# Observation of non-linear magnetic vortex gyration by X-ray microscopy and
micromagnetic simulations

A. Vansteenkiste1, B. Van de Wiele2, M. Weigand3, H. Stoll3, K.W. Chou4, T.
Tyliszczak4, G. Woltersdorf5, C. H. Back5, G. Schütz3, B. Van Waeyenberge 1 1
Department of Solid State Sciences, Ghent University, Kortrijksesteenweg
281-S1, 9000 Gent, Belgium.
2 Department of Electrical Energy, Systems and Automation, Ghent University,
Sint Pietersnieuwstraat 41, B-9000 Ghent, Belgium.
3 Max-Planck-Institut für Intelligente Systeme (ehemals Max-Planck-Institut
für Metallforschung), Heisenbergstr. 3, 70569 Stuttgart
4 Advanced Light Source, LBNL, 94720 Berkeley, CA, USA.
5 Institut für Experimentelle und Angewandte Physik, Universität Regensburg,
93040 Regensburg, Germany.

###### Abstract

Magnetic vortex gyration in a 500 nm wide, 50 nm thick Permalloy square
platelet was investigated using time-resolved scanning transmission x-ray
microscopy and micromagnetic simulations. The response of the vortex core on
an in-plane oscillating magnetic field was studied as a function of the
excitation frequency and amplitude. Non-linear behavior was observed in the
form of a _redshift_ of the gyrotropic resonance. I.e., when the excitation
amplitude was increased, the resonance frequency was found to decrease. The
nonlinearity is in agreement with extensive micromagnetic simulations. This
work complements previous reports on thinner nanostructures where a frequency
_blueshift_ was observed.

###### pacs:

75.75.+a, 76.50.+g, 07.85.Tt

##  1 Introduction

A magnetic vortex configuration occurs in, e.g., disk- and square-shaped soft
magnetic nanostructures of suited size [1, 2]. This configuration exhibits an
in-plane curling magnetization which forms a flux-closure (see Fig. 7). In the
case of a square platelet structure, four domains separated by 90∘ Néel walls
are formed, while no domain walls occur in disk-shaped structures. At the
center of the vortex, an out-of-plane magnetized vortex core appears. The
magnetization of the core has two possible directions (“up” or “down”), which
could enable vortices to be used as magnetic memory bits [3].

The vortex has a characteristic excitation mode: the vortex gyration mode [4].
This mode corresponds to a circular motion of the vortex core around the
center of the structure. It can be excited with, e.g., an in-plane alternating
[5] or rotating [6] magnetic field, by magnetic field pulses [7] or directly
by an electrical current [8]. The gyrotropic mode currently receives
considerable attention, since sufficient excitation causes the vortex core
polarization to switch its direction [3]. This happens whenever the motion of
the vortex core is accelerated beyond a threshold velocity in the order of 300
m/s [9, 10], independent of the excitation type. Alternating magnetic fields
with a frequency close to the gyrotropic frequency require only modest
amplitudes of the order of one mT to cause vortex core switching [3, 6]. A
precise determination of this frequency is thus a prerequisite to low-power
vortex core memories.

Subjected to a small excitation, the vortex gyration behaves linearly: the
dynamics can be well described by a two-dimensional harmonic oscillator model
[11, 12, 13, 14]. Such theoretical models describe the vortex core as a rigid
quasi particle that moves in a harmonic potential. The resulting gyration
radius as a function of the excitation frequency is a typical damped harmonic
oscillator curve, showing a single peak at the eigenfrequency which is
independent of the excitation amplitude.

At large excitations, on the other hand, these harmonic models break down and
the gyration dynamics show nonlinear behavior. Argyle was the first to
investigate the gyrotropic resonance and already noticed such nonlinearity in
1984 [4]. His group investigated a 15 $\mu$m wide garnet structure. The
optical Cotton-Mouton effect was used to image the displacement of the domain
walls while the vortex was excited by in-plane alternating fields. When the
excitation frequency was scanned, a peak response was found at the gyrotropic
eigenfrequency. When the same scan was repeated at a higher excitation
amplitude however, the resonance peak had moved towards a lower frequency and
also became broader. This frequency “redshift” was attributed to nonlinear
dynamics at large excitation amplitudes, without further investigation. More
recently, Ferromagnetic Resonance (FMR) techniques have been employed [15]. In
such studies, an ensemble of presumably identical nanostructures is positioned
on top of an electrical strip line through which an alternating current is
sent. This induces an in-plane alternating magnetic field in the radio
frequency range, which excites the gyrotropic mode. The resonance is then
studied by measuring the microwave absorption as a function of the excitation
frequency and amplitude. However, such experiments can not determine whether
vortex core switching occurs, making the interpretation difficult.

In this work, a more direct study of the frequency- and field dependence of
the gyrotropic resonance in Permalloy nanostructures is presented.
Experimental investigation with time-resolved magnetic Scanning Transmission
X-ray Microscopy (STXM) allowed for a single nanostructure, rather than an
array, to be investigated. The dynamics of the vortex core was imaged
directly, allowing its position and velocity to be determined. As vortex core
switching could be detected, the effects of switching could clearly be
separated from the non-linearities in the dynamics.

Additionally, micromagnetic simulations were carried out on Graphical
Processing Units using the in-house developed MuMax software [16]. The speedup
delivered by this hardware allowed an elaborate scan of the
frequency/amplitude parameter space. Previous micromagnetic studies like [17]
included only a relatively limited number of simulations.

Finally, previous works [17, 18] focused on relatively thin nanostructures, in
which case the vortex core can easily approach the nanostructure boundaries.
This was found to lead to a frequency _blueshift_ at high excitation
amplitudes — in contrast to the _redshift_ found in the thicker nanostructures
studied here.

##  2 Experiment

###  2.1 Experimental method

One element of an array of nominally 500 nm$\times$ 500 nm $\times$ 50 nm
square nanostructures, patterned in an evaporated Permalloy film was
investigated with the Scanning Transmission X-ray Microscope at the Advanced
Light Source (ALS, beamline 11.0.2) [19]. By orienting the sample plane
perpendicular to the photon beam and using the X-ray magnetic circular
dichroism (XMCD) effect [20], the out-of-plane component of the magnetization
was measured. This allowed for the vortex core to be directly imaged and its
polarization to be determined [5].

The nanostructures were positioned on top of a 2.5 $\mu$m wide, 150 nm thick
Cu strip line. To excite the gyrotropic mode, an in-plane alternating magnetic
field $B(t)=B_{0}\sin\left(2\pi ft\right)$, was induced by sending an
alternating current through the strip line. By taking advantage of the short,
intense X-ray flashes provided by the synchrotron, stroboscopic images of the
vortex core during a steady-state gyration could be recorded. In this
experiment, the synchronization requirements between the acquisition and the
synchrotron ring allowed only excitation frequencies $f=n\times$ 20 MHz with
$n$ no multiple of 5. Per period of the excitation, 25 images were acquired.
In order to collect sufficient photon counts, the responses of many gyration
periods were summed up, and consequently only a steady-state motion could be
recorded. Non-magnetic and non-dynamic contributions to the contrast were
removed by subtracting the average of all frames. The resulting images show
the gyrating vortex core as a white or black spot, depending on its
polarization (see Fig. 1).

Since the vortex core was imaged directly, vortex core switching could be
detected. When the magnetic excitation amplitude is sufficiently increased,
the vortex core starts toggling back and forth between the two possible
polarizations which causes the recorded contrast in the experiment to vanish
entirely (as only steady-state dynamics can be recorded). In such a case, the
switching could be confirmed by again lowering the excitation amplitude and
determining the core polarization. In about half of the cases, the
polarization was found to have switched, which can be expected when the core
has been constantly toggling back and forth for a long time [9]. Also, each
time the vortex core had switched, a short magnetic pulse was employed to
switch it back to its original polarization [7] before the experiment was
continued. In this way, all data was recorded for a vortex core pointing up,
and possible effects due to the small asymmetry between the core polarizations
[5, 6, 21] were excluded.

We have excluded the possibility that the observed changes in gyrotropic
frequency (see further) could be caused by sample heating due to the
electrical excitation. Sufficient heating could cause a frequency redshift due
to a corresponding reduction of the saturation magnetization. E.g., Kamionka
et al. found about a 5% decrease of resonance frequency when heating the
sample by 100∘C [22]. Therefore, we have determined the temperature of the
copper stripline by accurately measuring its electrical resistance while
sending different currents through. Even at the highest currents used in the
experiment, the temperature increase remained below 1∘C, so it can not be
responsible for the observed non-linearities. Furthermore, the micromagnetic
simulations presented in section 3 were carried out at $T=0$ and reproduce the
non-linearities.

###  2.2 Experimental results

Figure 1: Time-resolved STXM images of the vortex core gyration shown at five
phases of alternating magnetic fields. The excitation frequencies of 560 MHz
and 580 MHz are near the gyrotropic resonance frequency. The respective
gyration radii can easily be compared by from the rightmost images, which are
Fourier-filtered at a harmonic of the excitation frequency. Under the low-
field excitation of 0.48 mT, the gyration radius is larger for the 580 MHz
excitation. On the other hand, at the higher excitation of 1.19 mT, the
resonance has shifted downwards and the gyration radius is now larger at 560
MHz. The gyrotropic mode is thus anharmonic, as the resonance frequency
depends on the excitation strength.

First of all, the experiment shows qualitatively that the resonance frequency
decreases when the excitation amplitude is increased. This effect is
illustrated in Fig. 1. Five of the 25 recorded frames of the gyrating vortex
core are shown for different combinations of excitation frequency and
amplitude. The radius of gyration is highlighted in the Fourier-filtered
images. These show the magnitude of the image obtained by Fourier-filtering
the 25 time-dependent images at a harmonic of the excitation frequency. This
technique allows to easily observe the gyration radius with the bare eye, but
is not used in the further analysis. It can be observed that at the lowest
excitation amplitude the gyration radius is larger at 580 MHz, while at the
highest excitation amplitude the radius is larger at 540 MHz. The resonance
frequency thus shifts down while the field is increased. This frequency
“redshift” is a nonlinear effect that does appear in a perfect harmonic
oscillator.

Figure 2: Experimentally measured vortex core gyration velocity as a function
of excitation frequency and amplitude in a small region around the gyrotropic
resonance. The frequencies of maximal gyration are marked with a box,
indicating the resonance frequency for each amplitude. This shows that the the
gyrotropic resonance frequency shifts down as the amplitude of the excitation
field is increased. Above 1.5 mT, vortex core switching occurs (marked in
grey) at the resonance frequency. At even higher fields, switching also occurs
at frequencies further away from resonance.

For a more quantitative analysis, the vortex core position in each set of 25
recorded frames was determined by fitting two-dimensional Gaussian profiles to
the images. The subtraction of the average image was hereby taken into
account. From those positions, the average gyration radius and velocity during
one period was deduced. This procedure was repeated at four excitation
frequencies near the gyrotropic resonance frequency, as well as at several
different excitation amplitudes.

A full scan of excitation frequencies and amplitudes is shown in Fig. 2. Here,
the average vortex core velocity is shown. It was deduced by multiplying the
average gyration radius by the angular velocity of the core, which is fixed by
the excitation frequency $f_{0}$. For each magnetic excitation amplitude, the
frequency which lies closest to the resonance, and where the response is thus
maximal, has been marked with a box. It can again be observed that the
resonance shifts down as the field is increased.

At 1.5 mT and 540 MHz, the vortex core started toggling back and forth,
indicating that the critical switching velocity (about 300 m/s [23, 9]) was
reached. When the excitation amplitude was even further increased, the core
also switched at frequencies further away form the resonance. This results in
a _switching region_ with a typical triangle-like shape, marked in grey.
Although it was not yet identified as such, this kind of region is also
visible in the FMR spectra in [15].

##  3 Micromagnetic Simulations

###  3.1 Simulation Method

The experimental results were compared to micromagnetic simulations of a 500
nm$\times$ 500 nm $\times$ 50 nm Permalloy square structure (saturation
magnetization $M_{s}$=736 $\times$ 103 A/m (taken from [24]), exchange
constant $A$=13 $\times$ 10-12 J/m, damping parameter $\alpha$=0.01,
anisotropy constant $K_{1}$=0 and gyromagnetic ratio $\gamma$=2.211 $\times$
105 m/As). A 2D discretization with 3.9 $\times$ 3.9 nm2 finite difference
cells was used. The simulations were carried out with our in-house developed
MuMax micromagnetic simulator [16]. This program runs on Graphical Processing
Units (GPUs) rather than on standard CPUs, providing a computational speedup
of up to two orders of magnitude. This allowed a large parameter range to be
scanned with fine resolution. In total, 8000 separate field/amplitude
combinations were simulated. Each simulation started from the relaxed ground
state. The field was first applied for 20 periods to allow the system to
absorb energy and approach a stationary state. The next 20 periods of the
field were used for data-taking. The vortex core position and average velocity
were output during those 20 periods. We choose to start each simulation from
the ground state and not to, e.g., slowly scan the frequency or field. This
avoids possible effects due to a too high scan speed.

###  3.2 Simulation Results

Figure 3: Simulated frequency and field dependence of the vortex core gyration
velocity for a larger range than the experimental data in Fig. 2. The
simulated data qualitatively show the same behavior as the experimental data.
The simulations also reveal the triangular region in parameter space where
vortex core switching occurs. In this region the average core velocity is
lowered due to the continuous switching [9]. Experimentally, the velocity in
this region could not be determined.

The simulated gyration velocity as a function of excitation frequency and
amplitude is shown in Fig. 3. This simulation qualitatively agrees with the
experimental results: the frequency redshift at high excitations and the
typical shape of the switching region can here also be observed. The small
quantitative discrepancies with the experimental results are attributed to
uncertainties on the nanostructure size, thickness and material parameters. A
more detailed analyses is presented in the next section.

##  4 Detailed Analysis

###  4.1 Frequency redshift

First of all, we analyse frequency scans at a few fixed excitation amplitudes.
Fig. 1 already illustrated qualitatively that the resonance frequency drops as
the excitation amplitude is increased. This redshift is shown in more detail
in Fig. 4. Here, experimental and simulated frequency scans at two
representative excitation amplitudes are presented. One amplitude is chosen
well below the vortex core switching threshold, while the second amplitude
closely approaches it. It is clear that in both cases high amplitudes cause
the resonance peak to become broader and to shift to lower frequencies. The
simulated data also reveal that the resonance peak becomes asymmetric — a
typical property of anharmonic oscillators [15]. Fig. 5 shows the resonance
peaks for a wider range of excitation amplitudes. Here, the gradual
deformation and redshift of the resonance curve at increasingly higher
amplitudes can be clearly seen.

Figure 4: Experimental (left) and simulated (right) vortex core velocity $v$
as a function of the excitation frequency $f$ for a low and a high excitation
amplitude. The high excitation lies just below the vortex core switching
threshold. At high excitation, the peak of the response is clearly redshifted
towards a lower frequency. The detailed simulated curve also reveals that the
resonance peak shape becomes asymmetric.  Figure 5: Simulated vortex core
velocity as a function of the excitation frequency $f$ at various excitation
amplitudes until just below the vortex core switching threshold. At high
excitations, the resonance peak shifts towards lower frequencies and develops
an asymmetric shape.

###  4.2 Amplitude jumps

Figure 6: Experimental (left) and simulated (right) vortex core velocity $v$
as a function of the excitation amplitude $B$, for a frequency just below and
a frequency just above the respective resonances. At the lowest frequency, the
velocity increases suddenly around 1mT: while increasing the field amplitude,
the resonance redshifts towards the excitation frequency until the resonance
frequency suddenly is reached. The higher frequency lies already above the
resonance so the resonance peak redshifts away from the excitation frequency
as the amplitude increases. Hence, no sharp jump of the response occurs.

Secondly, we analyse amplitude scans at a few fixed frequencies. For a perfect
harmonic oscillator, the response (either velocity or displacement) should be
proportional to the excitation amplitude. Fig. 6 presents experimental and
simulated amplitude scans at two fixed frequencies, one slightly below and one
slightly above the resonance. At the lowest of these two frequencies, the
response exhibits a sharp jump around 1mT, which is clearly non-linear
behavior. This behavior can be explained by the redshift as follows: Let us
consider a frequency just below the resonance frequency at a relatively small
amplitude. By increasing the amplitude, not only the response naturally
becomes larger, but also the resonance frequency redshifts towards the
excitation frequency, causing an extra increase of the response. In contrary,
this does not happen when a frequency is considered above the (small-
amplitude) resonance frequency. In that case, increasing the amplitude causes
the resonance to shift away from the considered excitation and thus
counteracts the resonance increase.

##  5 Discussion

Until now, theoretical studies on thinner nanostructures predicted a
_blueshift_ of the gyrotropic frequency at high excitation amplitudes [17,
18]. In these studies, relatively thin nanostructures were considered, which
allows the vortex core to approach the edges of the magnet. A steepening of
the potential well in which the vortex core moves was attributed to the
expulsion of the core by the edge, hence the blueshift. In our case however,
the vortex core did not approach the edges of the system. The maximum gyration
radius was about 150 nm (see, e.g., Fig 1) — much smaller than the lateral
size of the magnet.

The redshift in this case can be explained by domain wall bending during the
vortex movement. Fig. 7 shows a simulated magnetization snapshot where such
bending occurs. It was taken at an excitation just below the threshold for
vortex core swichting — where there is a strong redshift. It can be seen that
the domain walls slightly bend so that the total dipole moment is reduced with
respect to straight domain walls. Hence the magnetostatic energy of the
potential well and thus the resonance frequency is reduced [14].

Nevertheless, this effect is not specific to square-shaped nanostructures.
Micromagnetic simulations on disks (not presented here) also revealed a
redshift. Although domain walls are absent in that case, the magnetization
adapted in a similar way, reducing the global dipole moment.

Figure 7: (a) Simulated magnetization snapshot at an excitation that leads to
gyrotropic redshift. The domain walls have slightly bended, reducing the
global dipole moment. As this lowers the dipolar energy, it weakens the
potential well in which the vortex core moves —hence decreasing the
eigenfrequency. (b) Exaggerated sketch for clarity.

##  6 Conclusion

We have performed a detailed analysis of the non-linearity of the magnetic
vortex gyration in a 50 nm thick, 500 nm wide Permalloy nanostructure. The
study has shown that a frequency _redshift_ occurs at high excitation
amplitudes. By using time-resolved magnetic STXM microscopy the gyrotropic
motion could be imaged directly, allowing for a straightforward interpretation
of the results. STXM also allows to detect the vortex core polarization and
switching. In this way, the region in frequency/amplitude space where the
vortex core switches could be determined. This was not possible in earlier FMR
studies like [15]. Extensive micromagnetic simulations were carried out on
Graphical Processing Units and support the experimental results.

## Acknowledgments

Financial support by The Institute for the promotion of Innovation by Science
and Technology in Flanders (IWT-Flanders) and by the Research Foundation
Flanders (FWO-Flanders) through the research grant 60170.06 are gratefully
acknowledged. The Advanced Light Source is supported by the Director, Office
of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy.
We would also like to cordially thank André Drews for the fruitful
discussions, including the explanation of the redshift by domain wall bending.

## References

## References

  * [1] J. Raabe, R. Pulwey, R. Sattler, T. Schweinbock, J. Zweck, and D. Weiss.  Magnetization pattern of ferromagnetic nanodisks.  J. Appl. Phys., 88:4437–4439, 2000.
  * [2] X. B. Zhu, Z. G. Liu, V. Metlushko, P. Grutter, and M. R. Freeman.  Broadband spin dynamics of the magnetic vortex state: Effect of the pulsed field direction.  Phys. Rev. B, 71:180408, 2005.
  * [3] B. Van Waeyenberge, A. Puzic, H. Stoll, K. W. Chou, T. Tyliszczak, R. Hertel, M. Fähnle, H. Bruckl, K. Rott, G. Reiss, I. Neudecker, D. Weiss, C. H. Back, and G. Schütz.  Magnetic vortex core reversal by excitation with short bursts of an alternating field.  Nature, 444:461–464, 2006.
  * [4] B. E. Argyle, E. Terrenzio, and J. C. Slonczewski.  Magnetic vortex dynamics using the optical Cotton-Mouton effect.  1984 Digests Of Intermag ’84. International Magnetics Conference (cat. No. 84ch1918-2), pages 350–350, 1984.
  * [5] K. W. Chou, A. Puzic, H. Stoll, D. Dolgos, G. Schütz, B. Van Waeyenberge, A. Vansteenkiste, T. Tyliszczak, G. Woltersdorf, and C. H. Back.  Direct observation of the vortex core magnetization and its dynamics.  Appl. Phys. Lett., 90:202505, 2007.
  * [6] M. Curcic, B. Van Waeyenberge, A. Vansteenkiste, M. Weigand, V. Sackmann, H. Stoll, H. Fahnle, T. Tyliszczak, G. Woltersdorf, C. H. Back, and G. Schütz.  Polarization selective magnetic vortex dynamics and core reversal in rotating magnetic fields.  Physical Review Letters, 101(19):197204, 2008.
  * [7] M. Weigand, B. Van Waeyenberge, A. Vansteenkiste, M. Curcic, V. Sackmann, H. Stoll, T. Tyliszczak, K. Kaznatcheev, D. Bertwistle, G. Woltersdorf, C. H. Back, and G. Schütz.  Vortex core switching by coherent excitation with single in-plane magnetic field pulses.  Physical Review Letters, 102(7):077201, February 2009.
  * [8] M. Bolte, G. Meier, B. Kruger, A. Drews, R. Eiselt, L. Bocklage, S. Bohlens, T. Tyliszczak, A. Vansteenkiste, B. Van Waeyenberge, K. W. Chou, A. Puzic, and H. Stoll.  Time-resolved x-ray microscopy of spin-torque-induced magnetic vortex gyration.  Phys. Rev. Lett., 100:176601, 2008.
  * [9] K. Yamada, S. Kasai, Y. Nakatani, K. Kobayashi, H. Kohno, A. Thiaville, and T. Ono.  Electrical switching of the vortex core in a magnetic disk.  Nat. Mater., 6:269–273, 2007.
  * [10] A. Vansteenkiste, K.W. Chou, M. Weigand, M. Curcic, V. Sackmann, H. Stoll, T. Tyliszczak, G. Woltersdorf, C.H. Back, G. Schütz, and B. Van Waeyenberge.  X-ray imaging of the dynamic magnetic vortex core deformation.  Nature Physics, 5:332–334, 2009.
  * [11] A. A. Thiele.  Steady-state motion of magnetic domains.  Phys. Rev. Lett., 30:230–233, 1973.
  * [12] D. L. Huber.  Equation of motion of a spin vortex in a two-dimensional planar magnet.  J. Appl. Phys., 53:1899–1900, 1982.
  * [13] V. Novosad, F. Y. Fradin, P. E. Roy, K. S. Buchanan, K. Y. Guslienko, and S. D. Bader.  Magnetic vortex resonance in patterned ferromagnetic dots.  Phys. Rev. B, 72:024455, 2005.
  * [14] B. Kruger, A. Drews, M. Bolte, U. Merkt, D. Pfannkuche, and G. Meier.  Harmonic oscillator model for current- and field-driven magnetic vortices.  Phys. Rev. B, 76:224426, 2007.
  * [15] K. S. Buchanan, M. Grimsditch, F. Y. Fradin, S. D. Bader, and V. Novosad.  Driven dynamic mode splitting of the magnetic vortex translational resonance.  Phys. Rev. Lett. (USA), 99:267201–1–267201–267201–4, 2007.
  * [16] A. Vansteenkiste and B. Van de Wiele.  Mumax: A new high-performance micromagnetic simulation tool.  Journal of Magnetism and Magnetic Materials, 323(21):2585–2591, November 2011.
  * [17] K. S. Lee and S. K. Kim.  Gyrotropic linear and nonlinear motions of a magnetic vortex in soft magnetic nanodots.  Applied Physics Letters, 91(13):132511, September 2007.
  * [18] K. Y. Guslienko, R. H. Heredero, and O. Chubykalo-Fesenko.  Nonlinear gyrotropic vortex dynamics in ferromagnetic dots.  Physical Review B, 82(1):014402, July 2010.
  * [19] ALD Kilcoyne, T Tyliszczak, WF Steele, S Fakra, P Hitchcock, K Franck, E Anderson, B Harteneck, EG Rightor, GE Mitchell, AP Hitchcock, L Yang, T Warwick, and H Ade.  Interferometer-controlled scanning transmission x-ray microscopes at the advanced light source.  Journal of Synchrotron Radiation, 10:125–136, 2003.
  * [20] G. Schütz, W. Wagner, W. Wilhelm, P. Kienle, R. Zeller, R. Frahm, and G. Materlik.  Absorption of circularly polarized x-rays in iron.  Phys. Rev. Lett., 58:737–740, 1987.
  * [21] A. Vansteenkiste, M. Weigand, M. Curcic, H. Stoll, G. Schütz, and B. Van Waeyenberge.  Chiral symmetry breaking of magnetic vortices by sample roughness.  New Journal Of Physics, 11:063006, 2009.
  * [22] Thomas Kamionka, Michael Martens, Andre Drews, Benjamin Krueger, Ole Albrecht, and Guido Meier.  Influence of temperature on the gyrotropic eigenmode of magnetic vortices.  Physical Review B, 83(22):224424, June 2011.
  * [23] K. Y. Guslienko, K. S. Lee, and S. K. Kim.  Dynamic origin of vortex core switching in soft magnetic nanodots.  Phys. Rev. Lett., 100:027203, 2008.
  * [24] A. Vansteenkiste, J. De Baerdemaeker, K. W. Chou, H. Stoll, M. Curcic, T. Tyliszczak, G. Woltersdorf, C. H. Back, G. Schütz, and B. Van Waeyenberge.  Influence of domain wall pinning on the dynamic behavior of magnetic vortex structures: Time-resolved scanning x-ray transmission microscopy in NiFe thin film structures.  Phys. Rev. B, 77:144420, 2008.


# On the Jacobian Conjecture and ideal membership for degree $d$-linear maps

Mario DeFranco

###### Abstract

We consider polynomial maps, which we call degree $d$-linear maps, that
satisfy the Jacobian condition. We prove that certain infinite families of
elements, which appear in the coefficients of the formal inverse of such maps,
are in the ideal determined by the Jacobian condition. Using the Cayley-
Hamilton theorem, we provide expressions for these elements in terms of the
generators of that ideal. We also give a combinatorial proof of the Cayley-
Hamilton theorem similar to that of Straubing [9]. We also include results of
Gröbner basis computations regarding other elements.

##  1 Introduction

The Jacobian Conjecture is a problem in algebraic geometry first stated by
Ott-Heinrich Keller in 1939 (see [8] and van den Essen [11]). It has several
formulations, and in this paper we consider the following.

For an integer $n\geq 1$, let $x=(x_{1},\dots,x_{n})\in\mathbb{C}^{n}$, define
a polynomial map to be a map of the form

$\displaystyle f\colon\mathbb{C}^{n}\rightarrow\mathbb{C}^{n}$
$\displaystyle(x_{1},\ldots,x_{n})\mapsto(f_{1}(x_{1},\ldots,x_{n}),\ldots,f_{n}(x_{1},\ldots,x_{n}))$

where each component function $f_{i}(x_{1},\ldots,x_{n})$ is a polynomials
function.

Jacobian Conjecture: Let $n\geq 2$. Suppose $f(x)$ is a polynomial map such
that

$\mathrm{Jac}(f)(x):=\det(D(f)(x))=1$ (1)

for all $x\in\mathbb{C}^{n}$, where $\mathrm{Jac}(f)$ is the Jacobian of
$f(x)$ and $D(f)(x)$ is the differential of $f(x)$. Then $f(x)$ has an inverse
function that is also a polynomial map.

Polynomial automorphisms appear in many topics in algebraic geometry and
dynamical systems (see van den Essen [11]). The Jacobian conjecture itself is
greatly generalized by the Generalized Vanishing Conjecture (see van den Essen
and Zhao [12]). A connection has also been proposed to quantum field theory
(see Abdesselam [1]).

One approach to the Jacobian Conjecure is to consider the formal inverse of
$f(x)$ (see Bass, Connell, Wright [2], Johnston [7], Singer [10], Wright
[14]). This is a priori an infinite power series in the indeterminates $x_{i}$
which may not converge on all of $\mathbb{C}^{n}$. If the condition (1)
implies that the coefficients of the monomials of the $x_{i}$ were zero for
all but finitely many monomials, then the formal series would truncate to a
polynomial, which would be the inverse function of $f(x)$, convergent
everywhere. The Jacobian conjecture thus claims that the vanishing of the non-
constant coefficients of the Jacobian (1) implies the vanishing of almost all
of the coefficients of the formal inverse.

In this paper we analyze the formal inverse of certain polynomial maps we call
degree $d$-linear maps:

######  Definition 1.

For integers $d,n\geq 1$, call a map $f$

$f\colon\mathbb{C}^{n}\rightarrow\mathbb{C}^{n}$

a degree $d$-linear map if the $i$-th component function is of the form

$f_{i}(x_{1},\ldots,x_{n})=x_{i}-(\sum_{j=1}^{n}a_{i,j}x_{r})^{d}$

for $a_{i,j}\in\mathbb{C}$.

To prove the Jacobian conjecture for all $n$, it is sufficient to prove it for
these maps for all $n$ when $d=3$ (Drużkowski [4]). In that case these maps
are called cubic-linear.

Now we describe the layout of this paper. We treat the $a_{i,j}$ as
indeterminates that generate a ring $R$. The non-constant coefficients of
$\mathrm{Jac}(f)(x)$ then generate an ideal $J_{d,n}$ of $R$. In Section 2 we
express each coefficient of the formal inverse as a sum of elements of $R$
over rooted plane trees. In Lemma 2, we show that portions of these sums
factorize using certain elements of $R$ which we denote by
$\mathrm{fern}_{d,n}$, taking this terminology from Johnston and Prochno [6].
In Theorem 1 we prove that certain infinite families of these elements are in
$J_{d,n}$, and we express them in the basis of the generators of that ideal.
We prove this formula by showing it is an instance of the Cayley-Hamilton
theorem applied to a certain matrix. We note that Abdesselam [1] interprets
the “$d$=1” Jacobian conjecture as the Cayley-Hamilton theorem. Thus Theorem 1
generalizes that result to all $d$. We give a combinatorial proof of the
Cayley-Hamilton theorem in Section 4 and note that it is very similar to that
of Straubing [9]. Finally in Section 5 we list some computations done with
Groebner bases to check more ideal membership questions. We specify certain
elements that are not in $J_{d,n}$ but are in its radical, and also consider
further nilpotent assumptions of Drużkowski [5].

##  2 Formal inverse

We express the formal inverse of a degree $d$-linear map using rooted plane
trees
(Lemma 2). First we present definitions.

######  Definition 2.

Let $a_{i,j},1\leq i,j\leq n$ be indeterminates that generate a polynomial
algebra $R$ over $\mathbb{Q}$. $R$ is a graded ring by total degree of a
monomial. Let $A$ denote the $n\times n$ matrix

$A_{i,j}=a_{i,j}.$

Monomials in the ring $R$ will be used as weights for rooted plane trees.

######  Definition 3.

A rooted plane tree $T$ is an acyclic graph with a distinguished vertex called
the root such that the children of each vertex are ordered, where a vertex $u$
of $T$ is a child of $v$ if $u$ is adjacent to $v$ and $v$ is on the path from
$u$ to the root. Let $V(T)$ denote the vertex set of $T$.

Define a labeled tree to be a pair $(T,\lambda)$ where $T$ is rooted plane
tree and $\lambda$ is a function

$\lambda\colon V(T)\rightarrow[1,n]$

from the vertex set of $T$ to the integers in $[1,n]$. Define a root-leaf
labeled tree to be a pair $(T,\mu)$ where $T$ is a rooted plane tree and $\mu$
is a function

$\mu\colon\\{\text{root and leaves of $T$}\\}\rightarrow[1,n].$

We say that a labeled tree agrees with a root-leaf labeled tree if they have
the same rooted plane tree and $\lambda=\mu$ when restricted to the root and
leaves. Define $L(T,\mu)$ to the set of all labeled trees that agree with the
root-leaf labeled tree $(T,\mu)$. For a labeled tree $(T,\lambda)$ and an edge
$e$ of $T$ going from parent vertex $v$ to child vertex $u$, define
$w(e,\lambda)\in R$ to be

$w(e,\lambda)=a_{\lambda(v),\lambda(u)}.$

Define

$w(T,\lambda)=\prod_{e\in T}w(e,\lambda),$

and for a root-leaf labeled tree $(T,\mu)$, define

$z(T,\mu)=\sum_{(T,\lambda)\in L(T,\mu)}w(T,\lambda)$

######  Lemma 1.

Suppose $f(x)$ is a degree $d$-linear map. Let $g(x)$ denote the formal
inverse of $f$, so as formal power series

$f_{i}(g_{1}(x_{1},\ldots,x_{n}),\ldots,g_{n}(x_{1},\ldots,x_{n}))=x_{i}.$

Then

$g_{i}(x_{1},\ldots,x_{n})=\sum_{(T,\lambda),\,\lambda(\mathrm{root}(T))=i}w(T)\prod_{v\in\mathrm{Leaves}(T)}x_{\lambda(v)}$
(2)

where the sum is over all labeled trees with root labeled $i$ and such that
every vertex of $T$ has degree $d$ or 0.

###### Proof.

Let $\tilde{g}(x_{1},\ldots,x_{n})$ denote the right side of (2). The term
$x_{i}$ appears in the series for $\tilde{g}(x_{1},\ldots,x_{n})$ because this
term corresponds to tree with one vertex, its root, labeled $i$, which is also
a leaf. Any other tree $T$ in this series has root of degree $d$ and is thus
determined by a $d$-tuple of the subtrees whose root is a child of the root of
$T$. Therefore the sum of all these other terms in this series has the
factored form

$(\sum_{j=1}^{n}a_{i,j}\tilde{g}_{j}(x_{1},\ldots,x_{n}))^{d},$

and so

$f_{i}(\tilde{g}(x_{1},\ldots,x_{n}))=x_{i}.$

This completes the proof. ∎

######  Definition 4.

Let the height of a tree $T$ denote the maximal number of edges in a path from
the root of $T$ to one of its leaf vertices. Denote the height by $h(T)$. If
$h(t)\geq n$, then define the leftmost $n$-path $LP_{n}(T)$ of $T$ to be the
leftmost path front the root that consists of $n$ edges. Given a labeling
$\lambda$ of $T$, let $K(T,\lambda)$ denote the set of all labelings
$\lambda^{\prime}$ such that $\lambda^{\prime}$ agrees with $\lambda$ on all
vertices except possibly on those in the interior of $LP_{n}(T)$.

We will consider certain trees we denote $\mathrm{fern}_{d,n}$ using the
terminology of Johnston and Prochno [6].

######  Definition 5.

Define $\mathrm{fern}_{d,n}$ to be a rooted plane tree that consists of $n$
vertices of degree $d$ or 0 such that each vertex of degree $d$ has at most
one child of degree $d$, and if such a child exists it is the leftmost of all
the children of its parent.

We therefore obtain a partition of the set of labelings of $T$ into
equivalence classes with each class consisting of those labelings that agree
on all vertices except possibly those on the interior of $LP_{n}(T)$.

######  Definition 6.

We say a rooted plane tree $T$ is $d$-regular if its vertices have degree
either $d$ or 0. For a rooted plane tree $T$ of height at least $n$, each of
the above mentioned equivalence classes determines a labeling
$\lambda_{\mathrm{fern}}$ of the tree $\mathrm{fern}_{d,n}$: the leaf vertices
of $\mathrm{fern}_{d,n}$ correspond to the vertices that are the children of
each vertex in $LP_{n}(T)$.

######  Lemma 2.

Let $T$ be a rooted plane tree with height at least $n$ and labeling
$\lambda$. Then

$\sum_{\eta\in
K(T,\lambda)}w(T,\eta)=z(\mathrm{fern}_{d,n},\lambda_{\mathrm{fern}})g$

for some $g\in R$ that depends on $\lambda$.

###### Proof.

This follows immediately from the construction. The element $g$ is

$g=\prod_{e}w(e,\lambda)$

where the product is over all edges of $T$ that are not adjacent to a vertex
in the interior of $LP_{n}(T)$ or to the root of $T$. ∎

##  3 Ideal membership

In this section we prove Theorem 1 that certain elements are in the ideal
determined by the Jacobian condition (1).

######  Definition 7.

Let $\mathrm{Jac}(f)$ denote the Jacobian of $f$:

$\mathrm{Jac}(f)=\det(D(f))$

where

$(D(f))_{i,j}=\frac{d}{dx_{j}}(f_{i}(x)).$

Now $\mathrm{Jac}(f)\in R[x_{1},\ldots,x_{n}]$. For a mult-degree $\alpha$, we
let $J_{\alpha}\in R$ be the coefficient of $x^{\alpha}$:

$\mathrm{Jac}(f)=\sum_{\alpha}J_{\alpha}x^{\alpha}.$

Suppose $d\geq 2$. Let $J$ denote the ideal generated by the $J_{\alpha}$ for
all $\alpha\neq\accentset{\rightharpoonup}{0}$, i.e. by all coefficients of
$\mathrm{Jac}(f)$ except for the constant term.

If $d=1$, then $\mathrm{Jac}(f)\in R$, and we let $J$ be the ideal generated
by all homogeneous components of $\mathrm{Jac}(f)$ except for the constant
term.

######  Lemma 3.

For integers $1\leq k,l\leq n$, let
$\alpha(k,l)=(\alpha_{1},\ldots,\alpha_{n})$ be a multi-degree such that
$\alpha_{l}=k(d-1)$ for some $1\leq k\leq n-1$, and $\alpha_{i}=0$ for all
other $i$. Then

$J_{\alpha(k,l)}=(-d)^{k}\sum_{(i_{1},\ldots,i_{k})}|A|_{(i_{1},\ldots,i_{k})}\prod_{r=1}^{k}a_{i_{r},l}^{d-1}$

where the sum is over all $k$-tuples $(i_{1},\ldots,i_{k})$ of distinct
integers in $[1,n]$ with $i_{r}<i_{r+1}$, and where
$|A|_{(i_{1},\ldots,i_{k})}$ denotes the principal minor of $A$ corresponding
to rows $i_{1},\ldots,i_{k}$.

###### Proof.

We have

$(D(f))_{i,j}=\delta_{i,j}-da_{i,j}(\sum_{r=1}^{n}a_{i,r}x_{r})^{d-1}$

where $\delta_{i,j}$ is the Kronecker delta. Apart from $\delta_{i,j}$, each
entry of $D(f)$ is thus homogeneous of degree $d-1$ in the variables $x_{i}$,
and in the expansion of the determinant each term of $x^{k(d-1)}$ is obtained
by choosing

$-da_{i,j}(\sum_{r=1}^{n}a_{i,r}x_{r})^{d-1}$ (3)

$k$ times and choosing 1 $n-k$ times in the main diagonal. The $k$ rows
$i_{1},\ldots,i_{k}$ from which a term (3) was chosen determine the rows of a
principal minor; the product of the coefficients $a_{i,j}$ outside the sum at
(3) summed over those rows yield the principal minor
$|A|_{(i_{1},\ldots,i_{k})}$, and taking the coefficient of $x_{l}$ from (3)
yields the product $\displaystyle\prod_{r=1}^{k}a_{i_{r},l}^{d-1}$. This
completes the proof. ∎

######  Definition 8.

For integers $i,j,l\in[1,n]$ not necessarily distinct, let $\mu(i,j,l)$ denote
the root-leaf labeling of $\mathrm{fern}_{d,n}$ such that the root is labeled
$i$, the vertex $v_{n}$ (the last vertex in the leftmost path) is labeled $j$,
and all other vertices are labeled $l$.

######  Lemma 4.

For integers $i,j,l\in[1,n]$,

$z(\mathrm{fern}_{d,n},\mu(i,j,l))=(B^{n})_{i,j}$

where $B$ is the $\times n$ matrix

$B_{i,j}=a_{i,j}a_{i,l}^{d-1}.$

###### Proof.

By construction

$z(\mathrm{fern}_{d,n},\mu(i,j,l))=\sum_{\lambda}\prod_{k=0}^{n-1}(a_{\lambda_{k},\lambda_{k+1}}a_{k,l}^{d-1})$

where the sum is over all $(n+1)$-tuples $\lambda$

$\lambda=(\lambda_{0},\ldots,\lambda_{n})$

where $\lambda_{0}=i$ and $\lambda_{n}=j$. But this is $(B^{n})_{i,j}$ by
definition. This completes the proof. ∎

######  Theorem 1.

For integers $i,j,l\in[1,n]$,

$z(\mathrm{fern}_{d,n},\mu(i,j,l))\in J_{d,n}$

and

$z(\mathrm{fern}_{d,n},\mu(i,j,l)=-\sum_{k=0}^{n-1}(A^{k})_{i,j}d^{-(n-k)}J_{\alpha(n-k,l)}.$

###### Proof.

Let $p_{B}(t)$ denote the characteristic polynomial of the matrix $B$ as in
Lemma 4:

$p_{B}(t)=\det(tI-B)=\sum_{k=0}^{n}c_{k}t^{k}$

where $c_{k}\in R$ and $c_{n}=1$. We claim that $c_{k}\in J_{d,n},0\leq k\leq
n-1$.

By expanding the determinant in the definition of the characteristic
polynomial we have

$c_{k}=(-1)^{n-k}\sum_{(i_{1},\ldots,i_{n-k})}|A|_{(i_{1},\ldots,i_{n-k})}\prod_{r=1}^{n-k}ka_{i_{r},l}^{d-1}$

with notation as in Lemma 3. By that lemma we thus have

$c_{k}=d^{-(n-k)}J_{\alpha(n-k,l)}.$

Thus $c_{k}\in J_{d,n}$ for $0\leq k\leq n-1$.

Now by the Cayley-Hamilton theorem $p_{B}(B)=0$, so

$\sum_{k=0}^{n}(B^{k})_{i,j}c_{k}=0.$

Since $c_{n}=1$, by Lemma 4 the above equation becomes

$z(\mathrm{fern}_{d,n},\mu(i,j,l)=-\sum_{k=0}^{n-1}(B^{k})_{i,j}c_{k}.$

This completes the proof. ∎

##  4 Combinatorial proof of Cayley-Hamilton theorem

Here we include a combinatorial proof of the Cayley-Hamilton Theorem which is
similar to that of Straubing [9]. We let $A$ denote the $n\times n$ matrix
whose entry $A_{i,j}$ is the indeterminate $a_{i,j}$.

######  Definition 9.

For a $k$-tuple $\lambda=(\lambda_{0},\ldots,\lambda_{k})$, we let
$C(\lambda)$ be the sub-string

$C(\lambda)=(\lambda_{l_{1}+1},\ldots,\lambda_{l_{2}})$

such that $\lambda_{l_{1}}=\lambda_{l_{2}},\lambda_{i}\neq\lambda_{j}$ for
$l_{1}<i<j$, if such a string exists. We say that $(l_{1},l_{2})$ are the
first-rep indices of $\lambda$.

######  Theorem 2.

Define elements $J_{i}\in R$ by

$\det(I-tA)=\sum_{i=0}^{n}J_{i}t^{i}.$

Then

$\sum_{i=0}^{n}J_{i}(A^{n-i})_{r,r}=0,$ (4)

and for $r\neq l$

$\sum_{i=0}^{n-1}J_{i}(A^{n-i})_{r,l}=0.$ (5)

###### Proof.

In the expansion of $(A^{n-i})_{u,v}$ we index each monomial by an
$(n-i+1)$-tuple $\lambda$ of integers in $[1,n]$

$\lambda=(\lambda_{0},\ldots,\lambda_{n-i})$

such that $\lambda_{0}=u$ and $\lambda_{n-i}=v$.

Each $J_{i}$ is a sum of terms indexed by a pair $(S,\sigma)$ where
$S=(s_{1},s_{2},\ldots,s_{i})$ is a subset of $[1,n]$, with $s_{j}<s_{j+1}$,
and $\sigma$ is a permutation of $S$. Thus

$J_{i}=\sum_{(S,\sigma)}(-1)^{|\sigma|}\prod_{j=1}^{i}a_{s_{j},\sigma(s_{j})}$

where $|\sigma|$ is the number of cycles of $\sigma$: the permutation $\sigma$
has signature

$\prod_{c\in\sigma}(-1)^{1+\text{length of }c}=(-1)^{i+\\#\text{cycles of
}\sigma}$

where the product is over the cycles of $\sigma$, and each factor of $t$
contributes a factor of $(-1)$.

We expand out each term in the sums (4) or (5) into a sum monomials in $R$ and
construct a sign-reversing involution $I$ on those monomials.

To a monomial in the $i$-th term we give the index

$(\lambda,S,\sigma)$

where $|S|=i$, $\lambda$ is empty if $i=n$, and $S$ and $\sigma$ are empty if
$i=0$.

Suppose $i=n$. This may only occur for the sum (4). Then $\lambda$ is empty.
Take the cycle $c$ of $\sigma$ that contains $r$:

$c=(c_{1},c_{2},\dots c_{m})$

where $c_{1}=r$. Let $\lambda^{\prime}$ be determined by $\lambda_{j}=c_{j+1}$
for $0\leq j\leq m-1$ and $\lambda_{m}=r$. Let $\sigma^{\prime}$ be the
permutation that results by removing the cycle $c$ from $\sigma$ and let
$S^{\prime}$ be the resulting set. Then $I$ maps

$(\lambda,S,\sigma)\mapsto(\lambda^{\prime},S^{\prime},\sigma^{\prime}).$

Now suppose $i\neq n$, so $\lambda$ is not empty. Let $h$ be the greatest
integer such that $\lambda_{h}\in S$.

If no such $h$ exists, then we claim $C(\lambda)$ exists. If
$\lambda_{0}=\lambda_{n-i}$, then $C(\lambda)$ exists because we have at least
one pair of indices $l_{1},l_{2}$ where $\lambda_{l_{1}}=\lambda_{l_{2}}$,
namely $l_{1}=0$ and $l_{2}=n-i$. If $\lambda_{0}\neq\lambda_{n-i}$ and
$C(\lambda)$ does not exist, then the $n-i+1$ integers in $\lambda$ are
distinct, and since $h$ does not exist by assumption, this set and $S$ (which
consists of $i$ distinct integers by construction) are disjoint. This
contradicts the fact that the integers are in $[1,n]$. This proves the claim.
Adjoin this cycle $C(\lambda)$ to $\sigma$ to obtain a permutation
$\sigma^{\prime}$ on the resulting set $S^{\prime}$, and remove
$C(\accentset{\rightharpoonup}{\lambda})$ from
$\accentset{\rightharpoonup}{\lambda}$ to obtain a labeling $\lambda^{\prime}$
on $P_{j}$. Then $I$ maps

$(\lambda,S,\sigma)\mapsto(\lambda^{\prime},S^{\prime},\sigma^{\prime}).$

Now suppose such an $h$ does exist and consider the cycle $c$ of $\sigma$

$c=(c_{1},\ldots,c_{m})$

where $c_{1}=\lambda_{h}$. If $C(\lambda)$ exists, then let $(l_{1},l_{2})$ be
the first-rep indices. If $C(\lambda)$ exists and $h>l_{1}$, or if
$C(\lambda)$ does not exist, then let $\lambda^{\prime}$ be

$\lambda^{\prime}=(\lambda_{0},\ldots,\lambda_{h},c_{2},\ldots,c_{m},\lambda_{h},\lambda_{h+1},\ldots,\lambda_{n-i}),$

and remove $c$ from $\sigma$ to obtain $\sigma^{\prime}$ and the resulting set
$S^{\prime}$. Then $I$ maps

$(\lambda,S,\sigma)\mapsto(\lambda^{\prime},S^{\prime},\sigma^{\prime}).$

Otherwise assume $h<l_{1}$ ($h$ cannot be equal to $l_{1}$, for then it would
also be equal to $l_{2}$, contradicting the assumption on the maximality of
$h$). Remove the substring $C(\lambda)$ from $\lambda$ to obtain
$\lambda^{\prime}$ and adjoin $C(\lambda)$ as a cycle to $\sigma$ to obtain
$\sigma^{\prime}$ and the resulting set $S^{\prime}$. Then $I$ maps

$(\lambda,S,\sigma)\mapsto(\lambda^{\prime},S^{\prime},\sigma^{\prime}).$

It is straightforward to check that $I$ is an involution. This completes the
proof. ∎

##  5 Gröbner basis computations

We consider other elements of the form

$z(\mathrm{fern}_{d,n},\mu)$

for certain $d,n$ and $\mu$. We have From Theorem 1

$z(\mathrm{fern}_{1,n},\mu)\in J_{1,n}$

for all $n$ and $\mu$. (Here $\mu$ is labeling of the root and the one leaf
vertex.) For the following $d$ and $n$ we have used Gröbner bases implemented
by Mathematica [13] and Singular [3] to determine membership in $J_{d,n}$ or
$\sqrt{J_{d,n}}$.

For $n=2$, we have

$z(\mathrm{fern}_{d,2},\mu)\in J_{1,2}\,\,\,d=2,3$

for all root-leaf labelings $\mu$.

We denote a root-leaf labeling of $\mathrm{fern}_{d,n}$ by

$(r,t_{0},\ldots,t_{n-2},t_{n-1})$

where $r$ is the label of the root, $t_{i}$ is a $(d-1)$-tuple of the labels
of the vertex $v_{i}$ on the leftmost path for $0\leq i\leq n-2$, and
$t_{n-1}$ is a $d$-tuple of the labels of the next-to-last vertex $v_{n-1}$ on
the leftmost path.

For $n=3$ and $2\leq d\leq 4$ we have

$z(\mathrm{fern}_{d,3},\mu)\in J_{1,3}$

for all for all root-leaf labelings $\mu$ except

$\displaystyle(1,(2),(3),(1,1))\text{ and }(1,(3),(2),(1,1))\notin J_{2,3}$
$\displaystyle(1,(2,2),(3,3,),(1,1,1))\text{ and
}(1,(3,3),(2,2),(1,1,1))\notin J_{3,3}$
$\displaystyle(1,(2,2,2),(3,3,,3),(1,1,1,1))\text{ and
}(1,(3,3,3),(2,2,2),(1,1,1,1))\notin J_{4,3}$

However we do have

$\displaystyle
z(\mathrm{fern}_{d,3},(1,(2),(3),(1,1)))+z(\mathrm{fern}_{d,3},(1,(3),(2),(1,1)))\in
J_{1,3}$ $\displaystyle
z(\mathrm{fern}_{d,3},(1,(2,2),(3,3),(1,1,1)))+z(\mathrm{fern}_{d,3},(1,(3,3),(2,2),(1,1,1)))\in
J_{3,3}$ $\displaystyle
z(\mathrm{fern}_{d,3},(1,(2,2,2),(3,3,3),(1,1,1,1)))+z(\mathrm{fern}_{d,3},(1,(3,3,3),(2,2,2),(1,1,1,1)))\in
J_{4,3}$

and

$\displaystyle z(\mathrm{fern}_{d,3},(1,(2),(3),(1,1)))^{2}\in J_{1,3}$
$\displaystyle z(\mathrm{fern}_{d,3},(1,(2,2),(3,3),(1,1,1)))^{2}\in J_{3,3}$
$\displaystyle z(\mathrm{fern}_{d,3},(1,(2,2,2),(3,3,3),(1,1,1,1)))^{2}\in
J_{4,3}$

If we assume that the matrix $a$ satisfies $A^{2}=0$, then let
$I_{\mathrm{nil-2}}$ denote the ideal of $R$ generated by elements of the form

$(A^{2})_{i,j}$

for $1\leq i,j\leq n$. Then

$\displaystyle z(\mathrm{fern}_{d,3},(1,(2),(3),(1,1)))\in J_{2,3}\cup
I_{\mathrm{nil-2}}$

Let $I_{\mathrm{char}}$ denote the non-constant coefficients of the
characteristic polynomial of $A$. Then

$\displaystyle z(\mathrm{fern}_{d,3},(1,(2),(3),(1,1)))\in J_{2,3}\cup
I_{\mathrm{char}}$

##  6 Further work

  * •

See if ideal membership results can be obtained for $n=2$ and all $d$.

  * •

Consider the maps $f(x)=x+H$ where the component functions of $H$ are linear
forms raised to different degrees.

  * •

See if there are corresponding results for the different differential
operators of the General Vanishing Conjecture.

  * •

Consider ideal membership questions using the Jacobian condition along with
the nilpotent assumptions of Drużkowski.

  * •

For a matrix $A$, see how to express the coefficients of the characteristic
polynomial or their powers in terms of the entries of $A^{k}$.

## References

  * [1] Abdesselam, Abdelmalek. “The Jacobian Conjecture as a Problem of Perturbative Quantum Field Theory” Ann. Henri Poincare 4 (2003) 199 - 215 Birkhauser Verlag, Basel, 2003
  * [2] Bass, Hyman, Connell, Edwin H., and Wright, David.“The Jacobian conjecture: Reduction of degree and formal expansion of the inverse”, Bull. Amer. Math. Soc. (N.S.) 7(2): 287-330 (September 1982).
  * [3] Decker, W., Greuel, G.-M., Pfister, G., and Schönemann, H. (2019). Singular 4-1-2 A computer algebra system for polynomial computations.
  * [4] Drużkowski, Ludwig M. “An Effective Approach to Keller’s Jacobian Conjecture” Mathematische Annalen Volume: 264, page 303-314 (1983)
  * [5] Drużkowski, Ludwig M. “New Reduction in the Jacobian Conjecture” Zeszyty Naukowe Uniwersytetu Jagielloskiego. Universitatis Iagellonicae Acta Mathematica (2001) Volume: 1255(39), page 203-206
  * [6] Johnston, Samuel G.G. and Prochno, Joscha. “Faà di Bruno’s formula and inversion of power series” (2019) https://arxiv.org/abs/1911.07458
  * [7] Johnston, Samuel G.G. “A degree bound for strongly nipotent polynomial automorphisms”. (2021) https://arxiv.org/abs/2110.12462
  * [8] Keller, O.H., “Ganze-Cremona Transformationen”, Monatsch. Math. Phys., 47 (1939), 299-306.
  * [9] Straubing, Howard. “A combinatorial proof go the Cayley-Hamilton theorem”. Discrete Mathematics, Volume 43, Issues 2-3, (1983) pages 273-279.
  * [10] Singer, Dan. “Towards a combinatorial proof of the Jacobian Conjecture!” Volume 18, Issue 2 (2011-2) (The Zeilberger Festschrift volume) P27
  * [11] van den Essen, A. Polynomial Automorphisms and the Jacobian Conjecture Progress in Mathematics Volume 190, Birkhauser Verlag, Basel, 2000.
  * [12] van den Essen, A. and Zhao, W. “Two results on homogenous Hessian nilpotent polynomials”. J. Pure Appl. Algebra 212(10), 2190-2193 (2008)
  * [13] Wolfram Research, Inc., Mathematica, Version 12.3.1, Champaign, IL (2021).
  * [14] Wright, David. “The Jacobian conjecture: ideal membership questions and recent advances” (2005). Mathematics Faculty Publications. 6. https://openscholarship.wustl.edu/math facpubs/6