# Intense harmonics with time-varying orbital angular momentum from
relativistic plasma mirrors

Jingwei Wang<EMAIL_ADDRESS>State Key Laboratory of High Field Laser
Physics and CAS Center for Excellence in Ultra-intense Laser Science, Shanghai
Institute of Optics and Fine Mechanics(SIOM), Chinese Academy of
Sciences(CAS), Shanghai 201800, China  Collaborative Innovation Center of
IFSA, Shanghai Jiao Tong University, Shanghai 200240, China  Matt Zepf
Helmholtz Institute Jena, Fröbelstieg 3, 07743 Jena, Germany  Faculty of
Physics and Astronomy, Friedrich-Schiller-Universität Jena, 07743 Jena,
Germany  Yuxin Leng  Ruxin Li  State Key Laboratory of High Field Laser
Physics and CAS Center for Excellence in Ultra-intense Laser Science, Shanghai
Institute of Optics and Fine Mechanics(SIOM), Chinese Academy of
Sciences(CAS), Shanghai 201800, China  Sergey G. Rykovanov
<EMAIL_ADDRESS>High Performance Computing and Big Data Laboratory,
Center for Computational and Data-Intensive Science and Engineering, Skolkovo
Institute of Science and Technology, Moscow, 121205, Russia

###### Abstract

In this Letter using three-dimensional particle-in-cell simulations and
analytical considerations we demonstrate intense high-order plasma surface
harmonics carrying a time-varying orbital angular momentum (OAM) – the self-
torque. We show that by using two laser beams with different OAMs $l_{1}$ and
$l_{2}$ and a certain delay between each other and shooting them obliquely on
an overdense plasma target, one can generate harmonics with OAM spanning
$nl_{1}$ to $nl_{2}$, where $n$ is the order of the harmonic. Such intense
self-torqued harmonics can offer new possibilities in ultrafast spectroscopy.

###### pacs:

52.38.Ph, 42.65.Ky, 52.27.Ny

Angular momentum (spin or orbital) is one of the properties of light, along
with the intensity, frequency, etc. Historically there were several important
milestones in the development of understanding and usage of light angular
momentum. In 1909 Poynting showed using the mechanical analogy that a
circularly polarized light carries a spin angular momentum (SAM) Poynting1909
. In 1992 Allen et al. suggested that a light beam with a helical phase-front
carries an orbital angular momentum (OAM) Allen1992 . Over the past few
decades considerable attention has been given to developing methods to control
and manipulate the OAM of light beam, such as imparting OAM onto light
Biener2002 ; Sueda2004 , transfer between SAM and OAM Marrucci2006 ;
WangNC2019 , and generating extreme-ultraviolet (EUV) high-order harmonics
with well-defined OAM Hernandez2013 ; Gariepy2014 ; Zhang2015 ; Geneaux2016 ;
Gauthier2017 ; Denoeud2017 ; Leblanc2017 . The rapid development of OAM light
beams is driven by their important applications in optical manipulation
Padgett2011 , optical microscopy Maurer2010 ; furhapter2005 , and optical
communications Wang2012 ; Willner2015 .

The transverse phase structure of a beam with OAM, also known as a vortex
beam, is typically described by a certain dependence of an electromagnetic
field component on azimuthal angle $\phi$ proportional to $\exp(-il\phi)$,
where integer $l$ is the so-called topological charge number. The OAM of a
vortex beam is then characterized by $l\hbar$. For a long time the OAM of
light was recognized as a static quantity without time-dependence, until
recently L. Rego et al. discovered a new class of OAM light beams: the self-
torqued light beams carrying time-varying OAM Rego2019 . The self-torqued
light beam was produced from the nonlinear process of high-order harmonic
generation in gases driven by two time-delayed laser pulses carrying different
values of OAM. The self-torque $\hbar\xi$ is defined as $\hbar\xi=\hbar
dl(t)/dt$, where $l(t)$ is the time-dependent OAM of the light beam.
Therefore, the term self-torque refers to the angular acceleration of the
light beam, in an analogy with other physical systems that possess a self-
induced time variation of the angular momentum, such as the gravitational
self-fields Dolan2007 . This inherent property of light opens new routes for
creating structured light beams. The self-torqued light beams could be used
for investigating the systems with time-varying OAM, such as imaging magnetic
and topological excitations, launching selective and chiral excitation of
quantum matter, imprinting OAM centrifuges as well as for ultrafast
spectroscopy.

Figure 1: (a) Schematic of generation of self-torqued harmonics from the
oscillating plasma mirrors driven by two relativistic lasers with different
values of OAM $l_{1}$ and $l_{2}$. The embedded three slices are the intensity
distributions of the harmonic at different longitudinal positions. (b) The
spectrum of the reflected laser field at some fixed radial coordinate. (c)
Temporal evolution of the OAM of the 5-th order harmonic. The white dashed
line is the theoretical result calculated from Eq. (8). The parameters of
lasers and target are given in the simulation setup in the text.

In Rego2019  the self-torqued EUV harmonics are driven by moderately intense
laser pulses (with intensities $\sim 10^{14}~{}\rm{W/cm}^{2}$) in gas. The low
intensity of the generated self-torqued EUV beams would be a limitation for
their future applications. Plasma surface harmonics, on the other hand, could
be much more intense as they are driven by relativistic lasers (with
intensities more than $10^{18}~{}\rm{W/cm}^{2}$) Teubner2009 . Generation of
intense surface harmonics carrying OAM in the relativistic region has already
been experimentally demonstrated Denoeud2017 ; Leblanc2017 . However, it is
not clear whether it is possible to generate self-torqued high-order harmonics
in the relativistic region, where the involved physics is quite different from
the gas harmonics generation. In this Letter, we reveal that self-torqued EUV
beams naturally arise when the plasma surface harmonics generation is driven
by two relativistic and time-delayed laser pulses with different values of
OAM, via the well-known relativistic oscillating mirror (ROM) mechanism
Bulanov1994 ; Lichters1996 .

The interaction schematic is demonstrated in Fig. 1(a): two $p$-polarized,
time-delayed intense vortex laser beams carrying different OAM values ($l_{1}$
and $l_{2}$) are obliquely incident onto a solid target. The driving laser
field oscillates the plasma surface with a velocity close to the light speed.
At the same time, the oscillating plasma surface reflects the driving field
and modulates it, which in turn brings multiple harmonics into the reflected
field, as shown in Fig. 1(b). In the hand-waiving view of quantum physics, the
harmonics generation can be understood as a process of absorbing $q$ photons
with a frequency $\omega$ and emitting one photon with a frequency $q\omega$.
According to the conversation of angular momentum, the emitted photon will
carry an OAM of $ql$, given that each absorbed photon carries an OAM of $l$
($\hbar$ is dropped here and also in the following text). For the present
scheme, there are simultaneously two kinds of photons in the overlapping
region of the time-delayed lasers, in terms of the photon OAM values: $l_{1}$
and $l_{2}$. Assuming that the emitted photon with a frequency of $q\omega$
comes from $m$ photons with OAM $l_{1}$ and $(q-m)$ photons with OAM $l_{2}$,
then each photon in the $q$-th harmonic will carry an OAM of
$ml_{1}+(q-m)l_{2}$. Since $m$ is related with the ratio of the instantaneous
intensities of the two pulses, $m$ is time-dependent, and consequently, the
OAM of the $q$-th harmonic is also time-dependent. This reasoning gives an
intuitive understanding of the generation of time-varying OAM in high order
harmonics.

The self-torque can also be explained from the theory of ROM mechanism
Bulanov1994 ; Lichters1996 ; Tsakiris2006 . The oblique incidence in the
laboratory frame can firstly be reduced to a normal incidence in a moving
frame by the Bourdier transform Bourdier1983 . In the moving frame we assume
that on the target at some radial coordinate the electric fields of the two
$p$-polarized driving lasers are
$E_{1}=u_{1}(t)\exp[i(\omega_{L}t-k_{L}z+l_{1}\phi)]$ and
$E_{2}=u_{2}(t)\exp[i(\omega_{L}t-k_{L}z+l_{2}\phi)]$, where $\omega_{L}$ is
the laser circular frequency, $k_{L}$ is the wave vector and $u_{1}(t)$ and
$u_{2}(t)$ are the field amplitudes. The total field can be written as

$E(\phi,t)=E_{1}+E_{2}=u_{0}(t)e^{i\omega_{L}t-k_{L}z}[(1-\eta)e^{il_{1}\phi}+\eta
e^{il_{2}\phi}],$ (1)

where we have defined $u_{0}(t)=u_{1}(t)+u_{2}(t)$ and
$\eta=u_{2}(t)/u_{0}(t)$. For convenience, we factorize $E(\phi,t)$ as

$E(\phi,t)=|E(\phi,t)|e^{i\omega_{L}t-k_{L}z+i\varphi(\phi,t)},$ (2)

with the phase
$\varphi(\phi,t)\approx(1-2\eta)(l_{1}-l_{2})\phi/2+(l_{1}+l_{2})\phi/2=[(1-\eta)l_{1}+\eta
l_{2}]\phi$, while the amplitude $|E(\phi,t)|$ will not be discussed in the
present problem. According to the ROM theory, the reflected field observed at
($t,z<0$) is emitted at a retarded time $t^{\prime}=t-Z(t^{\prime})/c+z/c$
from the oscillating source located at $Z(t^{\prime})$, where $Z(t^{\prime})$
is the longitudinal trajectory of the plasma mirror. Thus the reflected field
at the observer can be written as

$E_{r}(\phi,t)\propto|E_{(}\phi,t)|e^{i(\omega_{L}t-2k_{L}Z(t^{\prime}))+i\varphi(\phi,t)}.$
(3)

Here we have dropped the fixed phase $k_{L}z$. The plasma mirror driven by the
laser field will oscillate as
$k_{L}Z(t^{\prime})=\varepsilon[(1-\eta)\sin(\omega_{L}t^{\prime}+l_{1}\phi))+\eta\sin(\omega_{L}t^{\prime}+l_{2}\phi)]$,
where $\varepsilon$ is the amplitude. With the expression of $Z(t^{\prime})$
we then rewrite Eq. (3) as

$\displaystyle E_{r}(\phi,t^{\prime})\propto$
$\displaystyle|E(\phi,t^{\prime})|e^{i\omega_{L}t^{\prime}+i[(1-\eta)l_{1}+\eta
l_{2}]\phi}e^{-i\varepsilon[(1-\eta)\sin(\omega_{L}t^{\prime}+l_{1}\phi)+\eta\sin(\omega_{L}t^{\prime}+l_{2}\phi)]}.$
(4)

We then work in the $t^{\prime}$ coordinate for convenient. Employing the
Jacobi-Anger identityCuyt2008 , we can rewrite the last exponential term of
Eq. (4) as
$\sum_{n=-\infty}^{n=+\infty}J_{n}[\varepsilon(1-\eta)]\exp[in(\omega_{L}t^{\prime}+l_{1}\phi+\pi)]\sum_{m=-\infty}^{m=+\infty}J_{m}(\varepsilon\eta)\exp[im(\omega_{L}t^{\prime}+l_{2}\phi+\pi)]$,
$J_{n}$ and $J_{m}$ are the Bessel functions of the first kind. Putting this
expression into Eq. (4), we get the field of the $(q+1)$-th order harmonic as

$\displaystyle E_{r}^{q+1}$ $\displaystyle\propto
e^{i(q+1)\omega_{L}t^{\prime}}e^{i[(1-\eta)l_{1}+\eta
l_{2}]\phi}\sum_{n=0}^{q}J_{n}[\varepsilon(1-\eta)]J_{q-n}(\varepsilon\eta)e^{i[nl_{1}+(q-n)l_{2}]\phi}.$
(5)

From Eq. (5) we see that each term in the sum over $n$ has a definite OAM of
$nl_{1}+(q-n)l_{2}$, and each term contributes to the mean OAM with a weight
of $J_{n}[\varepsilon(1-\eta)]J_{q-n}(\varepsilon\eta)$. Then the mean OAM of
the $(q+1)$-th order harmonic at time $t^{\prime}$ can be calculated as

$\displaystyle\overline{l}_{q+1}(t^{\prime})$
$\displaystyle=[(1-\eta)l_{1}+\eta
l_{2}]+\frac{\sum_{n=0}^{q}J_{n}[\varepsilon(1-\eta)]J_{q-n}(\varepsilon\eta)[nl_{1}+(q-n)l_{2}]}{\sum_{n=0}^{q}J_{n}[\varepsilon(1-\eta)]J_{q-n}(\varepsilon\eta)}.$
(6)

Since $\varepsilon<1$, we can use the approximation
$J_{v}(x)\approx(1/v!)(x/2)^{v}$. Eq. (6) can be written as

$\displaystyle\overline{l}_{q+1}(t^{\prime})$
$\displaystyle=[(1-\eta)l_{1}+\eta
l_{2}]+\frac{\sum_{n=0}^{q}\binom{q}{n}(1-\eta)^{n}\eta^{q-n}[nl_{1}+(q-n)l_{2}]}{\sum_{n=0}^{q}\binom{q}{n}(1-\eta)^{n}\eta^{q-n}},$
(7)

where $\binom{q}{n}=q!/[n!(q-n)!]$ is the binomial coefficient. Considering
that $\sum_{n=0}^{q}\left({}^{q}_{n}\right)(1-\eta)^{n}\eta^{q-n}=1$ and
$\sum_{n=0}^{q}\left({}^{q}_{n}\right)(1-\eta)^{n}\eta^{q-n}n=q(1-\eta)$, we
get the final result of the mean OAM for the $(q+1)$-th order harmonic as

$\overline{l}_{q+1}(t^{\prime})=(q+1)[(1-\eta)l_{1}+\eta l_{2}].$ (8)

This result shows that the $(q+1)$-th harmonic contains a time-varying OAM,
changing from $(q+1)l_{1}$ to $(q+1)l_{2}$ over time.

The self-torque has been verified by three-dimensional particle-in-cell
simulations using the code LAPINE lapine2002 . Two time-delayed laser pulses
carrying OAM $l_{1}=1$ and $l_{2}=2$ are obliquely incident on a solid target
with an incident angel $\alpha=45^{\circ}$. The envelopes of the two incident
laser pulses are shaped with the same function $\sin^{2}(\pi t/T_{0})~{}(0\leq
t\leq T_{0})$, with the pulse full duration $T_{0}=20T_{L}$.
$T_{L}=\lambda_{L}/c$ is the laser period, $\lambda_{L}=0.8~{}\mu{\rm m}$ is
the laser wavelength and $c$ is the light speed in the vacuum. The time delay
between the two pulses is chosen as $t_{d}=T_{0}/2$ for the main results. The
normalized amplitudes of the two laser fields $E_{L}$ are both
$a_{0}=e|E_{L}|/m_{e}\omega_{L}c=1.68$, where $m_{e}$ is the electron mass and
$-e$ is the electron charge. The target is plane with a thickness of
$2\lambda_{L}$ and a density of $5n_{c}$, where $n_{c}=1.1\times
10^{21}\lambda_{L}(\mu{\rm m})/{\rm cm}^{3}$ is the critical density. The size
of the simulation box is $20\lambda_{L}(x)\times 20\lambda_{L}(y)\times
20\lambda_{L}(z)$ corresponding to grids $1000(x)\times 1000(y)\times
1000(z)$, with 8 macro-particles per cell. A detector in the $x-z$ plane is
placed at $y=0.5\lambda_{L}$ to collect the reflected field.

Fig. 1(b) shows the harmonic spectrum of the reflected field at some fixed
radial coordinate. The intensity of each harmonic can also be calculated. For
instant, the intensity of the $5$-th order harmonic (with a wavelength of
$160\rm~{}{nm}$) is $1.37\times 10^{17}~{}\rm{W/cm}^{2}$, which shows the
harmonics are indeed intense. We choose the $5$-th order harmonic as an
example to check the self-torque. The distribution of its OAM during the
overlapping time is presented in Fig. 1(c). One can see that the harmonic
contains all the OAM values between $l=5$ and $l=10$. The most probable value
of OAM (i. e. the OAM value which has the peak intensity at a time) increases
with time. The time evolution agrees well with the theoretical result from Eq.
(8), shown by the white dashed line in Fig. 1(c). The intensities of each OAM
are calculated from the Fourier analysis of the electric field upon the
azimuth angel. An animation describing the temporal evolution of the field
profile of the $5$-th order harmonic can be found in the Supplemental Material
and it shows how the OAM value of $5$-th harmonic changes from $l=5$ to
$l=10$.

Figure 2: The distinctions between a self-torqued beam and a fractional vortex
beam. The OAM spectrums at different times for a self-torqued beam (a) and a
fractional vortex beam (b). The intensity distribution of a self-torqued beam
(c) and a fractional vortex beam (d). The self-torque beam corresponds to the
5-th harmonic in Fig. 1. The fractional vortex beam corresponds to two time-
delayed vortex beams with $l_{1}=5$, $l_{2}=10$, pulse duration
$T_{0}=20T_{L}$ and time delay $t_{d}=T_{0}/2$.

It is important to emphasize that such a self-torqued beam differs essentially
from the so-called fractional OAM beams. A fractional OAM beam, which is a
superposition of two time-delayed vortex beams with OAM $ql_{1}$ and $ql_{2}$,
does not contain a self-torque, although the temporal evolution of its average
OAM is similar with that of a self-torqued beam. An important evidence is
that, a fractional OAM beam does not contain physical intermediate OAM states
(i.e., photons with OAM other than $ql_{1}$ and $ql_{2}$), while a self-
torqued beam contains all the intermediate OAM states between $ql_{1}$ and
$ql_{2}$. This is demonstrated by the OAM content at different times in Fig.
2(a) and (b). In Fig. 2(a) we plot the OAM spectra for the self-torqued $5$-th
order harmonic, while Fig. 2(b) is for the overlapping of two time-delayed OAM
beams with $l=5$ and $l=10$. One can clearly see that there are six OAM values
(from $l$=5 to $l$=10) during the overlapping time for a self-torque beam,
while there are only two OAM values during the whole interaction time for the
fractional OAM beam. The intermediate OAM values of the self-torqued beam come
from the interactions of the two driving lasers with different OAMs, which is
indicated in Eq. (5). Another distinction between a self-torqued beam and a
fractional beam is the intensity distribution. Fig. 2(c) and (d) show the
intensity distribution of the $5$-th order harmonic of the self-torque beam
and a fractional beam with $l=5$ and $l=10$. One can see that the intensity of
a self-torqued beam exhibits a distinctive “crescent” shape.

Figure 3: The azimuthal frequency chirp. (a) The distribution of harmonic
intensity in the space of frequency and azimuthal angel for the $5$-th, $7$-th
and $9$-th order harmonics. The dependences of frequency on the azimuthal
angel from the theoretical model are demonstrated by the white lines. The
self-torque here is the average over the overlapping time. (b) The harmonic
spectrums at different azimuthal angles. The simulation parameters here are
the same as Fig. 1.

A consequence of the self-torque is the presence of an azimuthal frequency
chirp, which means that the same order harmonic at different azimuthal angles
will carry slightly different frequencies. The phase of the $q$-th order
harmonic can be written as $\varphi_{q}=q\omega_{L}t+l_{q}\phi$. Since $l_{q}$
is time-dependent, the frequency of the $q$-th order harmonic at an azimuthal
angle $\phi$ will then be shifted as

$\omega_{q}(\phi)=\frac{d\varphi_{q}}{dt}\approx
q\omega_{L}+\frac{d\overline{l}_{q}(t)}{dt}\phi=q\omega_{L}+\xi_{q}\phi.$ (9)

The self-torque here is $\xi_{q}=d\overline{l}_{q}(t)/dt$, where
$\overline{l}_{q}(t)$ is the mean OAM calculated by Eq. (8). Fig. 3(a) shows
the intensity distributions of the harmonics in the space of frequency and
azimuthal angle. For each harmonic, the frequency almost linearly increases
with the azimuthal angel $\phi$. The simulation results agree well with the
theoretical results demonstrated by the white dashed lines in Fig. 3(a). One
can also find that the self-torque is proportional to the harmonic order. The
spectra in Fig. 3(b) show that the frequencies at $\phi=\pm 0.7$ are about
$0.05\omega_{L}$ shifted and their intensities are a bit lower, as compared to
$\phi=0$. In experiments, the azimuthal frequency chirp can be confirmed by
the phenomenon that photons will have different energies at different
azimuthal angles.

The self-torque can be controlled by changing the pulse duration and the
delay. In Fig. 4 we plot the dependence of the self-torque on the pulse
duration and delay. In Fig. 4(a) the delay of the two pulses changes from
$-12.5T_{L}$ to $12.5T_{L}$, while other parameters are the same as Fig. 1. In
Fig. 4(b) the pulse duration $T_{0}$ varies from $5T_{L}$ to $40T_{L}$ and the
delay is fixed as $t_{d}=T_{0}/2$. In simulations the self-torque is
calculated from the frequency shift divided by the azimuthal angle offset, as
shown in Fig. 3(b). Both the simulations and theoretical model show that the
self-torque increases with the delay $t_{d}$ but decreases with the pulse
duration $T_{0}$. These can be understood from that: the OAM changing content
$q(l_{2}-l_{1})$ keeps constant for a given harmonic; therefore, the
overlapping time $\tau=T_{0}-t_{d}$ determines the self-torque.

Figure 4: The dependencies of the self-torque on the time delay (a) and pulse
duration (b). The red triangle line is obtained from the simulations and the
blue line is obtained from the theoretical model. In (a) the pulsed duration
is fixed $T_{0}=20T_{L}$ and in (b) the pulse delay is set as $t_{d}=T_{0}/2$.
Other simulation parameters are the same as Fig. 1.

In this Letter we have presented a single example of using two laser pulses
with OAM difference of 1 ($l_{1}=1$, $l_{2}=2$). In such a case the average
OAM for the 5-th harmonic goes through all integer values from 5 to 10. In the
case of OAM difference of $p$, for example $l_{1}=1$ and $l_{2}=p+1$, the
average OAM will go from 5 to $5(p+1)$ with the spacing of $p$ according to
Eq. (5). Another option for creating self-torqued harmonics can potentially be
the usage of two pulses with same value of OAM but with different colors. We
also suspect that other mechanisms of plasma harmonics generation like
coherent wake emission Quere2006  or coherent synchrotron emission Brugge2010
; Dromey2012  would be feasible as well (far-fetched conjecture is that all
harmonics generation mechanisms in crystals, gases, plasmas, electron beams,
etc. would yield self-torque for proper conditions). These questions can
become interesting future studies.

In conclusion, we have theoretically and numerically demonstrated that intense
EUV harmonics with time-varying OAM can be produced from the relativistic
plasma mirror driven by two time-delayed laser pulses with different OAM
values. The generated self-torqued light beam is essentially different from
the fractional vortex beam. An important feature of the self-torqued harmonics
is the azimuthal frequency chirp. Such an intense self-torqued EUV source may
find their applications in launching selective and chiral excitation of
quantum matter, imprinting OAM centrifuges and studying the transient
processes sensitive to OAM.

This work was supported by the National Natural Science Foundation of China
(NSFC 11674341, 11991074), the Strategic Priority Research Program of Chinese
Academy of Sciences (Grant Nos. XDA25051100, XDB1603), the Chinese Academy of
Sciences President’s International Fellowship Initiative (No. 2018VMC0012),
and the joint laser laboratory between Shanghai Institute of Optics and Fine
Mechanics of Chinese Academy of Sciences and Institute of Applied Physics of
Russian Academy of Sciences. The authors gratefully acknowledge the computing
time granted by the National Supercomputer Center Tianhe-2 in Guangzhou and
Skoltech CDISE supercomputer “Zhores” Zacharov2019 .

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# Matrix factorizations of generic polynomials

Daniel Erman  Department of Mathematics, University of Wisconsin, Madison, WI
<EMAIL_ADDRESS>
###### Abstract.

We prove that the Buchweitz-Greuel-Schreyer Conjecture on the minimal rank of
a matrix factorization holds for a generic polynomial of given degree and
strength. The proof introduces a notion of the secondary strength of a
polynomial, and uses a variant of the ultraproduct technique of Erman, Sam,
and Snowden.

##  1\. Introduction

Buchweitz-Greuel-Schreyer have conjectured [bgs, Conjecture A]:

######  Conjecture 1.1.

Let $P$ be a regular local ring with algebraically closed residue field and
let $f\in P$. Let
$e:=\lfloor\frac{1}{2}(\operatorname{codim}_{P/f}\operatorname{Sing}(P/f)-2)\rfloor$
where $\operatorname{Sing}(P/f)$ denotes the singular locus of $P/f$. The
minimal rank of a nonfree maximal Cohen-Macaulay module on $P/f$ is $\geq
2^{e}$.

A matrix factorization of $f\in P$ is a pair $(\phi,\psi)=(\phi\colon F\to
G,\psi\colon G\to F)$ where $F$ and $G$ are finite rank free $P$-modules, and
where $\phi\circ\psi=f\cdot\operatorname{id}_{G}$ and
$\psi\circ\phi=f\cdot\operatorname{id}_{F}$. Based on the equivalence between
matrix factorizations of $f$ and maximal Cohen–Macaulay (MCM) modules on the
corresponding hypersurface ring [eisenbud], this conjecture is closely related
to the study of matrix factorizations of $f$. Matrix factorizations were
introduced by Eisenbud in [eisenbud], though they arise in Shamash’s [shamash]
and related ideas appear in [G1]. Matrix factorizations now arise in
connection with a huge range of topics: for a sampling, see [aspinwall,
avramov-infinite, bfk, MP-infinite, orlov] and the discusssion and references
within.

Yet despite this wide literature on matrix factorizations, there has been
limited progress on Conjecture 1.1. Since its origin, the conjecture has been
known for quadratic polynomials and simple hypersurface singularities
[knorrer]. Eisenbud, Peeva, and Schreyer have shown that the conjecture holds
for certain generic combinations of quadrics [eps, Corollary 2.2], while
Ravindra and Tripathi have proven the conjecture for homogeneous polynomials
when $e\leq 2$ [tripathi, rt1]; see also [beh, blaser-eisenbud-schreyer, mrr1,
mrr2, rt2, rt3].

Conjecture 1.1 has both algebraic and geometric motivations. Buchweitz,
Greuel, and Schreyer produced certain explicit matrix factorizations from
Koszul complexes [bgs, §2.1], and their conjecture roughly proposes that these
are the smallest possible. This echoes the Buchsbaum-Eisenbud-Horrocks
Conjecture on minimal free resolutions (see [be, p. 405] and [hartshorne-vb,
Problem 24])–which proposes that the Koszul complex is the smallest possible
resolution–as well as related rank conjectures of Carlsson, Avramov-Buchweitz-
Iyengar, and Avramov-Buchweitz [carlsson, AB, ABI]. In a geometric direction:
if $V(f)\subseteq\mathbb{P}^{n}$ is a general hypersurface, then
$e(f)=\lfloor\frac{n}{2}\rfloor-1$ and the construction from [bgs, §2.1] is
built on a nontrivial cycle on the hypersurface; by the Lefschetz Theorem,
this cannot happen before the middle codimension.

Many of the conjectures listed above have seen significant recent
breakthroughs due to the results of Walker [walker] and the counterexamples of
Iyengar-Walker [iyengar-walker]. But Conjecture 1.1 has remained untouched by
this progress.111However, in light of the counterexamples of [iyengar-walker],
Eisenbud, Peeva and Schreyer have used Conjecture 1.1 to update the Betti
Degree Conjecture of [AB]; see [eps, Conjecture 4.2].

We prove that Conjecture 1.1 holds for “generic” homogeneous polynomials,
though we emphasize that our genericity notion is based on a less standard way
of parametrizing polynomials. Instead of fixing the degree and number of
variables of $f$, we will fix the degree and the strength of the homogeneous
polynomial. Recall that the strength of $f$ is the minimal $s$ for which we
can write $f=\sum_{i=0}^{s}h_{i}g_{i}$ with
$1\leq\deg(h_{i}),\deg(g_{i})<\deg(f)-1$, or $\infty$ if no such decomposition
exists. For example, $x_{0}y_{0}+x_{1}y_{1}+\cdots+x_{s}y_{s}$ has strength
$s$. And while the Buchweitz-Greuel-Schreyer Conjecture was not previously
known for any homogeneous polynomial $f$ of degree $d$ with $d>2$ and
$e(f)>2$, our results imply that the conjecture holds and is sharp for some
polynomial with every $d$ and $e(f)$ value; see Example 4.2.

We define the rank of a matrix factorization $(\phi,\psi)$ to be the rank of
the free module $F$. For a homogeneous polynomial $f\in
R=k[z_{0},\dots,z_{n}]$, we say that a matrix factorization is reduced if all
entries belong to the maximal ideal. We use $\operatorname{MF-rank}(f)$ to
denote the minimal rank of a reduced, homogeneous matrix factorization, and we
let
$e(f):=\lfloor\tfrac{1}{2}\operatorname{codim}_{R/f}(\operatorname{Sing}(R/f)-2)\rfloor$.
The equality
$\operatorname{rank}\operatorname{coker}\phi+\operatorname{rank}\operatorname{coker}\psi=\operatorname{rank}F$
relates the rank of the corresponding MCM modules to that of $F$. Since our
proof involves passing to non-noetherian rings, and we want to avoid
discussing MCM modules in that setting, we now express the rank of the
corresponding MCM module entirely in terms of the matrix factorization. As we
recall in §2, if $f$ is irreducible, then $\det(\phi)$ is a scalar multiple of
$f^{r}$ where $r$ is the rank of the corresponding MCM module
$\operatorname{coker}\phi$. We thus define

$\operatorname{MCM-rank}(f):=\min\left\\{\begin{matrix}r\text{ where
$(\phi,\psi)$ is a reduced, homogeneous matrix}\\\ \text{factorization of $f$
with }\det(\phi)\text{ a scalar multiple of $f^{r}$}\end{matrix}\right\\}.$

Using this notation, the Buchweitz-Greuel-Schreyer Conjecture for homogeneous
polynomials $f$ is that $\operatorname{MCM-rank}(f)\geq 2^{e(f)}$; and this
implies the weaker conjecture that $\operatorname{MF-rank}(f)\geq 2^{e(f)+1}$.
While $\operatorname{MCM-rank}(f)$ is the subject of Conjecture 1.1,
$\operatorname{MF-rank}(f)$ has arisen in connection with other conjectures
(see, for example [eps, §4]) and so we will consider both invariants. Since
$\operatorname{MF-rank}(f),e(f)$ and $\operatorname{MCM-rank}(f)$ are all
invariant under adjoining extra variables (see Lemma 4.1), we will consider
polynomials in $S:=k[z_{0},z_{1},\dots]$ where $k$ is an algebraically closed
field. Let $\mathbf{S}_{d,s}$ be the set of homogeneous degree $d$ and
strength $\leq s$ polynomials in $S$. We will show:

######  Theorem 1.2.

Fix $d\geq 2$ and $s\geq 1$. There is a dense open subset
$(\mathbf{S}_{d,s})^{\circ}\subseteq\mathbf{S}_{d,s}$ where, for any
$f\in(\mathbf{S}_{d,s})^{\circ}$, we have that $\operatorname{MF-rank}(f)\geq
2^{e(f)+1}$ and $\operatorname{MCM-rank}(f)\geq 2^{e(f)}$.

The main idea behind the proof is a spin on the Ananyan–Hochster Principle
which was implicitly at the heart of [ananyan-hochster] and followup works
like [stillman, hartshorne]; see [bulletin] for an expository discussion of
the principle, which we briefly review. The collective strength of homogeneous
polynomials $f_{1},\dots,f_{r}\in k[z_{0},z_{1},\dots]$ is the minimal
strength of a $k$-linear combination of $f_{1},\dots,f_{r}$. The
Ananyan–Hochster Principle says: homogenous polynomials $f_{1},\dots,f_{r}$ of
sufficiently high collective strength will behave increasingly like
independent variables $x_{1},\dots,x_{r}$. We use a variant of this principle
based on the following definition:

######  Definition 1.3.

If $f$ is a homogeneous polynomial of strength $s$, then the secondary
strength of $f$ is the maximal collective strength of
$g_{0},g_{1},\dots,g_{s},h_{0},h_{1},\dots,h_{s}$ where
$f=\sum_{i=0}^{s}g_{i}h_{i}$.

For instance, in characteristic $0$, the power sum
$g_{d,n}:=\sum_{i=0}^{n}z_{i}^{d}$ has strength $\geq\frac{n-1}{2}$.222We do
not know the precise strength of $g_{d,n}$ but this inequality follows from
two simple computations. First, $g_{d,n}$ defines a smooth hypersurface in
$\mathbb{P}^{n}$. Second, if $f$ is a polynomial of strength $s$, then the
corresponding hypersurface is singular in codimension $\leq 2s+1$. Thus
$g_{1,n}g_{6,n}+g_{2,n}g_{5,n}+g_{3,n}g_{4,n}$ has strength $2$ but secondary
strength at least $\frac{n-1}{2}$.

Following the logic of the Ananyan–Hochster Principle: as the secondary
strength of $f$ goes to infinity, the $g_{i}$ and $h_{i}$ will behave
increasingly like independent variables, and so $f$ should behave increasingly
like the quadratic form $\sum_{i=0}^{s}x_{i}y_{i}$. In other words, we
propose:

(1) $\begin{matrix}\text{{\em A polynomial $f$ of strength $s$ will behave
increasingly like}}\\\ \text{\em{ a quadratic form of rank $2s+2$ as the
secondary strength $\to\infty$.}}\end{matrix}$

This is the central idea behind Theorem 1.2. If $q=\sum_{i=0}^{s}x_{i}y_{i}$
then Knörrer’s work implies that $\operatorname{MF-rank}(q)=2^{s}$ and
$\operatorname{MCM-rank}(q)=2^{s-1}$. We prove that the same holds for any $f$
of high enough secondary strength:

######  Theorem 1.4.

Fix $d\geq 2$ and $s\geq 1$. There exists $N=N(d,s)$ where, for any
homogeneous polynomial $f$ of degree $d$, strength $s$, and secondary strength
$\geq N$, we have that $\operatorname{MF-rank}(f)=2^{s}$ and
$\operatorname{MCM-rank}(f)=2^{s-1}$.

By noting that $s\geq e(f)+1$ and that the locus in $\mathbf{S}_{d,s}$ of
secondary strength $\geq N$ contains a dense open subset, Theorem 1.4 implies
Theorem 1.2.

The upper bound $\operatorname{MF-rank}(f)\leq 2^{s}$ was known. In fact, a
decomposition $f=\sum_{i=0}^{s}g_{i}h_{i}$ yields an explicit matrix
factorization of $f$ by specializing the rank $2^{s}$ matrix factorization of
the quadric $\sum_{i=0}^{s}x_{i}y_{i}$. For instance, if $s=1$ then

$\begin{pmatrix}g_{0}&-g_{1}\\\ h_{1}&h_{0}\end{pmatrix}\text{ and
}\begin{pmatrix}h_{0}&g_{1}\\\ -h_{1}&g_{0}\end{pmatrix}$

is such a matrix factorization of $f=g_{0}h_{0}+g_{1}h_{1}$. This
specialization argument is similar to the essential idea behind the
construction of matrix factorizations in [bgs, §2.1]. Via a similar idea, one
easily also obtains $\operatorname{MCM-rank}(f)\leq 2^{s-1}$. So the content
of Theorem 1.4 are the inequalities $\operatorname{MF-rank}(f)\leq 2^{s}$ and
$\operatorname{MCM-rank}(f)\leq 2^{s-1}$.

To prove these, we apply ultraproduct machinery from [stillman]. The first key
observation is that ultraproducts play well with constructing matrix
factorizations of a fixed rank and with fixed determinant; therefore
invariants like $\operatorname{MF-rank}(f)$ and $\operatorname{MCM-rank}(f)$
behave well under passage to ultraproduct limits. The second key observation
is that, in the ultraproduct limit, an element of strength $s$ and infinite
secondary strength will essentially equal a quadric of the form
$x_{0}y_{0}+\cdots+x_{s}y_{s}$; this is like the limit of the principle from
(1), and it is largely a consequence of the results of [stillman]. Combining
these, Theorem 1.4 is then reduced to Knörrer’s result about matrix
factorizations of quadric polynomials [knorrer]

Since Ulrich modules are a special type of maximal Cohen-Macaulay modules,
Theorem 1.2 also trivially implies that the same lower bound holds for Ulrich
modules on the hypersurface ring $k[z_{0},\dots,z_{n}]/f$, or equivalently for
Ulrich sheaves on $V(f)$. In particular, Theorem 1.2 shows that the conjecture
of Bläser-Eisenbud-Schreyer [blaser-eisenbud-schreyer, Conjecture 0.3] on the
Ulrich complexity of a hypersurface is also true for generic polynomials. We
add that Ravindra and Tripathi have proven bounds on Ulrich bundles on general
hypersurfaces in interesting cases, including degree $d\geq 4$ hypersurfaces
in $\mathbb{P}^{4}$ [rt3].

There has been a huge array of recent work on the properties of high strength
polynomials with applications to algebra and algebraic geometry [ah2, bbov,
bbov2, bv, bdov, bde, bdde, bdes, cmpv, ds, DES, cohom, HS, hartshorne, kz-
high-rank, kz-ranks, kp1, kp2, kp3, snowden-relative] as well as connections
with arithmetic questions [birch, cook-magyar, schmidt, xy, y].333As strength
is an elementary notion with a long history, terminology is inconsistent.
While the term strength originated in [ananyan-hochster], it is sometimes
known as Schmidt’s “$h$-invariant” in arithmetic settings; and it predates
even Schmidt, appearing at least as early as [davenport-lewis]. By comparison,
the central novelty in our proof is to fix the strength but allow the
secondary strength to be very large; this raises new Questions like 4.5 and
4.6

The use of graded ultraproducts of polynomial rings in a fixed number of
variables has precursors in [schoutens, van-den-Dries-schmidt] and more. Our
use of ultraproducts, which requires polynomials in increasing numbers of
variables, stems from [stillman, hartshorne]. Of course, the use of an
ultraproduct means that we obtain no effective bound in Theorem 1.4.

######  Remark 1.5.

Invariants like $\operatorname{MF-rank}(f)$ and $\operatorname{MCM-rank}(f)$
can only decrease under extension of scalars, so the Buchweitz-Greuel-Schreyer
Conjecture for polynomials can be reduced to the algebraically closed case.
But strength and secondary strength can also decrease under extension of
scalars. The algebraically closed assumption is not necessary for the part of
Theorem 1.4 on $\operatorname{MF-rank}(f)$; in fact, the proof goes through
essentially verbatim if one works with the ultraproduct over
$\\{k_{i}[x_{1},x_{2},\dots]\\}_{i\in\mathbb{N}}$ where the $k_{i}$ are
arbitrary fields and one applies [imperfect, Theorem 6.17]. For the part on
$\operatorname{MCM-rank}(f)$, we do require the algebraically closed condition
when reducing to the case where $\deg(\phi)=f^{r}$ on the nose, and not just
up to a scalar. More substantially, we use the algebraically closed hypothesis
in our set-theoretic arguments that the locus in $\mathbf{S}_{d,s}$ of
secondary strength $\geq N$ contains an open set, and we thus use that
assumption in the passage from Theorem 1.4 to Theorem 1.2.

## Acknowledgments

We thank Christine Berkesch, Michael K. Brown, David Eisenbud, Jordan
Ellenberg, Daniele Faenzi, Yeongrak Kim, Irena Peeva, Joan Pons-Llopis,
Claudiu Raicu, R.V. Ravindra, Eric Riedl, Steven V Sam, Andrew Snowden, Amit
Tripathi, and Mark Walker for helpful comments and conversations.

##  2\. Notation

We let $k$ be an algebraically closed field of arbitrary characteristic. Let
$S=k[z_{0},z_{1},\dots]$ with the standard grading and let
$\mathfrak{m}=\langle z_{0},z_{1},\dots\rangle$ be the homogeneous maximal
ideal. We summarize some basic facts about matrix factorizations; details and
background can be found in [eisenbud]. A matrix factorization of $f$ is a pair
of graded morphisms $(\phi,\psi)=(\phi\colon F\to G,\psi\colon G\to F)$ where
$F$ and $G$ are finite rank free graded $S$-modules, and where
$\phi\circ\psi=f\cdot\operatorname{id}_{G}$ and
$\psi\circ\phi=f\cdot\operatorname{id}_{F}$. We say that $(\phi,\psi)$ is
reduced if $\operatorname{im}\phi\subseteq\mathfrak{m}G$ and
$\operatorname{im}\psi\subseteq\mathfrak{m}F$. Throughout the remainder of
this paper, we focus entirely on reduced matrix factorizations. We set the
rank of a matrix factorization to be the rank of $F$, which equals the rank of
$G$. The map $(\phi,\psi)\mapsto\operatorname{coker}\phi$ yields a bijection
between reduced matrix factorizations of $f$ and maximal Cohen-Macaulay
modules of $S/f$, up to isomorphism.

The rank of $\operatorname{coker}\phi$ can be determined from properties of
the matrix factorization $(\phi,\psi)$. If $f=gh$ is reducible then $(g)$ and
$(h)$ form a rank $1$ matrix factorization, and we will ignore this case from
here on. If $f$ is irreducible then $\det(\phi)$ is a scalar multiple of
$f^{r}$, where $r$ is the rank of $\operatorname{coker}\phi$ over $S/f$; see
[bgs, Remark 2.5(3)]. Since we are over an algebraically closed field, we can
rescale $\phi$ so that $\det\phi=f^{r}$ on the nose, and we will assume this
throughout.

Our main proof uses the graded ultraproduct ring as defined in [stillman,
§4.1], and we refer the reader to [stillman] for a more detailed discussion or
to [bulletin] for an expository treatment. Let $\mathcal{F}$ be a non-
principal ultrafilter on $\mathbb{N}$. Following conventions [stillman], we
refer to subsets of $\mathcal{F}$ as neighborhoods of $\ast$. Let $\mathbf{R}$
denote the graded ultraproduct of
$\\{k[z_{0},z_{1},,\dots]\\}_{i\in\mathbb{N}}$, where each polynomial ring is
given the standard grading.

######  Remark 2.1.

The key points about ultrafilters and the ring $\mathbf{R}$ are the following:

  1. (1)

Every degree $d$ element $g\in\mathbf{R}$ can be represented by a sequence
$(g_{i})_{i\in\mathbb{N}}$ of degree $d$ elements $g_{i}\in S$. This is a
many-to-one relation, i.e. there are many sequences which correspond to the
same element. The equivalence relation is determined by the ultrafilter.
Specifically, if we represent $g$ as $(g_{i})$ then we have that $g=0$ if and
only if $g_{i}=0$ for all $i$ in some neighborhood of $\ast$.

  2. (2)

For any decomposition of $\mathbb{N}$ into a finite number of disjoint subsets
$\mathbb{N}=\sqcup_{i=1}^{r}\mathcal{N}_{i}$, one of the $\mathcal{N}_{i}$ is
a neighborhood of $\ast$.

For example, say we have a sequence of elements $(a_{i})_{i\in\mathbb{N}}$ and
each $a_{i}\in S$ is homogeneous of degree $\leq D$. We can decompose
$\mathbb{N}=\sqcup_{d=0}^{D}\mathcal{N}_{d}$ where $\mathcal{N}_{d}$ is the
set of indices $i$ such that $\deg(a_{i})=d$. Then one of the
$\mathcal{N}_{d}$ is a neighborhood of $\ast$. Thus, for some $d$, we have
$\deg(a_{i})=d$ for all $i$ in some neighborhood. We will use this argument
several times.

##  3\. Basics on $\mathbf{S}_{d,s}$

Throughout, $d$ will denote a positive integer of degree $\geq 2$. We let
$\mathbf{S}_{d}$ be the space of degree $d$ polynomials in
$S=k[z_{0},z_{1},\dots]$ and let $\mathbf{S}_{d,s}\subseteq\mathbf{S}_{d}$ be
the sublocus consisting of polynomials of strength $\leq s$. We will need some
basic facts about $\mathbf{S}_{d,s}$. We will avoid the more powerful
framework of ${GL}$-varieties introduced in [bdes] because those results are
(at the moment) only developed in characteristic zero, and because the facts
we need are elementary; the interested reader should see [bdes] as well
[draisma] for much more general results on polynomial functors, and [bde,
Theorem 4] [kz-high-rank, Theorem 1.9] for results on strength conditions
which contain invariant subloci.

Since every element of $S$ of degree $\geq 2$ has finite strength, we have
$\mathbf{S}_{d}=\bigcup_{s\geq 0}\mathbf{S}_{d,s}$. One can also realize
$\mathbf{S}_{d}$ as the direct limit $n\to\infty$ of the affine spaces
$\mathbf{S}^{(n)}_{d}$ of degree $d$ polynomials in $k[z_{0},\dots,z_{n}]$,
with respect to the natural inclusions $k[z_{0},\dots,z_{n}]\subseteq
k[z_{0},\dots,z_{n+1}]$; we use the Zariski topology on those spaces to induce
the direct limit topology on $\mathbf{S}_{d}$ and on $\mathbf{S}_{d,s}$.

######  Remark 3.1.

We choose to the space $\mathbf{S}_{d}$ of degree $d$ polynomials in
$k[z_{0},z_{1},\dots]$, which is a direct limit of the $\mathbf{S}_{d}^{(n)}$
as $n\to\infty$. One can also naturally construct an inverse limit of the
$\mathbf{S}_{d}^{(n)}$, and that would lead to studying degree $d$ elements in
the inverse limit of polynomial rings that arises in [stillman, §1.1]. Since
[genstillman, Proposition 2.1] allows one to relate the invariant closed
subsets in the inverse and direct limits, there is no significant distinction
between these two limits for the results we will be using.

Polynomials of degree $d$ and strength $\leq s$ come in different flavors
depending on the degrees of the polynomials arising in the decomposition. Let
$\mu=(\mu_{0},\mu_{1},\dots,\mu_{s})$ be an integer vector satisfying
$1\leq\mu_{0}\leq\mu_{1}\leq\cdots\mu_{s}\leq\lfloor\frac{d}{2}\rfloor$. Let
$f$ be a homogeneous polynomial of degree $d$; a strength decomposition of
type $\mu$ is an expression $f=\sum_{i=0}^{s}g_{i}h_{i}$ where
$\deg(g_{i})=\mu_{i}$. For instance, the polynomial
$f=x_{0}x_{1}^{5}+x_{2}x_{3}^{4}+x_{4}^{3}x_{5}^{3}$ admits a strength
decomposition of type $\mu=(1,2,3)$. We write $\mathbf{S}_{d,\mu}$ for the
subset of $\mathbf{S}_{d}$ consisting of degree $d$ polynomials that admit a
strength decomposition of type $\mu$. The locus $\mathbf{S}_{d,\mu}$ is the
image of the map

(2)
$\pi_{\mu}\colon\mathbf{S}_{\mu_{0}}\times\mathbf{S}_{\mu_{1}}\times\cdots\times\mathbf{S}_{\mu_{s}}\times\mathbf{S}_{d-\mu_{0}}\times\mathbf{S}_{d-\mu_{1}}\times\cdots\times\mathbf{S}_{d-\mu_{s}}\to\mathbf{S}_{d}$

where
$(g_{0},g_{1},\dots,g_{s},h_{0},h_{1},\dots,h_{s})\mapsto\sum_{i=0}^{s}g_{i}h_{i}$.
In particular, $\mathbf{S}_{d,\mu}$ is irreducible.

######  Lemma 3.2.

Fix $d\geq 2$ and $s\geq 0$. Any irreducible component of $\mathbf{S}_{d,s}$
has the form $\mathbf{S}_{d,\mu}$ for some $\mu=(\mu_{1},\dots,\mu_{s})$ with
$1\leq\mu_{1}\leq\mu_{2}\leq\cdots\leq\mu_{s}\leq\lfloor\frac{d}{2}\rfloor$.

###### Proof.

Given $d$ and $s$, there are only finitely many possibilities for $\mu$, and
the union of these $\mathbf{S}_{d,\mu}$ (each of which is irreducible) equals
the space of $\mathbf{S}_{d,s}$. ∎

######  Lemma 3.3.

Let $d_{1},\dots,d_{r}$ be any positive integers. Let $W_{<N}$ be the sublocus
of $\mathbf{S}_{d_{1}}\times\cdots\times\mathbf{S}_{d_{r}}$ of tuples
$(f_{1},\dots,f_{r})$ of collective strength $<N$. Then $W_{<N}$ is contained
in a proper Zariski closed subset of
$\mathbf{S}_{d_{1}}\times\cdots\times\mathbf{S}_{d_{r}}$.

###### Proof.

Write $W^{(n)}_{<N}$ for the locus of tuples $(f_{1},\dots,f_{r})$ of degrees
$d_{1},\dots,d_{r}$ in $k[z_{0},\dots,z_{n}]$ of collective strength $<N$.
Since collective strength is unchanged under adjoining an extra variable, we
have $W_{<N}=\cup_{n}W^{(n)}_{<N}$. Write $\mathbf{S}_{d_{i}}^{(n)}$ for the
space of polynomials of degree $d_{i}$ in $k[z_{0},\dots,z_{n}]$. It suffices
to show that $W^{(n)}_{<N}$ belongs to a proper, Zariski closed subset of
$\mathbf{S}_{d_{1}}^{(n)}\times\cdots\times\mathbf{S}_{d_{r}}^{(n)}$ for all
$n\gg 0$.

Consider the $r\times(n+1)$ Jacobian matrix with $(i,j)$ entry $\frac{\partial
f_{i}}{\partial z_{j}}$ and let $J$ be the ideal of $r\times r$ minors of this
matrix. This ideal is invariant under replacing $f_{1},\dots,f_{r}$ by
elements which generate the same ideal. So if $f_{1},\dots,f_{r}$ has
collective strength $<N$, then some $k$-linear combination of the $f_{i}$ has
strength $<N$; and so we may assume that $f_{1}$ has strength $<N$ without
altering $J$. Write $f_{1}=\sum_{i=0}^{N-1}g_{i}h_{i}$. Each partial
derivative of $f_{1}$ belongs to the ideal generated by the $g_{i}$ and
$h_{i}$ (by the product rule) and thus $J$ also belongs to that ideal. Thus
$J$ has codimension $\leq 2N$ in $S$, because it belongs to an ideal with
$\leq 2N$ generators.

We let $Z\subseteq\mathbf{S}_{d_{1}}\times\cdots\times\mathbf{S}_{d_{r}}$ be
the locus of tuples in $k[z_{0},z_{1},\dots]$ where this ideal of Jacobian
minors $J$ has codimension $\leq 2N$; and we let
$Z^{(n)}\subseteq\mathbf{S}_{d_{1}}^{(n)}\times\cdots\times\mathbf{S}_{d_{r}}^{(n)}$
be the locus of tuples in $k[z_{0},\dots,z_{n}]$ where this ideal $J$ has
codimension $\leq 2N$. Since the codimension of the ideal $J$ is unchanged
under adjoining an extra variable, we have $Z=\cup_{n}Z^{(n)}$.

It thus suffices to prove that $Z^{(n)}$ is Zariski closed and nontrivial for
$n\gg 0$. We consider the universal family
$\mathcal{U}\subseteq\mathbf{S}_{d_{1}}^{(n)}\times\cdots\times\mathbf{S}_{d_{r}}^{(n)}\times\mathbb{P}^{n}$
consisting of tuples $(f_{1},\dots,f_{r})$ of degrees $d_{1},\dots,d_{r}$ in
$k[z_{0},\dots,z_{n}]$ and points $x\in\mathbb{P}^{n}$ such that $x\in
V(f_{1},\dots,f_{r})$. We have a projection

$p\colon\mathcal{U}\to\mathbf{S}_{d_{1}}^{(n)}\times\cdots\times\mathbf{S}_{d_{r}}^{(n)},$

and $Z^{(n)}$ is the locus in the base where the fiber has codimension $\leq
2N$. Since $p$ is proper, $Z^{(n)}$ is closed by upper semicontinuity of fiber
dimensions, e.g. [stacks, 05F6]. For any $n\geq r$, there exist complete
intersections $V(f_{1},\dots,f_{r})$ which are smooth in codimension $n$ (by
the Bertini theorem, for example) and so $Z^{(n)}$ is a proper closed subset
whenever $n>2N$. We conclude that $Z$ is a proper, Zariski closed subset which
contains $W_{<N}$. ∎

Finally we turn to the corollary that we will use to connect Theorems 1.4 and
1.2.

######  Corollary 3.4.

For any $N,d$ and $\mu$, the subset of $f\in\mathbf{S}_{d,\mu}$ having
secondary strength $\geq N$ contains a dense open subset of
$\mathbf{S}_{d,\mu}$.

###### Proof.

Let $U\subseteq\mathbf{S}_{d,\mu}$ be the locus of $f$ of secondary strength
$\geq N$. Let $U^{\prime}$ be the dense open subset of
$\mathbf{S}_{d_{1}}\times\cdots\times\mathbf{S}_{d_{r}}$ constructed in Lemma
3.3 which misses the locus $W_{<N}$ of collective strength $<N$. Then $U$ is,
by definition, the image of $U^{\prime}$ under the map $\pi_{\mu}$ from (2).
So it suffices to check that $\pi_{\mu}(U^{\prime})$ contains a dense open
subset. As in the previous argument, we can check this after restricting to a
polynomial ring involving a finite number of variables.

Let $\mathbf{S}_{d}^{(n)}$ as defined in the proof of Lemma 3.3 and let
$\mathbf{S}^{(n)}_{d,\mu}$ be the intersection of $\mathbf{S}_{d,\mu}$ and
$\mathbf{S}^{(n)}_{d}$. In other words, $\mathbf{S}_{d,\mu}^{(n)}$ is the set
of degree $d$ polynomials in $k[z_{0},\dots,z_{n}]$ which admit a strength
decomposition of type $\mu$. Let
$U^{\prime(n)}=U^{\prime}\cap\left(\mathbf{S}_{d_{1}}^{(n)}\times\cdots\times\mathbf{S}_{d_{r}}^{(n)}\right)$
and let $\pi_{\mu}^{(n)}$ be the natural analogue of the map $\pi_{\mu}$. We
have that $U=\cup_{n}\pi_{\mu}^{(n)}(U^{\prime(n)})$ and so it suffices to
show that $\pi_{\mu}^{(n)}(U^{\prime(n)})$ contains a dense open subset of
$\mathbf{S}_{d,\mu}^{(n)}$ for $n\gg 0$. The proof of Lemma 3.3 showed that
$U^{\prime(n)}$ contains a dense open subset for $n\gg 0$, and so
$\pi_{\mu}^{(n)}(U^{\prime(n)})$ is a dense and constructible subset of
$\mathbf{S}_{d,\mu}^{(n)}$ and thus contains a dense open subset. ∎

##  4\. Proof of Main Results

######  Lemma 4.1.

Let $f\in k[z_{0},\dots,z_{n}]$. Neither $e(f)$ nor $\operatorname{MF-
rank}(f)$ nor $\operatorname{MCM-rank}(f)$ changes if we view $f\in
S=k[z_{0},z_{1},\dots]$.

###### Proof.

The extension $k[z_{0},\dots,z_{n}]\subseteq S=k[z_{0},z_{1},\dots]$ preserves
the codimension of ideals; a detailed proof of this fact is in [stillman,
Proposition 3.3]. Since $e(f)$ is defined in terms of the codimension of the
Jacobian ideal $J=\langle\frac{\partial f}{\partial
z_{i}}|i=0,1,\dots,n\rangle$ it follows that $e(f)$ is invariant upon passage
to $S$. Turning now to $\operatorname{MF-rank}(f)$: if we have a reduced
matrix factorization $(\phi,\psi)$ of $f$ over $k[z_{0},\dots,z_{n}]$ then the
same matrices yield a reduced matrix factorization of $f$ over $S$.
Conversely, consider a reduced matrix factorization $(\phi,\psi)$ of $f$ over
$S$. Let $(\overline{\phi},\overline{\psi})$ be the images of those matrices
under the quotient map $S\to k[z_{0},\dots,z_{n}]$ where $z_{i}=0$ for $i>n$.
Then $(\overline{\phi},\overline{\psi})$ is a reduced matrix factorization of
$f$ over $k[z_{0},\dots,z_{n}]$. It follows $\operatorname{MF-rank}(f)$ is the
same over $k[z_{0},\dots,z_{n}]$ or $S$. Since $\det(\phi)=f^{r}$ in $S$ if
and only if $\det(\overline{\phi})=f^{r}$ in $k[z_{0},\dots,z_{n}]$, we obtain
the desired invariance of $\operatorname{MCM-rank}(f)$. ∎

###### Proof of Theorem 1.4.

In the introduction, we noted that we have $\operatorname{MF-rank}(f)\leq
2^{s}$ and $\operatorname{MCM-rank}(f)\leq 2^{s-1}$; while the essential idea
is that we can simply specialize the matrix factorization of the quadric
$x_{0}y_{0}+\cdots+x_{s}y_{s}$, we provide details here for completeness. For
any $t$, we choose any homogeneous polynomials $g_{0},\dots,g_{t}$ and
$h_{0},\dots,h_{t}$ where $\deg(g_{i})+\deg(h_{i})=d$ for all $i=0,1,\dots,t$;
we will build a reduced, homogeneous matrix factorization of
$g_{0}h_{0}+\dots+g_{t}h_{t}$. For $t=0$ we set $F_{0}=S$ and $B_{0}=S(-\deg
g_{0})$ and we define $\alpha_{0}=(g_{0})$ and $\beta_{0}=(h_{0})$. We have
that $F_{0}(-d)\overset{\beta_{0}}{\to}G_{0}\overset{\alpha_{0}}{\to}F_{0}$ is
then a homogeneous, reduced matrix factorization of $g_{0}h_{0}$. For $t>1$,
we inductively define $F_{t}=F_{t-1}\oplus G_{t-1}(\deg h_{t})$ and
$G_{t}=G_{t-1}\oplus F_{t-1}(-\deg g_{t})$, and we define $\alpha_{t}$ and
$\beta_{t}$ as

$\bordermatrix{&G_{t-1}&F_{t-1}(-\deg g_{t})\cr
F_{t-1}&\alpha_{t-1}&g_{t}\cdot\operatorname{id}\cr G_{t-1}(\deg
h_{t})&h_{t}\cdot\operatorname{id}&-\beta_{t-1}\cr}\ \ \text{ and }\ \
\bordermatrix{&F_{t-1}&G_{t-1}(-\deg g_{t})\cr
G_{t-1}&\beta_{t-1}&g_{t}\cdot\operatorname{id}\cr F_{t-1}(\deg
h_{t})&h_{t}\cdot\operatorname{id}&-\alpha_{t-1}\cr}.$

By induction, one verifies that: the product of $\alpha_{t}$ and $\beta_{t}$
is $g_{0}h_{0}+\dots+g_{t}h_{t}$ times the identity matrix; the above matrices
induce homogeneous maps; and the ranks of $F_{t}$ and $G_{t}$ are $2^{t}$.
Since each nonzero entry of $\alpha_{t}$ or $\beta_{t}$ is one of the $g$’s or
$h$’s, this matrix factorization is also reduced, and we have thus produced a
homogeneous, reduced matrix factorization of $g_{0}h_{0}+\dots+g_{t}h_{t}$ of
rank $2^{t}$ for all $t$. This shows that if $f$ has strength $s$, then
$\operatorname{MF-rank}(f)$ is at most $2^{s}$. Moreover, since
$\det(\alpha_{t})=\det(\beta_{t})$ and
$\det(\alpha_{t}\beta_{t})=f^{\operatorname{rank}F_{t}}=f^{2^{t}}$ we conclude
that $\det(\alpha_{t})=f^{2^{t-1}}$. It follows that if $f$ has strength $s$
then $\operatorname{MCM-rank}(f)$ is at most $2^{s-1}$.

We now turn to showing that there is some $N$ where the following holds: if
the collective strength of $g_{0},\dots,g_{s},h_{0},\dots,h_{s}$ is at least
$N$, then $\operatorname{MF-rank}(f)\geq 2^{s}$. Assume for contradiction that
this is not the case. Then we can choose a sequence of collections of
polynomials $\mathcal{P}_{j}:=(g_{0,j},\dots g_{s,j},h_{0,j},\dots h_{s,j})$
with $j\in\mathbb{N}$ where:

  * •

The collective strength of $\mathcal{P}_{j}$ goes to infinity as $j\to\infty$
and

  * •

$f_{j}:=\sum_{i=0}^{s}g_{i,j}h_{i,j}$ has $\operatorname{MF-
rank}(f_{j})<2^{s}$.

We let $F,G_{0},\dots,G_{s},H_{0},\dots,H_{s}$ denote the elements in the
ultraproduct ring $\mathbf{R}$ corresponding to the sequences
$(f_{j}),(g_{0,j}),\dots(g_{s,j}),(h_{0,j}),\dots,(h_{s,j})$. Since there are
only finitely many positive integers $m<2^{s}$, by Remark 2.1(2), after
passing to a neighborhood of $\ast$, we can assume that $\operatorname{MF-
rank}(f_{j})=m<2^{s}$ for some fixed $m$ and all $j$ in the chosen
neighborhood of $\ast$. For each $j$ in this neighborhood, we choose any
reduced matrix factorization $(\phi_{j},\psi_{j})$ of $f_{j}$ of rank $m$.
Each of the entries of $\phi_{j}$ and $\psi_{j}$ must be homogeneous of degree
at least $1$ and most $d-1$ (because we started with reduced matrix
factorizations); thus by again applying Remark 2.1(2), after passing to a
neighborhood of $\ast$, we can assume that the degree of each entry of
$\phi_{j}$ and $\psi_{j}$ is indepdendent of $j$.

Now we get to the key point: we can take an ultraproduct of these matrix
factorizations $(\phi_{j},\psi_{j})$ of $f_{j}$ to obtain a matrix
factorization of $F$. Specifically, we let $\Phi=(\phi_{j})$ and
$\Psi=(\psi_{j})$ be the $m\times m$ matrices whose entries are the
ultraproducts of the entries of the $\phi_{j}$ and $\psi_{j}$ matrices. Since
$\phi_{j}\cdot\psi_{j}=f_{j}\cdot\operatorname{id}_{S^{m}}=\psi_{j}\cdot\phi_{j}$
for all $j$ in our neighborhood, these equations pass to the ultraproduct and
we have
$\Phi\cdot\Psi=F\cdot\operatorname{id}_{\mathbf{R}^{m}}=\Psi\cdot\Phi.$ In
particular, $(\Phi,\Psi)$ is a homogeneous, reduced matrix factorization of
$F$ of rank $m<2^{s}$.

We now produce the contradiction. The ultrapower ring $\mathbf{R}$ is a
polynomial ring by [stillman, Theorem 1.3]. Moreoever, since the collective
strength of $\mathcal{P}_{j}$ went to infinity, the elements
$G_{0},\dots,G_{s},H_{0},\dots,H_{s}$ have infinite collective strength. These
elements are thus abstract variables in the polynomial ring $\mathbf{R}$ by
[stillman, Theorem 1.3], and we have $F=G_{0}H_{0}+\cdots+G_{s}H_{s}$. The
degree of $G_{i}$ is $\mu_{i}$ and the degree of $H_{i}$ is $d-\mu_{i}$. But
the classification of matrix factorizations of quadrics from [knorrer] does
not make use of the grading, and so Knörrer’s classification (when the
characteristic of $k$ is not $2$) implies that $\operatorname{MF-rank}(F)\geq
2^{s}$. When the characteristic of $k$ is $2$, one can instead apply [beh,
Proposition 3.2] to obtain $\operatorname{MF-rank}(F)\geq 2^{s}$. We have thus
contradicted the conclusion of the previous paragraph.

For the statement about $\operatorname{MCM-rank}(f)$, the argument will be
quite similar. We must show that there is some $N$ such that if the collective
strength of $g_{0},\dots,g_{s},h_{0},\dots,h_{s}$ is at least $N$, then
$\operatorname{MCM-rank}(f)\geq 2^{s-1}$. Assume that this is not the case.
Then we can choose a sequence of collections of polynomials
$\mathcal{P}_{j}:=(g_{0,j},\dots g_{s,j},h_{0,j},\dots h_{s,j})$ with
$j\in\mathbb{N}$ where:

  * •

The collective strength of $\mathcal{P}_{j}$ goes to infinity as $j\to\infty$
and

  * •

$f_{j}:=\sum_{i=0}^{s}g_{i,j}h_{i,j}$ has $\operatorname{MCM-
rank}(f_{j})<2^{s-1}$.

The latter condition implies that, for each $j$, there is a matrix
factorization $(\phi_{j},\psi_{j})$ of $f_{j}$ where
$\det(\phi_{j})=f_{j}^{r_{j}}$ and $r_{j}<2^{s-1}$. From here, the argument is
nearly identical to the one above: by passing to a neighborhood of $\ast$, we
can produce homogeneous matrices $(\Phi,\Psi)$ over $\mathbf{R}$ where
$\det(\Phi)=F^{r}$ for some $r<2^{s-1}$. But $F$ is a quadric in the
polynomial ring $\mathbf{R}$, and this would contradict the known results for
quadrics [knorrer, beh]. ∎

###### Proof of Theorem 1.2.

By Lemma 3.2 it suffices to prove the claim for an open subset of
$\mathbf{S}_{d,\mu}$ for each $\mu=(\mu_{0},\dots,\mu_{s})$ satisfying
$1\leq\mu_{0}\leq\mu_{1}\leq\cdots\mu_{s}\leq\lfloor\frac{d}{2}\rfloor$. In
other words, we may assume that $f$ has a decomposition
$f=\sum_{i=0}^{s}g_{i}h_{i}$ with $\deg(g_{i})=\mu_{i}$ and prove that the
claim holds for a generic such $f$.

By a simple application of the product rule, the Jacobian ideal
$\langle\frac{\partial f}{\partial z_{i}}|i=0,1,\dots\rangle$ belongs to the
ideal $\langle g_{0},\dots,g_{s},h_{0},\dots,h_{s}\rangle$; this ideal has
codimension $\leq 2s+2$ in $S$ by the Principal Ideal Theorem, and thus it has
codimenion $\leq 2s+1$ in $S/f$. It follows that the codimension of the
singular locus of $S/f$ in $S/f$ is at most $2s+1$ and thus that $s\geq
e(f)+1$. It therefore suffices to prove $\operatorname{MF-rank}(f)\geq 2^{s}$.

Theorem 1.4 implies that there is some $N$ such that: $\operatorname{MF-
rank}(f)\geq 2^{s}$ and $\operatorname{MCM-rank}(f)\geq 2^{s-1}$ for any
$f\in\mathbf{S}_{d,\mu}$ where the collective strength of
$g_{0},\dots,g_{s},h_{0},\dots,h_{s}$ is at least $N$, and Corollary 3.4
implies the locus of secondary strength $\geq N$ in $\mathbf{S}_{d,\mu}$
contains a dense open subset. Since $s\geq e(f)+1$, this implies the desired
result. ∎

######  Example 4.2.

Theorem 1.4 implies that, for any given $d$ and $e$, there is a polynomial $F$
of degree $d$ and $e(F)=e$ where the Buchweitz-Greuel-Schreyer Conjecture
holds and is sharp. Let $F=\sum_{i=0}^{s}x_{i}g_{i}$ where each $g_{i}\in
k[y_{i,0},y_{i,1},\cdots,y_{i,N}]$ is a generic polynomial of degree $d-1$ for
some $N\gg 0$. Since the $x$’s and the $g$’s all involve different set of
variables, we can assume that the secondary strength of $F$ is at least the
strength of $g_{0}$, which is at least $\frac{N-1}{2}$ because
$V(g_{i})\subseteq\mathbb{P}^{N}$ is smooth. So Theorem 1.4 applies to $F$ for
$N\gg 0$. We always have $s\leq e(F)+1$ and we must check that $s=e(F)+1$. We
omit the details, but the basic point is that the Jacobian ideal of $F$
contains $g_{0},\dots,g_{s}$ and–after taking ideal quotients with respect to
the Jacobian ideals of $g_{0},\dots,g_{s}$–it contains $x_{0},\dots,x_{s}$ as
well. To give a specific example in characteristic $0$, we could let
$g_{i}=y_{i,0}^{d-1}+\cdots+y_{i,N}^{d-1}$ for $0\leq i\leq s$.

Theorem 1.2 has an implication for arithmetically Cohen-Macaulay vector
bundles on hypersurfaces, as noted in [bgs, Conjecture B].

######  Corollary 4.3.

There exists $N$ depending on $d$ and $s$ such that: if $f\in
k[z_{0},\dots,z_{n}]$ has degree $d$, strength $s$, and secondary strength
$\geq N$, then any vector bundle $\mathcal{E}$ of rank $<2^{s-1}$ on the
hypersurface $V(f)\subseteq\mathbb{P}^{n}$ which satsifies

$H^{i}(X,\mathcal{E}(e))=0\text{ for all }0<i<n-1\text{ and all
}e\in\mathbb{Z}$

must be a direct sum of line bundles
$\mathcal{E}=\oplus_{j}\mathcal{O}_{X}(a_{j})$.

See also [mrr1, mrr2, rt1, rt2, rt3, tripathi] where a number of cases of
[bgs, Conjecture B] are shown for hypersurfaces in $\mathbb{P}^{n}$ with
$n\leq 6$.

######  Remark 4.4.

Theorem 1.4 shows that strength determines $\operatorname{MF-rank}(f)$ and
$\operatorname{MCM-rank}(f)$ for generic $f$, but Faenzi and Pons–Llopis have
pointed out that this cannot hold for abitrary $f$. For instance, they noted
the Pfaffian of a general skew-symmetric $6\times 6$ matrix of linear forms,
in $\geq 6$ variables, will have strength $\geq 3$; but the skew-symmetric
matrix is part of a rank $6$ matrix factorization, and $6<2^{3}$. Similarly,
the cokernel of the skew-symmetric matrix is a rank $2<2^{3-1}$ MCM module (an
Ulrich module, in fact) on the hypersurface.

This provides a counterexample to a question from a prior draft of this paper
and raises a question: if $f\in R=k[x_{1,1},\dots,x_{n,n}]$ is the determinant
of a generic $n\times n$ matrix of linear forms, then $f$ has a matrix
factorization of rank $n$, namely, the matrix $(x_{i,j})$ and its adjoint. By
Laplace expansion along the top row, one can see that the strength of $f$ is
at most $n-1$. Is the strength equal to $n$? If so, this would show that
$\operatorname{MF-rank}(f)$ and $\operatorname{MCM-rank}(f)$ can be
arbitrarily far away from $2^{\operatorname{strength}(f)}$. A similar question
can be asked about Pfaffians.

The proof technique raises some new geometric and arithmetic directions to
consider. For instance, if the strength of a polynomial $f$ is sufficiently
large, then it is known that $V(f)$ is unirational; this follows from the main
result of [hmp] and various results relating strength with the codimension of
the singular locus c.f. [ananyan-hochster, kz-ranks]. But what happens if the
strength is small–so that $V(f)$ is highly singular–but $f$ has large
secondary strength?

######  Question 4.5.

Let $f$ be a degree $d$, strength $s$ polynomial. If the secondary strength of
$f$ is sufficiently large relative to $s,d$, will $V(f)$ be unirational? For a
specific example, consider

$f=(x_{0}+\cdots+x_{n})(x_{0}^{4}+\cdots+x_{n}^{4})+(x_{0}^{2}+\cdots+x_{n}^{2})(x_{0}^{3}+\cdots+x_{n}^{3}).$

The hypersurface $V(f)$ is singular in codimension $4$ and $f$ has strength
$1$, but its secondary strength will go to $\infty$ as $n\to\infty$. If $n\gg
0$, will $V(f)\subseteq\mathbb{P}^{n-1}$ be unirational?

One can also ask arithmetic questions in this vein.

######  Question 4.6.

Let $f\in\mathbb{Q}[z_{0},\dots,z_{n}]$ be a homogeneous polynomial of degree
$d$ and strength $s$ and sufficiently high secondary strength. Does $V(f)$
have nice arithmetic properties, similar to those of high strength polynomials
as observed in [birch, cook-magyar, schmidt, xy, y]? For example: does $V(f)$
satisfy the Hasse principle?

## References