and 16O
obtained from either ab initio Projected Generator Coordinate Method (PGCM)
Frosini:2021fjf ; Frosini:2021sxj ; Frosini:2021ddm  or Nuclear Lattice
Effective Field Theory (NLEFT) Lee:2008fa ; Lahde:2019npb ; Lee:2020meg
calculations, as presented in Ref. Giacalone:2024luz . The PGCM nucleons are
sampled either independently from the intrinsic nuclear densities (labeled
hereafter Independent configurations) or by enforcing two protons and two
neutrons to sit close to $\alpha$-cluster centers (labeled Cluster
configurations). For the NLEFT calculation, we do not take into consideration
the moderate sign problem that affects the Monte Carlo sampling, and construct
expectation values in the final state by assigning a positive weight to each
event, which is done by considering only positive-sign nuclear configurations
in the hydro calculations (see Giacalone:2024luz  for a detailed discussion of
this matter). This approximation is good enough for the qualitative nature of
the present study. For 208Pb, we assume independent nucleons within a matter
distribution, $\rho(r)$, corresponding to the measured charge density in
Woods-Saxon (W-S) form:

$\rho(r)\propto\left(1+\exp[(r-R)/a]\right)^{-1},$ (1)

with $R=6.62$ fm, and $a=0.55$ fm DeVries:1987atn .

Details on the three-dimensional energy- and baryon-density deposition model
within the Glauber picture Shen:2022oyg  are provided in the Supplemental
Material (SM). The MUSIC code Schenke:2010nt ; Paquet:2015lta ;
Denicol:2018wdp  solves 3+1D relativistic hydrodynamic equations by evolving
the initialized densities with lattice QCD based equation of state
Monnai:2019hkn . When the local energy density drops below 0.45 GeV$/{\rm
fm^{3}}$, we switch to a hadron gas phase described within the hadron cascade
model UrQMD Bass:1998ca ; Bleicher:1999xi . For each nuclear structure
scenario, we run 200k minimum bias events for both Pb+Ne and Pb+O collisions.
For each collision event, the particlization to hadrons and the UrQMD
evolution are repeated, so that the event is oversampled until we obtain
$10^{6}$ hadrons per hydro-surface, irrespective of centrality.

Final state observables are evaluated using all oversampled UrQMD events,
which effectively erases non-flow contributions and statistical fluctuations.
The model parameters used in the calculations are the same as in Ref.
Zhao:2022ugy . They lead to a good description of the pseudorapidity ($\eta$)
dependence of the charged yields in $p/d$/3He+Au collisions at top RHIC
energy. The simulations are performed in the centre-of-mass frame, where
$\sqrt{s_{\rm NN}}=68.5$ GeV. The final-state hadrons are then boosted by
$\Delta y=y_{\rm beam}({\rm 5.02~{}TeV})-y_{\rm beam}({\rm 68.5~{}GeV})$ to
the lab frame. The event centrality is determined from the distribution of the
charged hadron multiplicity within the acceptance of the LHCb spectrometer,
$2<\eta<5$, with 0% corresponding to the high-multiplicity limit.

Figure 1: (a): Probability density of the rescaled energy, $E/\langle
E\rangle$, in Pb+Ne and Pb+O collisions at $\sqrt{s_{\rm NN}}=$ 68.5 GeV.
Lines are hydrodynamic predictions for different nuclear structure inputs.
Symbols are experimental data from the LHCb Collaboration LHCb:2021ysy . (b):
Hydrodynamic prediction for the charged multiplicity in the LHCb acceptance as
a function of the collision centrality. We plot as well the ratio of charged
multiplicity taken between Pb+Ne and Pb+O collisions in panel (c). Errors are
statistical only and of the same size as the shown lines when not vibisle.

To start, we calculate the total energy, $E$, of the final-state hadrons in
the lab frame. In Fig. 1(a), the minimum-bias distribution of $E$, rescaled by
the mean value of the full sample, is compared to the histogram of the
rescaled energy collected in the electromagnetic calorimeter of LHCb in Pb+Ne
collisions LHCb:2021ysy , which should be strongly correlated with the total
event $E$. Excellent agreement between data and our 3+1D model result is
found. We emphasize that agreement is as good as that obtained by the LHCb
collaboration via a dedicated Glauber fit of the energy histogram (Fig. 18 in
LHCb:2021ysy ).

Next, we predict the centrality dependence of the average charged
multiplicity, $\langle N_{\rm ch}\rangle$. In Fig. 1(b)-(c), it displays a
steeper trend as a function of centrality in Pb+Ne collision than in Pb+O
collisions. We understand this as follows. Due to the larger 20Ne size, Pb+Ne
collisions have a higher nucleus-nucleus cross section, which implies larger
impact parameters at the same centrality percentile. The size of the 208Pb
nucleus is however the same in both systems. Therefore, the number of nucleons
that do not hit the incoming 208Pb increases more rapidly with centrality for
a 20Ne target than for a 16O target, explaining the steeper trend in Figs.
1(b)-(c).

The qualitative behavior is robust against variations in the nuclear structure
input. The NLEFT curve has the steepest decrease because it predicts a larger
Ne/O ratio for the nuclear radius than the PGCM calculations Giacalone:2024luz
. For completeness, we repeat our calculation with spherical 20Ne and 16O
nuclei parametrized via Eq. (1) with $R=$ 2.8 fm, $a=$ 0.57 fm for 20Ne, and
$R=$ 2.61 fm, $a=$ 0.51 fm for 16O DeVries:1987atn . This leads to essentially
the same prediction in Fig. 1(c), confirming that the trend is driven by a
size effect rather than the 20Ne deformation.

Moving to results involving the anisotropy of the overlap area in Pb+Ne
collisions, the novelty and uniqueness of this configuration can be grasped by
analyzing the initial-state ellipticity, $\varepsilon_{2}$, which sources the
final-state elliptic flow. This is quantified via the quadrupole moment of the
energy-density field, $\tau e(r,\phi_{r})$ [GeV/fm2], at the beginning of
hydrodynamics Teaney:2010vd :

$\varepsilon_{2}=\frac{\left|\int rdrd\phi_{r}~{}r^{2}e^{i2\phi_{r}}\tau
e(r,\phi_{r})\right|}{\int rdrd\phi_{r}~{}r^{2}\tau e(r,\phi_{r})},\hskip
20.0pt\varepsilon_{2}\\{2\\}^{2}\equiv\langle\varepsilon_{2}^{2}\rangle,$ (2)

where the average is over events at a given centrality.

Due to the peculiar shape of 20Ne, deformed into a bowling-pin-like
16O+$\alpha$ configuration Giacalone:2024luz , performing Pb+Ne collisions
amounts to playing bowling with the ball (lead) thrown towards the bowling pin
(neon) target. When the hit neon lies fully within the area of the lead
nucleus, its entire shape is resolved. Naturally, we expect the eccentricity
of the overlap region to be nearly constant for those impact parameters that
correspond to such a configuration. Given the large size of 208Pb, this
explains why the variation of $\varepsilon_{2}\\{2\\}$ observed in Fig. 2 in
Pb+Ne collisions is so small up to impact parameters of order $b\sim 5$ fm.
This behavior seems solid and genuinely induced by the large quadrupole
deformation of 20Ne: the calculation for a spherical W-S 20Ne nucleus leads to
a steeper (though, predictably, still mild compared to a symmetric, e.g.,
oxygen-oxygen configuration) centrality dependence for
$\varepsilon_{2}\\{2\\}$, consistent with the trend of Pb+O collisions for
both deformed and spherical 16O nuclei. In terms of observable quantities, we
expect thus an enhancement of the elliptic flow of Pb+Ne collisions relative
to Pb+O collisions in a broad centrality range.

Figure 2: Playing ultra-relativistic bowling at the LHC. The rms initial-state
ellipticity, $\varepsilon_{2}\\{2\\}$ is plotted as a function of the
centrality percentile in Pb+Ne collisions (solid lines) and Pb+O collisions
(dashed lines) at $\sqrt{s_{\rm NN}}=$ 68.5 GeV, for various nuclear structure
models. Errors are statistical only and of the same size as the line width
when not visible. The sketches on top of the figure illustrate the collision
geometry based on the average impact parameters of the Pb+Ne collisions
(labeling the upper axis of the plot). We recall the 208Pb W-S radius is
around 6.6 fm, and about 2.8 fm for 20Ne DeVries:1987atn .

With this insight in mind we look at the anisotropic flow in momentum space,
defined by the set of Fourier harmonics that characterize the azimuthal angle
$\phi$ dependence of the charged hadron spectrum:

$\frac{dN_{\rm ch}}{d\phi}\propto
1+2\sum_{n=1}^{\infty}v_{n}\cos[n(\phi-\phi_{n})],$ (3)

where $v_{n}$ is the magnitude of the $n$-th order harmonic. At a given
centrality, we evaluate the root mean square value of the distribution of the
coefficient (see the SM for further derivations),

$v_{n}\\{2\\}=\sqrt{\langle v_{n}^{2}\rangle}.$ (4)

Figure 3(a) displays our predictions for the elliptic flow ($n=2$) of Pb+Ne
and Pb+O collisions. The centrality dependence of $v_{2}\\{2\\}$ in Fig. 3(a)
is rather flat, especially when a deformed 20Ne is considered. Computing the
Pb+Ne/Pb+O ratio, shown in Fig. 3(b), highlights instead the strong impact of
the shape of 20Ne, which enhances the elliptic flow in central Pb+Ne
collisions by more than 20%. The signal survives up to large centralities,
confirming the intuition from Fig. 2. This showcases the unique power of
SMOG2, and in general of asymmetric Pb+X collisions Bally:2022vgo , as a tool
to image nuclear ground states STAR:2024eky . For the spherical baseline, the
elliptic flow ratio is below unity. This is due to the larger mass number of
20Ne, which reduces initial-state fluctuations with respect to collisions
involving 16O nuclei. Indeed, in absence of nuclear deformation corrections, a
larger elliptic flow should be observed in Pb+O collisions.

Figure 3: Anisotropic flow coefficients in Pb+Ne collisions (solid lines) and
Pb+O collisions (dashed lines) at $\sqrt{s_{\rm NN}}=$ 68.5 GeV, calculated
within the LHCb acceptance, as a function of the collision centrality. Upper
panels show predictions for the rms elliptic flow, $v_{2}\\{2\\}$ (a),
triangular flow, $v_{3}\\{2\\}$ (c), and quadrangular flow, $v_{4}\\{2\\}$
(e). Lower panels show the corresponding ratios taken between Pb+Ne and Pb+O
collisions. Different line colors correspond to different nuclear structure
inputs. Errors are statistical only and of the same size as the shown lines
when not visible.

Figures 3(c)-(d) show our predictions for the rms triangular flow,
$v_{3}\\{2\\}$, in both systems. The predictions seem diametrically opposite
to those obtained for the elliptic flow: $v_{3}\\{2\\}$ depends strongly on
the centrality percentile in Fig. 3(c), while Fig. 3(d) indicates little
difference between Pb+Ne and Pb+O systems. This in agreement with the
predictions obtained for the ratio taken between symmetric Ne+Ne and O+O
collisions Giacalone:2024luz .

Moving on to the quadrangular flow ($n=4$), the same patterns seen in the case
of elliptic flow are recovered. In Fig. 3(e), $v_{4}\\{2\\}$ shows little
variation with centrality, though the ratio of quadrangular flow coefficients
in Fig. 3(f) is enhanced by as much as 20% in central Pb+Ne collisions
compared to Pb+O. As explicitly demonstrated in the SM, one can identify the
origin of such behavior in the non-linear mode coupling between the elliptic
and quadrangular flow vectors Gardim:2011xv ; Teaney:2012ke ; Yan:2015jma ;
Qian:2016fpi . Therefore, while the shape of 20Ne directly impacts the
magnitude of $v_{2}$, it impacts the $v_{4}$ indirectly via nonlinear
coupling. This extra layer of complexity leads to an even more stringent test
of the hydrodynamic model. It should be investigated in experiments, and
tested as well within a framework that is not hydrodynamic, such as transport
Jia:2022qrq .

Before concluding, we analyze a standard probe of hydrodynamic behavior in
high-energy collisions, the fourth-order cumulant of the elliptic flow
distribution:

$c_{2}\\{4\\}=\langle v_{2}^{4}\rangle-2\langle v_{2}^{2}\rangle^{2},$ (5)

where averages are again over events at a given centrality. As we explain in
the SM, this quantity measures the kurtosis of the elliptic flow vector
distribution at a given centrality Abbasi:2017ajp ; Bhalerao:2018anl , and can
have either negative or positive sign. The centrality dependence of
$c_{2}\\{4\\}$ is shown in Fig. 4(a). Surprisingly, in central collisions we
predict:

$c_{2}\\{4\\}_{\rm PbNe}<0<c_{2}\\{4\\}_{\rm PbO}.$ (6)

To get some understanding, we calculate in Fig. 4(b) the same quantity with
$v_{2}$ in Eq. (5) replaced by the initial-state ellipticity,
$\varepsilon_{2}$. The sign of the resulting cumulant is negative, and its
magnitude is much larger in Pb+Ne collisions than in Pb+O collisions, in
agreement with previous studies with deformed nuclei Giacalone:2018apa ;
Mehrabpour:2023ign . The hydrodynamic expansion adds a positive correction to
the value of $c_{2}\\{4\\}$ in a given centrality, changing even the sign of
the cumulant in Pb+O collisions, or when a spherical neon is used. Similar
results are found for $p$-$p$ collisions Zhao:2017rgg ; Zhao:2020pty .
However, the effect of the 20Ne shape is opposite: the impact of the
deformation reduces the kurtosis, causing $c_{2}\\{4\\}$ in Pb+Ne collisions
to preserve its negative sign after the hydrodynamic evolution. This interplay
between the deformation effect and hydrodynamic response provides a nontrivial
probe of the dynamics of the collisions, and so we urge the experiments to
verify this result.

Note that the parameters of the hydrodynamic model used here do not yet result
from a Bayesian analysis of the RHIC BES data. Though unlikely, we can not
exclude a priori that there may exist a plausible set of parameters (likely,
initial-state parameters) for which $c_{2}^{\varepsilon}\\{4\\}$ has a
different sign. In addition, the choice of the variable used to the define the
collision centrality affects the sign of the cumulant Zhou:2018fxx ;
ATLAS:2019peb . Systematic analyses to address these points are costly for
3+1D simulations, and outside the scope of present manuscript, but should be
carried out for future comparisons with the experimental results.

Figure 4:  Centrality dependence of $c_{2}\\{4\\}$ computed both from the
final-state hadron distributions (upper) and from the initial-state
eccentricity distribution (lower) in Pb+Ne collisions (solid lines) and Pb+O
collisions (dashed lines) at $\sqrt{s_{\rm NN}}=$ 68.5 GeV. Different line
colors correspond to different nuclear structure inputs. Errors are
statistical only and of the same size as the shown lines when not visible.

In summary, we have unveiled the great opportunities offered by studies of the
anisotropic flow in fixed-target collisions at the LHCb detector. Our
predictions may be amenable to experimental verification already with the next
LHC ion run. The availability of the bowling-pin nucleus 20Ne in SMOG2 enables
one to isolate strong effects of the hydrodynamic response. If the dramatic
enhancements of $v_{2}$ and $v_{4}$ relative to the Pb+O baseline will be
confirmed by the LHCb collaboration, they will hint at the validity of a QGP
description. This will pave the way to quantitative characterizations of the
matter formed in these experiments via future Bayesian analyses.

The predictions of this manuscript can indeed be improved in several
directions. The impact of short-range two-particle correlations (non flow) is
washed out in our simulations due to the oversampling of the UrQMD events,
which can be relaxed in future studies. In addition, a more detailed analysis
of the systematics of the theoretical model will elucidate whether theoretical
uncertainties cancel when normalizing Pb+Ne observables with those coming from
another collision species Giacalone:2024luz .

Additional observables can be studied to fully exploit the LHCb data. Focusing
on $\eta$-differential quantities will help constrain models of longitudinal
fluctuations, which are under intense study in the community Soeder:2023vdn ;
McDonald:2023qwc ; Garcia-Montero:2023gex ; Ipp:2024ykh ; Zhang:2024bcb . In
addition, probing hydrodynamic behavior via shape-induced modifications of the
collective flow can be done as well through the mean transverse momentum,
$\langle p_{T}\rangle$, of final-state hadrons, by measuring its event-by-
event fluctuations and the $v_{2}^{2}$-$\langle p_{T}\rangle$ correlation
Giacalone:2019pca ; Giacalone:2020awm ; Jia:2021wbq ; Jia:2021tzt ;
Bally:2021qys ; Jia:2021qyu ; ALICE:2021gxt ; ATLAS:2022dov ; Fortier:2023xxy
; Nielsen:2023znu ; STAR:2024eky ; Wang:2024vjf ; Fortier:2024yxs .

Finally, other species are available in SMOG2, notably, nitrogen (N), argon
(Ar), krypton (Kr), and xenon (Xe). Their clustered and deformed geometries
will have to be elucidated via future ab initio computations of nuclear
structure based on high-resolution chiral EFT interactions Epelbaum:2008ga ;
Piarulli:2019cqu , which have only recently been pushed to the description of
medium-mass deformed nuclei Elhatisari:2022qfr ; Frosini:2024ajq ; Sun:2024iht
; Hu:2024pee . We are thus looking forward to an exciting program of cross-
disciplinary QCD studies centered around the unique high-energy fixed-target
mode of the LHC.

The gas-injection system of LHCb has been so far used with a natural neon
source. Abundances of stable isotopes in the gas are 90.48% for 20Ne, 0.27%
for 21Ne, and 9.25% for 22Ne Meija2016a . We recommend a purified 20Ne sample
for use in the experimental campaigns for a transparent interpretation of the
physics results. This is especially important in the limit of ultra-central
collisions with the largest numbers of participant nucleons.

We thank the LHCb Collaboration for sharing with us the data shown in Fig. 1.
We thank in particular Giacomo Graziani and the participants of the
program+workshop “Exploring Nuclear Physics Across Energy Scales 2024” for
useful discussions. G.G. is funded by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) – Project-ID 273811115 – SFB 1225 ISOQUANT,
and under Germany’s Excellence Strategy EXC2181/1-390900948 (the Heidelberg
STRUCTURES Excellence Cluster). This material is based upon work supported by
the U.S. Department of Energy, Office of Science, Office of Nuclear Physics,
under DOE Contract No. DE-SC0012704 (B.P.S.) and Award No. DE-SC0021969
(C.S.), and within the framework of the Saturated Glue (SURGE) Topical Theory
Collaboration. C.S. acknowledges a DOE Office of Science Early Career Award.
W.B.Z. is supported by DOE under Contract No. DE-AC02-05CH11231, by NSF under
Grant No. OAC-2004571 within the X-SCAPE Collaboration, and within the
framework of the SURGE Topical Theory Collaboration. This research was done
using resources provided by the Open Science Grid (OSG) Pordes:2007zzb ;
Sfiligoi:2009cct , which is supported by the National Science Foundation award
#2030508. This work is supported in part by the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme (ERC AdG EXOTIC, grant agreement No. 101018170), by DFG and NSFC
through funds provided to the Sino-German CRC 110 “Symmetries and the
Emergence of Structure in QCD” (NSFC Grant No. 11621131001, DFG Grant No.
TRR110). The work of UGM was supported in part by the CAS President’s
International Fellowship Initiative (PIFI) (Grant No. 2018DM0034). R.R. is
supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) – Projektnummer 279384907 – SFB 1245. T.R.R. is supported by the
Spanish MCIU (PID2021- 127890NB-I00). The PGCM calculations were performed
using HPC resources from GENCI-TGCC (Contracts No. A0130513012 and
A0150513012) and CCRT (TOPAZE supercomputer). The NLEFT code development and
production calculations utilized the following computational resources: the
Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for computing time
on the GCS Supercomputer JUWELS at Jülich Supercomputing Centre (JSC) and
special GPU time allocated on JURECA-DC; the Oak Ridge Leadership Computing
Facility through the INCITE award “Ab-initio nuclear structure and nuclear
reactions”; and the TUBITAK ULAKBIM High Performance and Grid Computing Center
(TRUBA resources). J.N.H. acknowledges financial support from the US-DOE
Nuclear Science Grant No. DE-SC0023861 and within the framework of the
Saturated Glue (SURGE) Topical Theory.

##  I SUPPLEMENTAL MATERIAL

###  I.1 3D Glauber Monte Carlo initial conditions

The 3D Monte-Carlo Glauber model is a dynamical event-by-event initial
condition Shen:2017bsr ; Shen:2022oyg , that provides the space-time and
momentum distributions of the initial energy-momentum tensor and net baryon
charge current, employing a classical string deceleration model Bialas:2016epd
; Shen:2017bsr . The transverse positions of valence quarks and the soft
partonic cloud in wounded nucleons are sampled from a 2D Gaussian. After the
collision, the deposited energy density distribution and baryon charge have
Gaussian profiles in the transverse plane. The average rapidity loss function
of the valence quarks and the soft partonic cloud with their incoming rapidity
$y_{\mathrm{init}}$ in the collision pair rest frame is parameterized as
Shen:2018pty ; Shen:2022oyg ,

$\langle
y_{\mathrm{loss}}\rangle(y_{\mathrm{init}})=Ay_{\mathrm{init}}^{\alpha_{2}}[\tanh(y_{\mathrm{init}})]^{\alpha_{1}-\alpha_{2}}.$
(7)

The parameters are fixed by fitting the measured pseudo-rapidity distributions
of charged hadrons for Au+Au and light-ion collisions Shen:2018pty ;
Shen:2022oyg ; Zhao:2022ugy ; Zhao:2022ayk . The detailed implementation of
this initial condition model is discussed in Ref. Shen:2022oyg . The pre-
hydrodynamic flow is included by a finite transverse initial velocity
Zhao:2022ugy . Finally, the produced strings from individual nucleon-nucleon
collisions are treated as dynamical source terms for the hydrodynamic
evolution Okai:2017ofp ; Shen:2017ruz ; Shen:2017bsr ; Shen:2022oyg .

##  II Cumulants of flow fluctuations in the lab frame

We recall here the definition and the physical meaning of the cumulants of
anisotropic flow fluctuations, $v_{n}\\{2\\}$ and $c_{2}\\{4\\}$, analyzed in
this manuscript.

We start with the harmonic decomposition of the measured charged-hadron
spectrum,

$\frac{dN}{d\phi_{p}}=\sum_{n=-\infty}^{+\infty}\textbf{v}_{n}e^{in\phi_{p}},\hskip
20.0pt\textbf{v}_{-n}=\textbf{v}_{n}^{*},$ (8)

where $\phi_{p}$ is the azimuthal angle of transverse momentum and the set of
complex coefficients $\textbf{v}_{n}$ define the anisotropic flow. Note that
$\textbf{v}_{n}$ comes with a real and an imaginary part

$\textbf{v}_{n}=(v_{x},v_{y})$ (9)

which depends on the choice of the coordinate frame. Typically, $x$ and $y$
label either the transverse plane in the laboratory frame or the frame of the
reaction plane with $x$ chosen along the impact parameter direction.

Observables constructed from the anisotropic flow in heavy-ion collisions are
measured as statistical averages involving azimuthal angles, which are taken
over collision events, typically at a fixed final-state multiplicity
(centrality). In the sample of events, the flow vector, $\textbf{v}_{n}$, has
some underlying distribution:

$P(\textbf{v}_{n})=P(v_{x},v_{y}),\hskip
30.0ptv_{n}=\sqrt{v_{x}^{2}+v_{y}^{2}}.$ (10)

Experimentally, the aim is to characterize such probability distribution in a
given centrality class, which yields detailed information about the evolution
and the properties of the produced matter.

Cumulants fully characterize a probability distribution and are particularly
useful in this context. The cumulant generating function of the distribution
of $\textbf{v}_{n}$ is defined by:

$\ln G({\bf k})=\ln\int d\textbf{v}_{n}P(\textbf{v}_{n})e^{{\bf
k}\cdot\textbf{v}_{n}}=\ln\langle e^{{\bf
k}\cdot\textbf{v}_{n}}\bigr{\rangle},$ (11)

from which the cumulant of order $m$ is derived as $\partial^{(m)}_{\bf k}\ln
G({\bf k})|_{{\bf k}=0}$ . In nuclear collisions, while the magnitude of the
impact parameter can be controlled from the number of final-state particles,
its orientation is always random. Therefore, experiments only access the
angle-averaged information about the $\textbf{v}_{n}$ harmonics, i.e., moments
of the magnitude, $v_{n}$, but can not reconstruct information about the
individual $v_{x,y}$ components. In practice, we write the vector ${\bf
k}=(k\cos\varphi,k\sin\varphi)$, and we are forced to average over the angle
in Eq. (11), which yields:

$\ln
G(k)=\ln\langle\int_{0}^{2\pi}\frac{d\varphi}{2\pi}e^{\textbf{k}\cdot\textbf{v}_{n}}\biggr{\rangle}=\ln\langle
I_{0}(kv_{n})\bigr{\rangle}.$ (12)

Information is lost and we end up with a modified Bessel function, $I_{0}$,
which is an even function. The cumulants of the angle-averaged distribution of
the $\textbf{v}_{n}$ vector, whose standard notation is $v_{n}\\{m\\}^{m}$,
are then obtained from the equality:

$\ln\langle
I_{0}(kv_{n})\bigr{\rangle}=\sum_{m=0}^{\infty}c_{m}k^{m}v_{n}\\{m\\}^{m}.$
(13)

The integer $m\geq 2$ is an even number, while the coefficients $c_{m}$ have
to be matched to the power expansion of the left-hand side, that is:

$\ln\langle I_{0}(kv_{n})\bigr{\rangle}=\ln\biggl(1+\frac{\langle
v_{n}^{2}\rangle k^{2}}{4}-\frac{\langle v_{n}^{4}\rangle
k^{4}}{64}+\frac{\langle v_{n}^{6}\rangle k^{6}}{576}+\ldots\biggr{missing}).$
(14)

Expanding this logarithm in powers of $k$, and matching to the right-hand side
of Eq. (13), one obtains:

$\displaystyle v_{n}\\{2\\}^{2}$ $\displaystyle=\langle v_{n}^{2}\rangle,$
$\displaystyle v_{n}\\{4\\}^{4}$ $\displaystyle=2\langle
v_{n}^{2}\rangle-\langle v_{n}^{4}\rangle,$ $\displaystyle v_{n}\\{6\\}^{6}$
$\displaystyle=\frac{1}{4}\bigl{(}\langle v_{n}^{6}\rangle-9\langle
v_{n}^{4}\rangle\langle v_{n}^{2}\rangle+12\langle
v_{n}^{2}\rangle^{3}\bigr{)},$ (15)

and so on, where averages are over events at a given centrality. The cumulants
are thus expressed in terms of even moments of the distribution of $v_{n}$, of
the form $\langle v_{n}^{2k}\rangle$. These moments represent the quantities
that can be measured in the experiments from established multi-particle
correlation techniques Ollitrault:2023wjk .

As pointed out in Ref. Giacalone:2016eyu , it is insightful to relate the
observable quantities on the left-hand side of Eq. (II) to an expansion
involving the (non-measurable) cumulants of the individual $x$ and $y$
components of $\textbf{v}_{n}$. Up to fourth order, the cumulants of the
projections are:

$\displaystyle\kappa_{10}$ $\displaystyle=\langle v_{x}\rangle,\hskip
10.0pt\kappa_{01}=\langle v_{y}\rangle,$ $\displaystyle\kappa_{20}$
$\displaystyle=\left\langle(v_{x}-\kappa_{10})^{2}\right\rangle,\hskip
10.0pt\kappa_{02}=\left\langle(v_{y}-\kappa_{01})^{2}\right\rangle,$
$\displaystyle\kappa_{30}$
$\displaystyle=\left\langle(v_{x}-\kappa_{10})^{3}\right\rangle,\hskip
10.0pt\kappa_{03}=\left\langle(v_{y}-\kappa_{01})^{3}\right\rangle,$
$\displaystyle\kappa_{12}$
$\displaystyle=\langle(v_{x}-\kappa_{10})(v_{y}-\kappa_{01})^{2}\rangle,$
$\displaystyle\kappa_{21}$
$\displaystyle=\langle(v_{x}-\kappa_{10})^{2}(v_{y}-\kappa_{01})\rangle,$
$\displaystyle\kappa_{40}$
$\displaystyle=\left\langle(v_{x}-\kappa_{10})^{4}\right\rangle-3\kappa_{20}^{2},$
$\displaystyle\kappa_{04}$
$\displaystyle=\left\langle(v_{y}-\kappa_{01})^{4}\right\rangle-3\kappa_{02}^{2},$
$\displaystyle\kappa_{22}$
$\displaystyle=\langle(v_{x}-\kappa_{10})^{2}(v_{y}-\kappa_{01})^{2}\rangle-\kappa_{20}\kappa_{02}.$
(16)

If $x$ and $y$ parametrize the transverse plane in the lab frame, the odd
moments of the $v_{x}$ and $v_{y}$ distributions vanish (for an ideal
detector), meaning that we are only left with non-zero values for
$\kappa_{20}$, $\kappa_{02}$, $\kappa_{40}$, $\kappa_{04}$, $\kappa_{22}$. A
little algebra Giacalone:2016eyu ; Abbasi:2017ajp ; Bhalerao:2018anl  shows
that the first two cumulants in Eq. (II) can consequently be written as:

$\displaystyle v_{n}\\{2\\}^{2}$ $\displaystyle=\kappa_{20}+\kappa_{02},$ (17)
$\displaystyle v_{n}\\{4\\}^{4}$
$\displaystyle=-(\kappa_{04}+\kappa_{40}+2\kappa_{22}).$ (18)

This implies the following. The second-order cumulant $v_{2}\\{2\\}^{2}$
isolates the variance of the two-dimensional distribution. It tells us how a
change in the mean square value of the flow vector magnitude, $\langle
v_{n}^{2}\rangle$, is associated with an increase in the variances of its $x$
and $y$ components. Concerning the fourth-order cumulant, conventionally
denoted by 111The sign takes into account the negative sign of the $k^{4}$
term in the expansion of Eq. (14).

$c_{2}\\{4\\}\equiv-v_{2}\\{4\\}^{4},$ (19)

it isolates instead the kurtosis correction, and vanishes if $P(v_{x},v_{y})$
is a two-dimensional Gaussian.

We understand then the impact of the shape of 20Ne on the elliptic flow
vector, $\textbf{v}_{2}$:

  * •

The increase of $v_{2}\\{2\\}$ induced by the deformation of the nucleus is
due to an isotropic increase in spread of the two-dimensional $\textbf{v}_{2}$
distribution (see also Jia:2022qgl ).

  * •

The increase in $c_{2}\\{4\\}$ implies instead that larger elliptic flow
fluctuations lead as well to enhanced non-Gaussian tails. The kurtosis comes
with a positive sign in the expression of $c_{2}\\{4\\}$, therefore, nuclear
deformation reduces the kurtosis leading to a platykurtic distribution (tails
thinner than a Gaussian) at a given centrality.

##  III Understanding the enhancement of $v_{4}$

Here we explain the enhancement of $v_{4}\\{2\\}$ in central Pb+Ne collisions
relative to central Pb+O collisions observed in Fig. 3. We recall that the
quadrangular flow vector, $\textbf{v}_{4}=v_{4}e^{i4\phi_{4}}$, can by virtue
of symmetry be decomposed as follows Yan:2015jma ,

$\textbf{v}_{4}=\textbf{v}_{4L}+\chi_{4}\textbf{v}_{2}^{2},$ (20)

where $\chi_{4}$ is a coefficient that quantifies the strength of the
nonlinear coupling to the squared elliptic flow vector,
$\textbf{v}_{2}=v_{2}e^{i2\phi_{2}}$, while $\textbf{v}_{4L}$ is defined as
the vector statistically uncorrelated with $\textbf{v}_{2}$ in the considered
centrality class, that is,

$\langle\textbf{v}_{4L}\textbf{v}_{2}^{2*}\rangle=0.$ (21)

With these definitions, one obtains that the mean squared quadrangular flow is
the sum of two contributions that add in quadrature,

$\langle v_{4}^{2}\rangle=\langle v_{4L}^{2}\rangle+\chi_{4}^{2}\langle
v_{2}^{4}\rangle,$ (22)

where $v_{4}=|\textbf{v}_{4}|$, $v_{2}=|\textbf{v}_{2}|$,
$v_{4L}=|\textbf{v}_{4L}|$.

Now we take the ratio between Pb+Ne and Pb+O collisions, i.e.,

$\frac{\langle v_{4}^{2}\rangle_{\rm PbNe}}{\langle v_{4}^{2}\rangle_{\rm
PbO}}=\frac{\langle v_{\rm 4L}^{2}\rangle_{\rm PbNe}+\chi_{4,\rm
PbNe}^{2}\langle v_{2}^{4}\rangle_{\rm PbNe}}{\langle v_{\rm
4L}^{2}\rangle_{\rm PbO}+\chi_{4,\rm PbO}^{2}\langle v_{2}^{4}\rangle_{\rm
PbO}}.$ (23)

In central collisions, one expects the term coupling to elliptic flow to be
sub-leading compared to the uncorrelated contribution, $\langle v_{\rm
4L}^{2}\rangle\gg\chi_{4}\langle v_{2}^{4}\rangle$. This justifies a Taylor
expansion of the previous expression:

$\displaystyle\frac{\langle v_{4}^{2}\rangle_{\rm PbNe}}{\langle
v_{4}^{2}\rangle_{\rm PbO}}=$ $\displaystyle\frac{\langle v_{\rm
4L}^{2}\rangle_{\rm PbNe}}{\langle v_{\rm 4L}^{2}\rangle_{\rm
PbO}}+\frac{\chi_{4,\rm PbNe}^{2}\langle v_{2}^{4}\rangle_{\rm PbNe}}{\langle
v_{\rm 4L}^{2}\rangle_{\rm PbO}}-\frac{\langle v_{\rm 4L}^{2}\rangle_{\rm
PbNe}}{(\langle v_{\rm 4L}^{2}\rangle_{\rm PbO})^{2}}\chi_{4,\rm
PbO}^{2}\langle v_{2}^{4}\rangle_{\rm PbO}.$ (24)

Now, we show in Fig. 5 hydrodynamic results for $\chi_{4,22}$ [panel (a)] and
$\langle v_{\rm 4L}\\{2\\}\equiv\sqrt{\langle v_{\rm 4L}^{2}\rangle}$ [panel
(b)]. We see that both these quantities are rather universal and receive only
percent-level corrections if one varies the nuclear species/structure. For
simplicity, let us then consider that

$\displaystyle\langle v_{\rm 4L}^{2}\rangle_{\rm PbNe}=\langle v_{\rm
4L}^{2}\rangle_{\rm PbO}$ $\displaystyle\equiv\langle v_{\rm 4L}^{2}\rangle,$
(25) $\displaystyle\chi_{4,\rm PbNe}=\chi_{4,\rm PbO}$
$\displaystyle\equiv\chi_{4}.$ (26)

With this, we can rewrite Eq. (24) as follows:

$\frac{v_{4}^{2}\\{2\\}_{\rm PbNe}}{v_{4}^{2}\\{2\\}_{\rm
PbO}}\equiv\frac{\langle v_{4}^{2}\rangle_{\rm PbNe}}{\langle
v_{4}^{2}\rangle_{\rm PbO}}=1+\chi_{4}^{2}\frac{\langle v_{2}^{4}\rangle_{\rm
PbNe}}{\langle v_{\rm 4L}^{2}\rangle}\left(1-\frac{\langle
v_{2}^{4}\rangle_{\rm PbO}}{\langle v_{2}^{4}\rangle_{\rm PbNe}}\right).$ (27)

This equality should hold in central collisions if the departure from unity in
the quadrangular flow ratio is due to the nonlinear coupling term to the
elliptic flow. Plugging in numbers from the 0-5% class, we obtain the results
shown in Fig. 5(c). Agreement between the $v_{4}\\{2\\}$ ratio and the
prediction of Eq. (27) is excellent in central collisions, confirming that it
is the nonlinear coupling to elliptic flow that drives the nontrivial Pb+Ne to
Pb+O ratio.

Figure 5: Centrality dependence of the non-linear coupling coefficient
$\chi_{4}$ (a), of the linear flow mode $v_{\rm 4L}\\{2\\}$ (b), and of the
$v_{4}$ ratio (c) in Pb+Ne and Pb+O collisions at $\sqrt{s_{\rm NN}}=$ 68.5
GeV. The dashed line in panel (c) corresponds to the estimate of Eq. (27).
Errors are statistical only and of the same size as the shown lines when not
visible.

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# Companion matrix, Vandermonde matrix, Jordan form, Interpolating
Polynomials, and Linear Transformations

Chi-Kwong Li and Jephian C.-H. Lin

###### Abstract.

Let ${\mathbb{F}}$ be a field, and let $C$ be the $n\times n$ companion matrix
of the monic polynomial $f(x)\in{\mathbb{F}}[x]$ such that
$f(x)=\det(xI-C)=(x-\lambda_{1})^{n_{1}}\cdots(x-\lambda_{m})^{n_{m}}$ for $m$
distinct elements $\lambda_{1},\dots,\lambda_{m}\in{\mathbb{F}}$. It is shown
that there is a generalized Vandermonde matrix $V$ associated with $f(x)$ such
that $VCV^{-1}$ is in Jordan form, and the columns of $V^{-1}$ are connected
to the Hermite interpolating polynomials, whose higher derivatives will have
specific values at $\lambda_{1},\dots,\lambda_{m}$. If $m=n$ and
$n_{1}=\cdots=n_{m}=1$, then the results reduce to the fact that the
(classical) Vandermonde $V$ of $\lambda_{1},\dots,\lambda_{n}$ satisfies
$VCV^{-1}$ is a diagonal matrix and that the columns of $V^{-1}$ correspond to
the Lagrange interpolating polynomials. This shows that the results for real
polynomials and matrices also hold for polynomials and matrices over an
arbitrary field ${\mathbb{F}}$. Moreover, interpretations and insights of the
results are given in terms of linear transformation between ${\mathbb{F}}^{n}$
and the linear space of polynomials in ${\mathbb{F}}[x]$ with degree less than
$n$.

Keywords. Companion matrix, Vandermonde matrix, Jordan form, interpolating
polynomials.

AMS Classifications. 15A20, 15A21, 33C45.

##  1\. Introduction

Let $M_{n}({\mathbb{F}})$ be the set of $n\times n$ matrices with entries in
the field ${\mathbb{F}}$, and ${\mathbb{F}}[x]$ be the set of polynomials with
coefficients in ${\mathbb{F}}$. For any monic polynomial
$f(x)=x^{n}-a_{n-1}x^{n-1}-\cdots-a_{0}\in{\mathbb{F}}[x]$, the companion
matrix of $f(x)$ is

$C=\sum_{j=2}^{n}E_{j,j-1}+\sum_{j=0}^{n-1}a_{j}E_{j+1,n}=\begin{pmatrix}0&&&a_{0}\cr
1&\ddots&&a_{1}\cr&\ddots&0&\vdots\cr&&1&a_{n-1}\cr\end{pmatrix}\in
M_{n}({\mathbb{F}}),$

where $\\{E_{11},E_{12},\dots,E_{nn}\\}$ is the standard basis for
$M_{n}({\mathbb{F}})$. It is known and easy to check that $f(x)$ is the
characteristic polynomial $\det(xI-C)$ of the matrix $C$. The companion matrix
is a well-studied object that appears in many textbooks in linear algebra and
abstract algebra; e.g., see [2, 12]. This concept has connections to many
different topics such as canonical forms, matrix inequalities, interpolations,
differential equations and difference equations; e.g., see [3, 6, 8, 9, 14, 4,
16].

By the rational canonical form theorem, every $A\in M_{n}({\mathbb{F}})$ is
similar to a direct sum of companion matrices; see [5, 7] for short proofs for
this fact, and [4] for construction of such canonical form. If
$f(x)=\det(xI-C)$ has simple zeros
$\lambda_{1},\dots,\lambda_{n}\in{\mathbb{F}}$, which are the eigenvalues of
$C$, then it is known that the Vandermonde matrix of
$\\{\lambda_{1},\ldots,\lambda_{n}\\}$, defined by

(1.1) $V=\begin{pmatrix}1&\lambda_{1}&\cdots&\lambda_{1}^{n-1}\\\
1&\lambda_{2}&\cdots&\lambda_{2}^{n-1}\\\ \vdots&\vdots&~{}&\vdots\\\
1&\lambda_{n}&\cdots&\lambda_{n}^{n-1}\\\ \end{pmatrix},$

has nonzero determinant

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

and is invertible and satisfies
$VCV^{-1}=\operatorname{diag}(\lambda_{1},\dots,\lambda_{n})$. Moreover,
$V^{-1}$ is related to the Lagrange interpolating polynomials $g_{j}(x)$
satisfying $g_{j}(\lambda_{j})=1$ and $g_{j}(\lambda_{i})=0$ whenever $i\neq
j$, for $j=1,\dots,n$; see [14, 16].

If $f(x)=\det(xI-C)$ has zeros that are not simple, researchers considered the
generalized Vandermonde matrix $V$ (a.k.a. the confluent Vandermonde matrix)
such that $VCV^{-1}$ is in Jordan form. In particular, for real matrices
researchers have shown that $V^{-1}$ is related to the Hermite polynomials in
${\mathbb{F}}[x]$ with specific values at $\lambda_{1},\dots,\lambda_{m}$ and
their derivatives; see [8]. Efficient algorithms were derived for computing
$V^{-1}$; e.g., see [3, 6] and their references. Some of the derivations in
these papers used the concepts of limits and derivatives of real polynomial
functions.

In this paper, we show that the same results hold for matrices over an
arbitrary field ${\mathbb{F}}$. Using the definition of formal derivatives of
polynomials over ${\mathbb{F}}$, the matrix $V^{-1}$ is related to the Hermite
polynomials in ${\mathbb{F}}[x]$ associated with
$\lambda_{1},\dots,\lambda_{m}\in{\mathbb{F}}$. Furthermore, we provide
interpretation and insight for the results in terms of linear transformations
between ${\mathbb{F}}^{n}$ and the vector space of polynomials in
${\mathbb{F}}[x]$ with degree at most $n-1$.

In Section 2, we discuss the case when $f(x)=\det(xI-C)$ has simple zeros, and
treat the general case in Section 3.

In our discussion, we let
$\mathcal{E}=\\{\mathbf{e}_{1},\ldots,\mathbf{e}_{n}\\}$ as the standard basis
of $\mathbb{F}^{n}$, that is, $\mathbf{e}_{1},\ldots,\mathbf{e}_{n}$ are the
columns of the $n\times n$ identity matrix. Also, we let
${\mathcal{B}}=\\{1,x,\ldots,x^{n-1}\\}$ be the standard basis of
$\mathcal{P}_{n-1}$, the vector space of polynomials in ${\mathbb{F}}[x]$ with
degree at most $n-1$.

##  2\. The case when the zeros of $f(x)$ are simple

When $f(x)\in{\mathcal{P}}_{n-1}$ is a monic polynomial with simple zeros in
${\mathbb{F}}$, we have the following known result; e.g., see [14]. We include
a short proof for easy reference, and to facilitate our subsequent discussion.

######  Proposition 2.1.

Suppose $f(x)=x^{n}-a_{n-1}x^{n-1}-\cdots-a_{0}\in{\mathbb{F}}[x]$ and $C\in
M_{n}({\mathbb{F}})$ is the companion matrix of $f(x)$ such that $f(x)$ has
$n$ distinct zeros $\lambda_{1},\dots,\lambda_{n}\in{\mathbb{F}}$. Let $V$ be
the Vandermonde matrix of $\\{\lambda_{1},\dots,\lambda_{n}\\}$ defined as in
(1.1). Then $VCV^{-1}=\operatorname{diag}(\lambda_{1},\dots,\lambda_{n})$.
Moreover, if $V^{-1}=(c_{ij})$, then

$g_{j}(x)=c_{1j}+c_{2j}x+\cdots+c_{nj}x^{n-1}=\prod_{\begin{subarray}{c}1\leq\ell\leq
n\\\ \ell\neq
j\end{subarray}}\frac{(x-\lambda_{\ell})}{(\lambda_{j}-\lambda_{\ell})}$

is the Lagrange interpolating polynomial satisfying $g_{j}(\lambda_{j})=1$ and
$g_{j}(\lambda_{i})=0$ for $i\neq j$.

###### Proof.

Let $v_{j}$ be the $j$th row of $V$ for $j=1,\dots,n$. Then

$\displaystyle v_{j}C$
$\displaystyle=(1,\lambda_{j},\dots,\lambda_{j}^{n-1})C$
$\displaystyle=(\lambda_{j},\dots,\lambda_{j}^{n-1},a_{0}+a_{1}\lambda_{j}+\cdots+a_{n-1}\lambda_{j}^{n-1})$
$\displaystyle=(\lambda_{j},\dots,\lambda_{j}^{n})=\lambda_{j}v_{j}.$

Thus, $VC=DV$ with $D=\operatorname{diag}(\lambda_{1},\dots,\lambda_{n})$.
Consequently, $VCV^{-1}=D$.

The last assertion follows from the fact the $(i,j)$-entry of $VV^{-1}$ equals
$g_{j}(\lambda_{i})$.

There are explicit formulas for $V^{-1}$ in the literature [15, 10]. Here we
provide some insights about the equality $VCV^{-1}=D$ in terms of linear
transformations. Recall that $\mathcal{P}_{n-1}$ denotes the vector space of
polynomials over ${\mathbb{F}}$ with degree at most $n-1$. For a fixed monic
polynomial $f(x)=x^{n}-a_{n-1}x^{n-1}-\cdots-a_{0}\in\mathcal{P}_{n}$, define
the linear function $T\colon\mathcal{P}_{n-1}\rightarrow\mathcal{P}_{n-1}$
such that $T(1)=x,T(x)=x^{2},\dots,T(x^{n-2})=x^{n-1}$ and

$T(x^{n})=a_{0}+a_{1}x+\cdots+a_{n-1}x^{n-1}.$

Formally, $T$ has the form

(2.1) $p(x)\mapsto xp(x)\mod{f(x)}.$

######  Remark 2.2.

Let $T$ be the function defined in (2.1). Then the matrix representation
$[T]_{\mathcal{B}}^{\mathcal{B}}$ of $T$ with respect to ${\mathcal{B}}$ and
${\mathcal{B}}$ is the companion matrix $C$.

Next we turn our attention to the Vandermonde matrix. Fix a set of distinct
elements $\\{\lambda_{1},\ldots,\lambda_{n}\\}$ in ${\mathbb{F}}$. Define the
evaluation function $L\colon\mathcal{P}_{n-1}\rightarrow\mathbb{F}^{n}$ by

(2.2) $p(x)\mapsto\begin{pmatrix}p(\lambda_{1})\\\ \vdots\\\
p(\lambda_{n})\end{pmatrix}.$

######  Remark 2.3.

Let $L$ be the function defined in (2.2). Then the matrix representation
$[L]_{\mathcal{B}}^{\mathcal{E}}$ with respect to ${\mathcal{B}}$ and
$\mathcal{E}$ is the Vandermonde matrix $V$ of
$\\{\lambda_{1},\ldots,\lambda_{n}\\}$.

Since the Vandermonde matrix $V$ is invertible, the function $L$ is
invertible. The converse of $V^{-1}$ has connection with the Lagrange
interpolating polynomials as shown in Proposition2.1. In fact, the Lagrange
interpolating polynomials

$g_{1}(x)=\prod_{\begin{subarray}{c}1\leq j\leq n\\\ j\neq
1\end{subarray}}\frac{(x-\lambda_{j})}{(\lambda_{1}-\lambda_{j})},\quad\cdots\quad,\quad
g_{n}(x)=\prod_{\begin{subarray}{c}1\leq j\leq n\\\ j\neq
n\end{subarray}}\frac{(x-\lambda_{j})}{(\lambda_{n}-\lambda_{j})}$

satisfy $L(g_{i}(x))=\mathbf{e}_{i}$ for $i=1,\ldots,n$.

######  Remark 2.4.

Let $L$ be the function defined in (2.2) and $V$ the corresponding Vandermonde
matrix. Then $L$ is invertible with
$L^{-1}\colon\mathbb{F}^{n}\rightarrow\mathcal{P}_{n-1}$ satisfying

(2.3) $\mathbf{e}_{i}\mapsto g_{i}(x)\text{ for }i=1,\ldots,n.$

Hence, the $i$th column of $[L^{-1}]_{\mathcal{E}}^{\mathcal{B}}$ is the
vector representation $[g_{i}(x)]_{\mathcal{B}}$ of $g_{i}(x)$ with respect to
${\mathcal{B}}$.

Note that if $p(x)=r(x)+q(x)f(x)$, then $p(\lambda_{i})=r(\lambda_{i})$ as
$f(\lambda_{i})=0$ for $i=1,\ldots,n$. Therefore, we have

$\displaystyle L\circ T\circ L^{-1}(\mathbf{e}_{i})$ $\displaystyle=L\circ
T(g_{i}(x))$ $\displaystyle=L(xg_{i}(x)\mod{f(x)})$
$\displaystyle=\begin{pmatrix}\lambda_{1}g_{i}(\lambda_{1})\\\ \vdots\\\
\lambda_{n}g_{i}(\lambda_{n})\end{pmatrix}=\lambda_{i}\mathbf{e}_{i}$

for $i=1,\ldots,n$. In summary, this explains that

$VCV^{-1}=[L]_{\mathcal{B}}^{\mathcal{E}}[T]_{\mathcal{B}}^{\mathcal{B}}[L^{-1}]_{\mathcal{E}}^{\mathcal{B}}=[L\circ
T\circ
L^{-1}]_{\mathcal{E}}^{\mathcal{E}}=\operatorname{diag}(\lambda_{1},\ldots,\lambda_{n}).$

##  3\. The general case

We have seen that each row of a Vandermonde matrix represent an evaluation of
the form $p(x)\mapsto p(\lambda_{i})$. Formally, let $\mathcal{P}_{n-1}^{*}$
be the dual space of $\mathcal{P}_{n-1}$, which contains all the linear
functions from $\mathcal{P}_{n-1}$ to $\mathbb{F}$. For example, the
evaluation $\Delta(\lambda_{i})$ defined by $p(x)\mapsto p(\lambda_{i})$ is an
element in $\mathcal{P}_{n-1}^{*}$, and its matrix representation with respect
to ${\mathcal{B}}$ and $\\{1\\}$ (a basis of $\mathbb{F}$) is

$[\Delta(\lambda_{i})]_{\mathcal{B}}=(1,\lambda_{i},\cdots,\lambda_{i}^{n-1}).$

For elements in $\mathcal{P}_{n-1}^{*}$, we always use $\\{1\\}$ as the basis
for $\mathbb{F}$ and write $[\cdot]_{\mathcal{B}}$ instead of
$[\cdot]_{\mathcal{B}}^{\\{1\\}}$. Moreover, the fact that the Vandermonde
matrix of distinct elements $\\{\lambda_{1},\ldots,\lambda_{n}\\}$ is
invertible leads to the conclusion that
$\\{\Delta(\lambda_{i}):i=1,\ldots,n\\}$ form a basis of
$\mathcal{P}_{n-1}^{*}$.

To define the generalized Vandermonde matrix so that the elements in
$\\{\lambda_{1},\ldots,\lambda_{n}\\}$ might be repeated, we need some
standard definitions and notations in abstract algebra. For any polynomial
$p(x)=b_{0}+b_{1}x+\cdots+b_{n}x^{n}\in{\mathbb{F}}[x]$, define the derivative
of $p(x)$ by

$p^{\prime}(x)=D[p(x)]=b_{1}+2b_{2}x+\cdots+nb_{n}x^{n-1},$

and define the higher derivatives $p^{(\ell)}(x)=D^{(\ell)}(f(x))$ by the
composition of the linear map $D$ for $\ell$ times for
$\ell\in\\{2,\dots,n\\}$. We also use the convention that $f(x)=f^{(0)}(x)$.
Here, for any positive integer $k$ and $\mu\in{\mathbb{F}}$,
$k\mu=\mu+\cdots+\mu$ for $k$ so many times.

######  Definition 3.1.

Let $n>s\geq 0$ be integers and $\lambda\in{\mathbb{F}}$. Define the vector

$\mathbf{v}_{\lambda,n}^{(s)}=\left(\underbrace{0,\ldots,0}_{s},{s\choose
s},{s+1\choose s}\lambda,\dots,{n-1\choose s}\lambda^{n-s-1}\right),$

where $n$ is often omitted when the context is clear, and we just write
$\mathbf{v}_{\lambda}^{(s)}=\mathbf{v}_{\lambda,n}^{(s)}$

Observe that for $i=1,\dots,n$,
$\mathbf{v}_{\lambda_{i}}^{(0)}=(1,\lambda_{i},\ldots,\lambda_{i}^{n-1})=[\Delta(\lambda_{i})]_{\mathcal{B}}$
is the rows of a Vandermonde matrix of $\\{\lambda_{1},\dots,\lambda_{n}\\}$.
Evidently, for ${\mathcal{B}}=\\{1,x,\dots,x^{n-1}\\}$, we have
$\mathbf{v}_{\lambda}^{(0)}[p(x)]_{\mathcal{B}}=p(\lambda)$ if
$p(x)=b_{0}+\cdots+b_{n-1}x^{n-1}\in\mathcal{P}_{n-1}$. More generally, we
need the following definitions to define and establish the properties of the
generalized Vandermonde matrix.

######  Definition 3.2.

Let $p(x)\in b_{0}+b_{1}x+\cdots+b_{n}x^{n}\in{\mathbb{F}}[x]$, $s$ be a
nonnegative integer with $s\leq n$, and $\lambda\in{\mathbb{F}}$. Define
$\frac{1}{s!}p^{(s)}(\lambda)\in{\mathbb{F}}$ by

$\frac{1}{s!}p^{(s)}(\lambda)=\underbrace{0+\cdots+0}_{s}+b_{s}\binom{s}{s}+b_{s+1}\binom{s+1}{s}\lambda+\cdots+b_{n-1}\binom{n-1}{s}\lambda^{n-s-1},$

and define $\frac{1}{s!}p^{(s)}(x)\in{\mathbb{F}}[x]$ by

$\frac{1}{s!}p^{(s)}(x)=\underbrace{0+\cdots+0}_{s}+b_{s}\binom{s}{s}+b_{s+1}\binom{s+1}{s}x+\cdots+b_{n-1}\binom{n-1}{s}x^{n-s-1}.$

On the right sides of the definitions of $\frac{1}{s!}p^{(s)}(\lambda)$ and
$\frac{1}{s!}p^{(s)}(x)$, the “binomial coefficients”
$\binom{n}{k}\in{\mathbb{F}}$ are defined inductively for $n\geq k\geq 0$ by
$\binom{n}{0}={n\choose n}=1\in{\mathbb{F}}$, and
$\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}$ for $1<k<n$. So,

$\binom{n}{k}=\overbrace{1+\cdots+1}^{\binom{n}{k}\text{ times}},$

and the definitions of $\frac{1}{s!}p^{(s)}(\lambda)$ and
$\frac{1}{s!}p^{(s)}(x)$ are meaningful even if $(s!)$ is a multiple of $q$,
where $q>0$ is the the characteristic of ${\mathbb{F}}$.

For example, when $n=4$ and ${\mathbb{F}}=\mathbb{Z}_{2}$, the field of two
elements, we have

$\begin{array}[]{rll}p(x)&=b_{0}\binom{0}{0}+b_{1}\binom{1}{0}x+b_{2}\binom{2}{0}x^{2}+b_{3}\binom{3}{0}x^{3}+b_{4}\binom{4}{0}x^{4}&=b_{0}+b_{1}x+b_{2}x^{2}+b_{3}x^{3}+b_{4}x^{4},\\\
\frac{1}{1!}p^{(1)}(x)&=0+b_{1}\binom{1}{1}+b_{2}\binom{2}{1}x+b_{3}\binom{3}{1}x^{2}+b_{4}\binom{4}{1}x^{3}&=0+b_{1}+0+b_{3}x^{2}+0,\\\
\frac{1}{2!}p^{(2)}(x)&=0+0+b_{2}\binom{2}{2}+b_{3}\binom{3}{2}x+b_{4}\binom{4}{2}x^{2}&=0+0+b_{2}+b_{3}x+0,\\\
\frac{1}{3!}p^{(3)}(x)&=0+0+0+b_{3}\binom{3}{3}+b_{4}\binom{4}{3}x&=0+0+0+b_{3}+0,\\\
\frac{1}{4!}p^{(4)}(x)&=0+0+0+0+b_{4}\binom{4}{4}&=0+0+0+0+b_{4}.\\\
\end{array}$

######  Proposition 3.3.

For any $p(x)\in\mathcal{P}_{n-1}$ and nonnegative integer $s$,

$\mathbf{v}_{\lambda}^{(s)}[p(x)]_{\mathcal{B}}=\frac{1}{s!}p^{(s)}(\lambda).$

######  Definition 3.4.

Let $f(x)=(x-\lambda_{1})^{m_{1}}\cdots(x-\lambda_{m})^{n_{m}}$ for distinct
elements $\lambda_{1},\dots,\lambda_{m}\in{\mathbb{F}}$ and positive integers
$n_{1},\dots,n_{m}$ with $n=n_{1}+\cdots+n_{m}$ elements in ${\mathbb{F}}$.
Define the generalized Vandermonde matrix of
$\\{\lambda_{1}^{(n_{1})},\ldots,\lambda_{m}^{(n_{m})}\\}$ by

(3.1)
$V(\lambda_{1}^{(n_{1})},\ldots,\lambda_{m}^{(n_{m})})=\begin{pmatrix}\mathbf{v}_{\lambda_{1}}^{(0)}\\\
\vdots\\\ \mathbf{v}_{\lambda_{1}}^{(n_{1}-1)}\\\ \vdots\\\
\mathbf{v}_{\lambda_{m}}^{(0)}\\\ \vdots\\\
\mathbf{v}_{\lambda_{m}}^{(n_{m}-1)}\\\ \end{pmatrix}.$

Evidently, if $n=m$ and $n_{1}=\cdots=n_{m}=1$, then the definition reduces to
the usual Vandermonde matrix.

######  Theorem 3.5.

Let $C\in M_{n}({\mathbb{F}})$ be the companion matrix of the polynomial
$f(x)=x^{n}-a_{n-1}x^{n-1}-\cdots-a_{0}$. Suppose $\lambda$ is a zero of
$f(x)$ of multiplicity $k$ so that
$f(\lambda)=f^{\prime}(\lambda)=\dots=f^{(k-1)}(\lambda)=0$. Let
$v_{\ell}=\mathbf{v}_{\lambda}^{(\ell)}=\left(\underbrace{0,\dots,0}_{\ell},{\ell\choose\ell},{\ell\choose\ell}\lambda,\dots,{n-1\choose\ell}\lambda^{n-\ell-1}\right)$
for $\ell=0,\dots,k-1$. Then

$v_{0}C=\lambda v_{0}\qquad\hbox{ and }\qquad v_{\ell}C=v_{\ell-1}+\lambda
v_{\ell}\quad\hbox{ for }\ell=1,\dots,k-1.$

Hence, if $V(\lambda)=\begin{pmatrix}v_{0}\cr\vdots\cr
v_{k-1}\cr\end{pmatrix}\in M_{k,n},$ then
$V(\lambda)C=J_{k}(\lambda)V(\lambda)$, where

$J_{k}(\lambda)=\lambda
I_{k}+\sum_{j=2}^{k}E_{j,j-1}=\begin{pmatrix}\lambda&&\cr
1&\ddots&\cr&\ddots&\ddots\cr&&1&\lambda\cr\end{pmatrix}\in
M_{k}({\mathbb{F}})$

is the lower triangular Jordan block of $\lambda$.

###### Proof.

Note that
$v_{0}C=(\lambda,\dots,\lambda^{n-1},a_{0}+a_{1}\lambda+\cdots+a_{n-1}\lambda^{n-1})=(\lambda,\dots,\lambda^{n})=\lambda
v_{0}$. For $v_{1}=(0,1,2\lambda,3\lambda^{2},\dots,(n-1)\lambda^{n-2})$,

$\displaystyle v_{1}C$ $\displaystyle=$
$\displaystyle(1,2\lambda,3\lambda,(n-1)\lambda^{n-2},a_{1}+2a_{2}\lambda+\cdots+(n-1)a_{n-1}\lambda^{n-2})$
$\displaystyle=$
$\displaystyle(1,2\lambda,3\lambda,(n-1)\lambda^{n-2},n\lambda^{n-1})\hskip
36.135pt(\hbox{as
}0=f^{\prime}(\lambda)=n\lambda^{n-1}-\sum_{j=1}^{n-1}ja_{j}\lambda^{j-1})$
$\displaystyle=$
$\displaystyle(1,\lambda,\dots,\lambda^{n-1})+\lambda(0,1,2\lambda,3\lambda^{2},\dots,(n-1)\lambda^{n-2})$
$\displaystyle=$ $\displaystyle v_{0}+\lambda v_{1}.$

This argument can be extended to show that $v_{\ell}C=\lambda
v_{\ell}+v_{\ell-1}$ for $\ell\in\\{2,\dots,k-1\\}$. To do this, we need the
identity

(3.2)
$0={n\choose\ell}\lambda^{n-\ell}-\sum_{j=\ell}^{n-1}{j\choose\ell}a_{j}\lambda^{j-\ell}.$

To prove (3.2), one may apply an induction argument on $\ell=0,1,\dots,k-1$,
to show that if $f(x)=(x-\lambda)^{\ell+1}g(x)$ with
$g(x)=x^{n-\ell}+g_{n-\ell-1}x^{n-\ell-1}+\cdots g_{0}$, then

$\frac{1}{\ell!}f^{(\ell)}(x)={n\choose\ell}x^{n-\ell}-\sum_{j=\ell}^{n-1}{j\choose\ell}a_{j}x^{j-\ell}\in{\mathbb{F}}[x]$

has the form $(x-\lambda)h(x)$ for some $h(x)\in{\mathbb{F}}[x]$. Thus, (3.2)
holds. Now, by (3.2), we have

$\displaystyle v_{\ell}C$ $\displaystyle=$
$\displaystyle\left(\underbrace{0,\dots,0}_{\ell-1},{\ell\choose\ell},{\ell+1\choose\ell}\lambda,{\ell+2\choose\ell}\lambda^{2},\dots,{n-1\choose\ell}\lambda^{n-\ell-1},\sum_{j=\ell}^{n-1}{j\choose\ell}a_{j}\lambda^{j-\ell}\right)$
$\displaystyle=$
$\displaystyle\left(\underbrace{0,\dots,0}_{\ell-1},{\ell\choose\ell},{\ell+1\choose\ell}\lambda,{\ell+2\choose\ell}\lambda^{2},\dots,{n-1\choose\ell}\lambda^{n-\ell-1},{n\choose\ell}\lambda^{n-\ell}\right).$

Since ${\ell-1\choose\ell-1}={\ell\choose\ell}=1\in{\mathbb{F}}$ and
${p\choose\ell}\lambda={p-1\choose\ell-1}\lambda+{p-1\choose\ell}\lambda$ for
any nonnegative integers $p>\ell$, we have

$\displaystyle v_{\ell}C$ $\displaystyle=$
$\displaystyle\left(\underbrace{0,\dots,0}_{\ell-1},{\ell-1\choose\ell-1},{\ell\choose\ell-1}\lambda,{\ell+1\choose\ell-1}\lambda^{2},\dots,{n-1\choose\ell-1}\lambda^{n-\ell}\right)$
$\displaystyle\hskip 50.58878pt+\
\left(\underbrace{0,\dots,0}_{\ell},{\ell\choose\ell}\lambda,{\ell+1\choose\ell}\lambda^{2},\dots,{n-1\choose\ell}\lambda^{n-\ell}\right)$
$\displaystyle=$ $\displaystyle v_{\ell-1}+\lambda v_{\ell}.$

The last assertion is clear.

By Theorem 3.5, we have the following.

######  Corollary 3.6.

Let $C\in M_{n}({\mathbb{F}})$ be the companion matrix of the polynomial
$f(x)=(x-\lambda)^{n}\in{\mathbb{F}}[x]$. Suppose $V=V(\lambda^{(n)})\in
M_{n}({\mathbb{F}})$ is the generalized Vandermonde matrix of defined as in
(3.1). Then $VCV^{-1}=J_{n}(\lambda)=\lambda I_{n}+\sum_{j=2}^{n}E_{j,j-1}$ is
a lower triangular Jordan block of $\lambda$.

######  Remark 3.7.

Let $f(x)=x^{n}-a_{n-1}x^{n-1}-\cdots-a_{0}$ and $C$ its companion matrix. Let
$\lambda\in{\mathbb{F}}$ be a zero of $f(x)$ of degree $k$. Then Theorem 3.5
essentially says
$\mathbf{v}_{\lambda}^{(0)}C=\lambda\mathbf{v}_{\lambda}^{(0)}$ and
$\mathbf{v}_{\lambda}^{(s)}C=\mathbf{v}_{\lambda}^{(s-1)}+\lambda\mathbf{v}_{\lambda}^{(s)}$
for $s=2,\ldots,k-1$. If we let
$\frac{1}{s!}\Delta^{(s)}(\lambda)\in\mathcal{P}_{n-1}^{*}$ be defined by
$p(x)\mapsto\frac{1}{s!}p^{(s)}(\lambda)$ and

$[\frac{1}{s!}\Delta^{(s)}(\lambda)]_{\mathcal{B}}=\mathbf{v}_{\lambda}^{(s)}=\left(\underbrace{0,\ldots,0}_{s},{s\choose
s},{s+1\choose s}\lambda,\dots,{n-1\choose s}\lambda^{n-s-1}\right)$

by Proposition 3.3, then Theorem 3.5 also follows from the fact that for any
$p(x)=b_{0}+b_{1}x+\cdots+b_{n-1}x^{n-1}\in\mathcal{P}_{n-1}$,

$\mathbf{v}_{\lambda}^{(s)}C=[\frac{1}{s!}\Delta^{(s)}(\lambda)\circ
T]_{\mathcal{B}}$

and

$\displaystyle\frac{1}{s!}\Delta^{(s)}(\lambda)\circ T(p(x))$
$\displaystyle=\frac{1}{s!}\Delta^{(s)}(\lambda)(xp(x)\mod{f(x)})$
$\displaystyle=\frac{1}{s!}D^{(s)}(xp(x))|_{x=\lambda}$
$\displaystyle=\lambda\frac{1}{s!}p^{(s)}(\lambda)+\frac{1}{(s-1)!}p^{(s-1)}(\lambda)$
$\displaystyle=\left[\lambda\Delta^{(s)}(\lambda)+\Delta^{(s-1)}(\lambda)\right](p(x)),$

where $\frac{1}{s!}p^{(s)}(\lambda),\frac{1}{(s-1)!}p^{(s-1)}(\lambda)$, and
$\frac{1}{s!}D^{(s)}(xp(x))$ are defined as in Definition 3.2.

In general, we have the following.

######  Theorem 3.8.

Suppose $f(x)=x^{n}-a_{n-1}x^{n-1}-\cdots-
a_{0}=(x-\lambda_{1})^{n_{1}}\cdots(x-\lambda_{m})^{n_{m}}$ and $C\in
M_{n}({\mathbb{F}})$ is the companion matrix of $f(x)$ such that
$f(x)=\det(xI-C)=(x-\lambda_{1})^{n_{1}}\cdots(x-\lambda_{m})^{n_{m}}$ for $m$
distinct elements $\lambda_{1},\dots,\lambda_{m}\in{\mathbb{F}}$ and positive
integers $n_{1},\dots,n_{m}$, summing up to $n$. Let
$V=V(\lambda_{1}^{(n_{1})},\ldots,\lambda_{m}^{(n_{m})})$ be the generalized
Vandermonde matrix of defined as in (3.1). Then $V$ is invertible with

$\det(V)=\prod_{1\leq i<j\leq
m}(\lambda_{j}-\lambda_{i})^{n_{j}n_{i}}\quad\hbox{ and }\quad
VCV^{-1}=J_{n_{1}}(\lambda_{1})\oplus\cdots\oplus J_{n_{m}}(\lambda_{m}).$

Moreover, if $V^{-1}=(c_{ij})$ and
$h_{j}(x)=c_{1j}+c_{2j}x+\cdots+c_{nj}x^{n-1}$ for $j=1,\dots,n$, then
$h_{j}(x)$ is the unique Hermite polynomial satisfying

$h_{j}^{(k)}(\lambda_{i})=\begin{cases}1&\hbox{ if }(i,k)=(q,\ell-1)\hbox{
such that }j=n_{1}+\cdots+n_{q-1}+\ell\hbox{ with }\ell\leq n_{q},\\\
0&\hbox{otherwise}.\end{cases}$

###### Proof.

By Theorem 3.5, we see that $VC=JV$ if
$J=J_{n_{1}}(\lambda_{1})\oplus\cdots\oplus J_{n_{m}}(\lambda_{m})$. We will
show that $V$ is invertible by a row and column reduction argument so that
$VCV^{-1}=J$.

Let $R=I_{n}-\lambda_{1}\sum_{j=1}^{n-1}E_{j,j+1}$. Suppose
$VR=\begin{pmatrix}F_{1}\cr\vdots\cr F_{m}\cr\end{pmatrix}$ so that
$F_{j}=V(\lambda_{j})R$ is $n_{j}\times n$. If $c_{j}(X)$ denote the $j$th
column of $X\in M_{n}$. Then $c_{1}(VR)=c_{1}(V)$ and
$c_{\ell}(VR)=c_{\ell}(V)-\lambda_{1}c_{\ell-1}(V)$ for $\ell=2,\dots,n$.

The first row of $VR=(1,0,\dots,0)$. Delete the first row and the first column
of $VR$ to obtain
$\tilde{F}=\begin{pmatrix}\tilde{F}_{1}\cr\vdots\cr\tilde{F}_{m}\cr\end{pmatrix}\in
M_{n-1}({\mathbb{F}})$ so that $\tilde{F}_{1}$ is $(n_{1}-1)\times(n-1)$, and
$\tilde{F}_{j}$ is $n_{j}\times(n-1)$ for $j\in\\{2,\ldots,m\\}$. We have

$\det(V)=\det(VR)=\det(\tilde{F}).$

Now, $\tilde{F}_{1}$ is obtained by deleting the first row and first column of
$F_{1}=V(\lambda_{1})R$ and has rows

$(1,\lambda_{1},\dots,\lambda_{1}^{n-2}),(0,1,2\lambda_{1},\dots,(n-2)\lambda_{1}^{n-3}),\left(0,0,{2\choose
2},{3\choose 2}\lambda_{1},\dots,{n-2\choose
2}\lambda_{1}^{n-4}\right),\dots,$

so that $\tilde{F}_{1}$ can be obtained from $F_{1}$ by deleting its last row
and last column.

For $j\in\\{2,\dots,m\\}$, we will apply invertible row operations to
$\tilde{F}_{j}$ to get $G_{j}$ so that the matrix
$\tilde{V}=\begin{pmatrix}\tilde{F}_{1}\cr G_{2}\cr\vdots\cr
G_{m}\cr\end{pmatrix}\in M_{n-1}({\mathbb{F}})$ has the same form as $V$. By
an induction argument, one sees that $\tilde{V}$ is invertible. Thus,
$\tilde{F}$ is invertible and so is $V$.

Let $j\in\\{2,\dots,m\\}$. The matrix $\tilde{F}_{j}$ is obtained from
$V(\lambda_{j})R$ by deleting its first column. Thus, the first row of
$\tilde{F}_{j}$ has the form

$(\lambda_{j}-\lambda_{1},\lambda_{j}^{2}-\lambda_{j}\lambda_{1},\lambda_{j}^{3}-\lambda_{j}^{2}\lambda_{1},\dots,\lambda_{j}^{n-1}-\lambda_{j}^{n-2}\lambda_{1}).$

Multiply the first row of $\tilde{F}_{j}$ by $1/(\lambda_{j}-\lambda_{1})\neq
0$ so that the first row becomes $(1,\lambda_{j},\dots,\lambda_{j}^{n-2}).$

If $n_{j}\geq 2$, subtract the new first row from the second row of the
$\tilde{F}_{j}$, which has the form

$(1,2\lambda_{j}-\lambda_{1},3\lambda_{j}^{2}-2\lambda_{j}\lambda_{1},4\lambda_{j}^{3}-3\lambda_{j}^{2}\lambda_{1},\dots,(n-1)\lambda_{j}^{n-2}-(n-2)\lambda_{j}^{n-3}\lambda_{1}),$

to obtain

$(0,\lambda_{j}-\lambda_{1},2\lambda_{j}^{2}-2\lambda_{j}\lambda_{1},3\lambda_{j}^{3}-3\lambda_{j}^{2}\lambda_{1},\dots,(n-2)\lambda_{j}^{n-2}-(n-2)\lambda_{j}^{n-3}\lambda_{1}).$

Then multiply the resulting row by $(\lambda_{j}-\lambda_{1})^{-1}$. The
second row will become

$(0,1,2\lambda_{j},3\lambda_{j},\dots,(n-2)\lambda_{j}^{n-3}).$

If $n_{j}\geq 3$, subtract the new second row from the third row of
$\tilde{F}_{j}$, which has the form

$\left(0,{2\choose 2},{3\choose 2}\lambda_{j},\dots,{n-1\choose
2}\lambda_{j}^{n-3}\right)-\lambda_{1}\left(0,0,{2\choose 2},{3\choose
2}\lambda_{j},\dots,{n-2\choose 2}\lambda_{j}^{n-4}\right),$

to obtain

$\left(0,0,{2\choose 2}\lambda_{j},\dots,{n-2\choose
2}\lambda_{j}^{n-3}\right)-\lambda_{1}\left(0,0,{2\choose 2},{3\choose
2}\lambda_{j},\dots,{n-2\choose 2}\lambda_{j}^{n-4}\right).$

Then multiply the resulting row by $1/(\lambda_{j}-\lambda_{1})$. The third
row will become

$\left(0,0,{2\choose 2},{3\choose 2}\lambda_{j},{4\choose
2}\lambda_{j}^{2},\dots,{n-2\choose 2}\lambda_{j}^{n-4}\right).$

Now, subtract this row from the fourth row of $\tilde{F}_{j}$ if $n_{j}\geq
4$, and then multiply the row by $(\lambda_{j}-\lambda_{1})^{-1}$. The $j$th
row will become

$\left(0,0,0,{3\choose 3}\lambda_{j},{4\choose
3}\lambda_{j}^{2},\dots,{n-2\choose 3}\lambda_{j}^{n-5}\right).$

We can repeat this reduction process till the last row of $\tilde{F}_{j}$, and
let the resulting matrix be $G_{j}$. Then
$\tilde{V}=\begin{pmatrix}\tilde{F}_{1}\cr G_{2}\cr\vdots\cr
G_{m}\cr\end{pmatrix}\in M_{n-1}({\mathbb{F}})$ will be the generalized
Vandermonde matrix of
$\\{\lambda_{1}^{(n_{1}-1)},\ldots,\lambda_{m}^{(n_{m})}\\}$ So, $V$ is
invertible, and $VCV^{-1}=J$.

Keeping track on the row reductions used to change $\tilde{F}$ to $\tilde{V}$,
we see that

$\det(V)=\det(\tilde{F})=\prod_{j=2}^{m}(\lambda_{j}-\lambda_{1})^{n_{j}}\det(\tilde{V}).$

An easy induction argument shows that $\det(V)=\prod_{1\leq i<j\leq
m}(\lambda_{j}-\lambda_{i})^{n_{j}n_{i}}$ as asserted.

Finally, let $V^{-1}=(c_{ij})$, and let

$h_{j}(x)=c_{j1}+c_{j2}x+\cdots+c_{jn}x^{n-1}\quad\hbox{ for }j=1,\dots,n.$

It is easy to check that for $1\leq j\leq n_{1}$, the $j$th column of
$VV^{-1}=I$ equals ${\mathbf{e}}_{j}$ if and only if

$\frac{1}{\ell!}h^{(j-1)}(\lambda_{1})=1$, $h_{j}^{(\ell)}(\lambda_{1})=0$ for
$\ell\in\\{0,\dots,n_{1}-1\\}\setminus\\{j-1\\},$

and $h_{j}^{(\ell)}(\lambda_{i})=0$ for $i\in\\{1,\ldots,m\\}\setminus\\{1\\}$
and $0\leq\ell\leq n_{i}-1$.

Replacing $\lambda_{1}$ by $\lambda_{i}$ for other $i$, we get a unique
polynomial $h_{j}(x)$ for each column. Hence, $h_{j}(x)$ is the unieque
Hermite polynomial satisfying the specified conditions.

Clearly, if $m=n$ and $n_{1}=\cdots=n_{m}=1$, then Theorem 3.8 reduces to
Proposition 2.1. In particular, $\det(V)$ reduces to the well-known
determinant formula for the Vandermonde matrix [16], and the Hermite
polynomials reduce to the Lagrange polynomials. For the construction of
Hermite polynomials, one may see [1, Chapter 3] and also [11, 13].

Acknowledgment

The authors would like to thank Prof. Ngai-Ching Wong from some helpful
comments. Research of Li was supported by the Simons Foundation Grant 851334.
The research started while Li was visiting the National Sun Yat-sen University
in January, 2024, supported by the Mathematics Research Promotion Center,
National Science and Technology Council, Taiwan.

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(Li) Department of Mathematics, College of William & Mary, Williamsburg, VA
23187, USA.

<EMAIL_ADDRESS>
(Lin) Department of Applied Mathematics, National Sun Yat-sen University,
Kaohsiung 80424, Taiwan.

<EMAIL_ADDRESS>