# The Dynamics of Fluctuating Thin Sheets Under Random Forcing

Chanania Steinbock  Eytan Katzav  Racah Institute of Physics, The Hebrew
University, Jerusalem 9190401, Israel

###### Abstract

We study the dynamic structure factor of fluctuating elastic thin sheets
subject to conservative (athermal) random forcing. In Steinbock, Katzav &
Boudaoud, Phys. Rev. Research 4, 033096 (2022), the static structure factor of
a such a sheet was studied. In this paper, we recap the model developed there
and investigate its dynamic properties. Using the self-consistent expansion
(SCE), the time dependent two-point function of the height profile is
determined and found to decay exponentially in time. Despite strong nonlinear
coupling, the decay rate of the dynamic structure factor is found to coincide
with the effective coupling constant for the static properties which suggests
that the model under investigation exhibits certain quasi-linear behaviour.
Confirmation of these results by numerical simulations is also presented.

##  I Introduction

Thin sheets and surfaces are ubiquitous in everyday life yet the theory of
their physical properties remains incomplete. For instance, despite the fact
that crumpled paper balls take but a moment to make, the response of a thin
sheet to random forcing remains poorly understood. Since randomly driven thin
surfaces are relevant to a wide diversity of fields, ranging from the physics
of crumpled paper to the properties of graphene to the behaviour of cell
membranes, a theory of randomly driven surfaces derived from first principles
would have far reaching consequences.

Loosely speaking, we can distinguish between two types of random forcing,
completely uncorrelated white noise typical of thermal fluctuations and
deliberate correlated noise such as the type of forcing applied when crumpling
a sheet of paper. Here, we will focus on the latter kind of noise. Perhaps the
easiest way to probe the structure of an athermally fluctuating sheet is to
study the properties of the resultant crumpled sheet and indeed this is an
active field of research in its own right, both experimentally [1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11] and through mathematical or numerical modeling [1, 12, 13,
14]. The development of the theory of singular structures supported by thin
sheets such as d-cones and ridges [15, 16, 17, 18, 19, 20, 21] has gone some
way to bridging research into crumpled sheets with that of fluctuating sheets
however its applicability has been limited by the impracticality of
characterising sheets with more than a handful of ridges. Additionally, the
structure of crumpled sheets can at most inform us of the static unvarying
properties of fluctuating systems. To obtain insight into the complete dynamic
structure of a fluctuating thin sheet, we must tackle such a system directly.

Previous research into the time-dependent dynamic structure of fluctuating
surfaces is limited but has been studied in the context of tethered surfaces
[22, 23] and polymerised membranes [24, 25], though this research focused
exclusively on thermally driving white noise. In the context of tethered
surfaces [22, 23], the dynamics of phantom and self-avoiding flexible sheets
was studied though at the cost of neglecting the elastic properties of real
sheets. In contrast, [24] focuses on the dynamic character of elastic
polymerised membranes coupled to a random perturbing flowing fluid. Finally,
[25] uses a super-symmetric $\varepsilon$-expansion of a $D=4-\varepsilon$
dimensional membrane to obtain the dynamic exponent of an elastic thermally
fluctuating polymerised membrane.

Recently, we showed how the static properties of a fluctuating sheet can be
derived directly by applying techniques from out-of-equilibrium statistical
mechanics to the physics of elastic systems [26]. In particular, we developed
a dynamic variant of the Föppl-von Kármán equations which describes the
deformations of thin sheets and used this to obtain the static structure
factor of a fluctuating thin sheet driven by athermal noise. In the current
paper, we extend this approach to derive the time-dependent structure factor
of the fluctuating sheet. This dynamic structure should be of direct relevance
in understanding many features of the sheet, including its acoustic emissions
[27, 28, 29, 30], optical signature [31] and dissipative character [32, 33].
Further applications to diverse fields [34] such as the biophysics of cell
membranes [24, 35], the properties and stability of fluctuating graphene
sheets [36, 37, 38, 39, 40, 41] and wave turbulence [42] can also be
envisioned.

The paper is organised as follows. In Section II, we briefly recap the
derivation of the overdamped Föppl-von Kármán equations developed in [26] and
in Section III, we apply the self-consistent expansion (SCE) to these
equations to determine the dynamic structure factor of the fluctuating sheet.
In Section IV, the accuracy of our solution is confirmed by comparison with
numerical simulations. Finally, the implications of these results are
discussed in Section V.

##  II The Overdamped Dynamic Föppl-von Kármán Equations

In [26], we developed a model to describe fluctuating elastic thin sheets. In
this section, we recap the main ideas and relate them to the dynamic structure
factor of such a system.

The equilibrium out-of-plane displacement $\xi\left(x,y\right)$ of a thin
elastic sheet subject to an external pressure $P_{\textrm{ex}}$ is given by
the well known Föppl-von Kármán equations [43]

$\displaystyle P_{\textrm{ex}}$
$\displaystyle=\frac{Eh^{3}}{12\left(1-\nu^{2}\right)}\nabla^{4}\xi$
$\displaystyle-h\left(\frac{\partial^{2}\xi}{\partial
x^{2}}\frac{\partial^{2}\chi}{\partial y^{2}}+\frac{\partial^{2}\xi}{\partial
y^{2}}\frac{\partial^{2}\chi}{\partial x^{2}}-2\frac{\partial^{2}\xi}{\partial
x\partial y}\frac{\partial^{2}\chi}{\partial x\partial y}\right)$ (1)
$\displaystyle 0$
$\displaystyle=\nabla^{4}\chi+E\left[\frac{\partial^{2}\xi}{\partial
x^{2}}\frac{\partial^{2}\xi}{\partial
y^{2}}-\left(\frac{\partial^{2}\xi}{\partial x\partial
y}\right)^{2}\right]\,,$ (2)

where $h$, $E$ and $\nu$ denote the sheet thickness, Young’s modulus and
Poisson ratio respectively. The scalar field $\chi\left(x,y\right)$ denotes
the Airy stress potential of the deformation. By writing the out-of-plane
displacement $\xi\left(x,y\right)$ in the Monge parameterisation, ie. as a
function of $x$ and $y$, we assume that deformations of our sheet are mostly
flat and thus our focus here will be on weak fluctuations.

To explore the dynamics of a driven fluctuating sheet, we apply Newton’s
second law to each element of the sheet with density $\rho$

$h\rho\frac{\partial^{2}\xi}{\partial
t^{2}}=-P_{\textrm{ex}}+P_{\textrm{damping}}+P_{\textrm{driving}}\,.$ (3)

where $P_{\textrm{driving}}$ and $P_{\textrm{damping}}$ describe driving and
damping forces respectively. Though variations of this equation have been
studied in the context of wave turbulence [44, 45, 46, 47, 48, 49, 50, 51, 52,
53, 42], here we continue the approach introduced in [26] of a sheet subject
to ordinary fluid friction
$P_{\textrm{damping}}=-\alpha\frac{\partial\xi}{\partial t}$ being driven by
conserved Gaussian noise $P_{\textrm{driving}}=\eta\left(\vec{r},t\right)$
with noise amplitude $D$, that is,

$\displaystyle\left\langle\eta\left(\vec{r},t\right)\right\rangle$
$\displaystyle=0$ (4)
$\displaystyle\left\langle\eta\left(\vec{r},t\right)\eta\left(\vec{r}\,^{\prime},t^{\prime}\right)\right\rangle$
$\displaystyle=-D\delta\left(t-t^{\prime}\right)\nabla^{2}\delta\left(\vec{r}-\vec{r}\,^{\prime}\right)\,.$
(5)

Other driving forces could be considered, but as argued in [26], there is
value in studying the setup where the sheet’s center of mass does not wander
in space and hence we impose conserved noise on the sheet. More specific forms
of noise which are consistent with the conservation of center of mass could
also be considered but following the principle of parsimony, we consider only
the simplest possibility here.

Taking the overdamped limit and thus neglecting the inertia term
$h\rho\frac{\partial^{2}\xi}{\partial t^{2}}$, this approach provides a
concrete model for a driven fluctuating elastic sheet, namely, the overdamped
dynamic Föppl-von Kármán equation

$\alpha\frac{\partial\xi}{\partial
t}+\frac{Eh^{3}}{12\left(1-\nu^{2}\right)}\nabla^{4}\xi\\\
-h\left(\frac{\partial^{2}\xi}{\partial x^{2}}\frac{\partial^{2}\chi}{\partial
y^{2}}+\frac{\partial^{2}\xi}{\partial y^{2}}\frac{\partial^{2}\chi}{\partial
x^{2}}-2\frac{\partial^{2}\xi}{\partial x\partial
y}\frac{\partial^{2}\chi}{\partial x\partial
y}\right)=\eta\left(\vec{r},t\right)\,.$ (6)

where the Airy stress potential $\chi\left(x,y\right)$ is still determined by
equation (2).

The fundamental difference between the problem under study here and the one
studied in the wave turbulence community [44, 45, 46, 47, 48, 49, 50, 51, 52,
53, 42] is that they focus on the regime where inertia is very important,
while friction is present only at the smallest scales. Also, the forcing of
the sheet, which is often modeled as white noise, is applied only at the
largest scales. As a result, the main feature which is studied is the energy
cascade from the large scales (where the forcing is applied) to the smallest
scales (where it is dissipated). In fact, there exist concrete predictions
regarding this energy cascade depending on the specific scenario that drives
this cascade. In contrast, we focus on the dynamics of the structure of the
sheet under forcing and friction across all scales.

It is shown in [26] that for a sheet with dimensions $L\times L$, equations
(2) and (6) can be combined into a single equation for the Fourier components
$\tilde{\xi}_{\vec{n}}\left(t\right)$ of
$\xi\left(x,y,t\right)=\sum_{\vec{n}}\tilde{\xi}_{\vec{n}}\left(t\right)e^{i\frac{2\pi}{L}\vec{n}\cdot\vec{r}}$
where the sum is taken over all lattice points of $\mathbb{Z}^{2}$. After
nondimensionalising, one obtains the following Langevin equation

$\frac{\partial\bar{\xi}_{\vec{n}}}{\partial\bar{t}}+g\left|\vec{n}\right|^{4}\bar{\xi}_{\vec{n}}\\\
+\frac{1}{2}\sum_{\vec{\ell}_{1}\neq\vec{n}}\sum_{\vec{\ell}_{2}}\sum_{\vec{\ell}_{3}}V_{\vec{n},\vec{\ell}_{1},\vec{\ell}_{2},\vec{\ell}_{3}}\bar{\xi}_{\vec{\ell}_{1}}\bar{\xi}_{\vec{\ell}_{2}}\bar{\xi}_{\vec{\ell}_{3}}=\bar{\eta}_{\vec{n}}\left(\bar{t}\,\right)$
(7)

containing a single dimensionless parameter

$g=\frac{2\pi}{12\left(1-\nu^{2}\right)}\sqrt{\frac{\alpha h^{5}E}{D}}\,.$ (8)

The scaled time and Fourier components are given by

$\displaystyle\bar{t}$
$\displaystyle=\left[\left(2\pi\right)^{6}hDE/\left(\alpha^{3}L^{8}\right)\right]^{1/2}t$
(9) $\displaystyle\bar{\xi}_{\vec{n}}$
$\displaystyle=\left[\left(2\pi\right)^{2}\alpha
hE/D\right]^{1/4}\tilde{\xi}_{\vec{n}}$ (10)

and the dimensionless noise in Fourier space has mean 0 and variance

$\left\langle\bar{\eta}_{\vec{n}}\left(\bar{t}\,\right)\bar{\eta}_{\vec{n}^{\prime}}\left(\bar{t}\,^{\prime}\right)\right\rangle=\left|\vec{n}\right|^{2}\delta_{\vec{n},-\vec{n}^{\prime}}\delta\left(\bar{t}-\bar{t}\,^{\prime}\right)\,.$
(11)

Finally, the kernel $V_{\vec{n},\vec{\ell}_{1},\vec{\ell}_{2},\vec{\ell}_{3}}$
is simply the Fourier transform of the transverse projection operator of the
sheet deformation [34, 54] and is given by

$V_{\vec{n},\vec{\ell}_{1},\vec{\ell}_{2},\vec{\ell}_{3}}=\delta_{\vec{n},\vec{\ell}_{1}+\vec{\ell}_{2}+\vec{\ell}_{3}}\frac{\left|\vec{n}\times\vec{\ell}_{1}\right|^{2}\left|\vec{\ell}_{2}\times\vec{\ell}_{3}\right|^{2}}{\left|\vec{n}-\vec{\ell}_{1}\right|^{4}}\,,$
(12)

where we have denoted
$\left|\vec{n}\times\vec{\ell}\,\right|=n_{x}\ell_{y}-n_{y}\ell_{x}$, thus
equation (7) can be thought of as a type of $\phi^{4}$-field Langevin equation
with a non-trivial spatially varying kernel
$V_{\vec{n},\vec{\ell}_{1},\vec{\ell}_{2},\vec{\ell}_{3}}$ [55]. Accordingly,
in principle, equation (7) can be used to find structure factors such as the
time-dependent two-point function

$S_{\vec{n}}\left(\bar{t},\bar{t}\,^{\prime}\right)=\left\langle\bar{\xi}_{\vec{n}}\left(\bar{t}\,\right)\bar{\xi}_{-\vec{n}}\left(\bar{t}\,^{\prime}\right)\right\rangle$
(13)

which in steady-state will only depend on the difference
$\Delta\bar{t}=\left|\bar{t}-\bar{t}\,^{\prime}\right|$ and thus can be
written as a function of a single argument as

$S_{\vec{n}}\left(\bar{t}\,\right)=\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{-\vec{n}}\left(\bar{t}\,\right)\right\rangle\,.$
(14)

Unfortunately, the single dimensionless parameter $g$ in equation (7) is
coupled to its linear part and in [26], it is argued that $g$ is typically
small since $g\sim 0.1$ for a typical sheet of aluminum or steel. Indeed, the
scaling $g\sim h^{5/2}$ ensures that for any sufficiently thin sheet, $g$ will
be small and thus any expansion around the linear part of equation (7) which
treats the nonlinear part as a mere correction is guaranteed to fail. Instead,
following the success of [26], we will analyse equation (7) by application of
the self-consistent expansion (SCE).

##  III The Self-Consistent Expansion

As described in [26], the self-consistent expansion (SCE) can be thought of as
a renormalised perturbation theory [56] capable of providing series
approximations even in the presence of strong coupling. The method has found
previous application to the KPZ equation and its variations [57, 58, 59, 60,
61, 62, 63, 64, 65, 66], fracture and wetting fronts [67, 68] and turbulence
[69]. More relevant to our system, the SCE provides an extremely successful
solution to the zero-dimensional $\phi^{4}$-theory giving good results at low
orders and exact convergence at high orders [70, 71]. Accordingly, the success
of the SCE in determining the static structure factor of a fluctuating sheet
in [26] was not entirely unexpected and since the SCE has a natural extension
to dynamic quantities, we extend the approach taken in [26] here.

###  III.1 The Fokker-Planck Equation and the SCE

As in [26], we begin by writing the Fokker-Planck equation corresponding to
equation (7) [72]

$\frac{\partial
P}{\partial\bar{t}}=\frac{1}{2}\sum_{\vec{n}}\left|\vec{n}\right|^{2}\frac{\partial^{2}P}{\partial\bar{\xi}_{\vec{n}}\partial\bar{\xi}_{-\vec{n}}}+g\sum_{\vec{n}}\left|\vec{n}\right|^{4}\frac{\partial}{\partial\bar{\xi}_{\vec{n}}}\left(\bar{\xi}_{\vec{n}}P\right)\\\
+\frac{1}{2}\sum_{\vec{n}}\frac{\partial}{\partial\bar{\xi}_{\vec{n}}}\left[P\sum_{\vec{\ell}_{1}\neq\vec{n}}\sum_{\vec{\ell}_{2}}\sum_{\vec{\ell}_{3}}V_{\vec{n},\vec{\ell}_{1},\vec{\ell}_{2},\vec{\ell}_{3}}\bar{\xi}_{\vec{\ell}_{1}}\bar{\xi}_{\vec{\ell}_{2}}\bar{\xi}_{\vec{\ell}_{3}}\right]\,,$
(15)

where
$P=P\left(\left\\{\bar{\xi}_{\vec{n}}\left(\bar{t}\,\right)\right\\},\bar{t}\,\right)$
denotes the probability functional that the system will have a specific
configuration, as prescribed by the Fourier components
$\left\\{\bar{\xi}_{\vec{n}}\left(\bar{t}\,\right)\right\\}$ at time
$\bar{t}$. We can multiply this equation by a function of the Fourier
components
$\mathbb{F}\left(\left\\{\bar{\xi}_{\vec{n}}\left(\bar{t}\,\right)\right\\}\right)$
and integrate over all $\bar{\xi}_{\vec{n}}\left(\bar{t}\,\right)$ to obtain
the following equation for the expectations

$\frac{\partial\left\langle\mathbb{F}\right\rangle}{\partial\bar{t}}=\frac{1}{2}\sum_{\vec{n}}\left|\vec{n}\right|^{2}\left\langle\frac{\partial^{2}\mathbb{F}}{\partial\bar{\xi}_{\vec{n}}\partial\bar{\xi}_{-\vec{n}}}\right\rangle-g\sum_{\vec{n}}\left|\vec{n}\right|^{4}\left\langle\frac{\partial\mathbb{F}}{\partial\bar{\xi}_{\vec{n}}}\bar{\xi}_{\vec{n}}\right\rangle\\\
-\frac{1}{2}\sum_{\vec{n}}\sum_{\vec{\ell}_{1}\neq\vec{n}}\sum_{\vec{\ell}_{2}}\sum_{\vec{\ell}_{3}}V_{\vec{n},\vec{\ell}_{1},\vec{\ell}_{2},\vec{\ell}_{3}}\left\langle\frac{\partial\mathbb{F}}{\partial\bar{\xi}_{\vec{n}}}\bar{\xi}_{\vec{\ell}_{1}}\bar{\xi}_{\vec{\ell}_{2}}\bar{\xi}_{\vec{\ell}_{3}}\right\rangle$
(16)

where we have defined the expectation values

$\left\langle\mathbb{F}\right\rangle=\int\prod_{\vec{n}}d\bar{\xi}_{\vec{n}}\,\mathbb{F}\left(\left\\{\bar{\xi}_{\vec{n}}\right\\}\right)P\left(\left\\{\bar{\xi}_{\vec{n}}\right\\},\bar{t}\,\right)\,.$
(17)

Equation (16) can be used to obtain relationships between various moments. For
instance, in [26], it was shown that subbing in
$\mathbb{F}=\bar{\xi}_{\vec{n}}\bar{\xi}_{\vec{n}^{\prime}}$ results in an
equation relating the static two-point function
$\left\langle\bar{\xi}_{\vec{n}}\bar{\xi}_{\vec{n}^{\prime}}\right\rangle$ to
the static four-point function
$\left\langle\bar{\xi}_{\vec{n}_{1}}\bar{\xi}_{\vec{n}_{2}}\bar{\xi}_{\vec{n}_{3}}\bar{\xi}_{\vec{n}_{4}}\right\rangle$.
Similarly, to obtain relations for dynamic quantities, we can sub in time
dependent functions such as
$\mathbb{F}\left(\left\\{\bar{\xi}_{\vec{n}}\left(\bar{t}\,\right)\right\\}\right)=\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{n}^{\prime}}\left(\bar{t}\,\right)$
which results in the ODE

$\frac{\partial\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{n}^{\prime}}\left(\bar{t}\,\right)\right\rangle}{\partial\bar{t}}=-g\left|\vec{n}^{\prime}\right|^{4}\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{n}^{\prime}}\left(\bar{t}\,\right)\right\rangle\\\
-\frac{1}{2}\sum_{\vec{\ell}_{1}\neq\vec{n}^{\prime}}\sum_{\vec{\ell}_{2}}\sum_{\vec{\ell}_{3}}V_{\vec{n}^{\prime},\vec{\ell}_{1},\vec{\ell}_{2},\vec{\ell}_{3}}\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{\ell}_{1}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{2}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{3}}\left(\bar{t}\,\right)\right\rangle\,.$
(18)

This first order non-homogeneous ODE provides the time-dependent two-point
function
$\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{n}^{\prime}}\left(\bar{t}\,\right)\right\rangle$
if given the time-dependent four-point function
$\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{\ell}_{1}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{2}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{3}}\left(\bar{t}\,\right)\right\rangle$.
An ODE for the four-point function can of course be obtained by subbing
$\mathbb{F}\left(\left\\{\bar{\xi}_{\vec{n}}\left(\bar{t}\,\right)\right\\}\right)=\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{\ell}_{1}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{2}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{3}}\left(\bar{t}\,\right)$
into equation (16) though the resulting ODE would require knowledge of the
time-dependent six-point function. As observed in [26], this situation of
needing higher moments to find lower ones is similar to the BBGKY hierarchy
[73, 74, 75] and finding closure is in general non-trivial. The naive approach
would be to simply neglect the non-homogeneous part of equation (18) and then
attempt to perturbatively correct for it however since the small parameter $g$
is coupled to the homogeneous part of equation (18), the non-homogeneous
contribution is large and non-negligible and thus such an approach is
guaranteed to fail. Since this occurs at every level of the hierarchy, a more
sophisticated approach is required.

Following the approach taken in [26], we apply the SCE to equation (16) by
introducing a free parameter $\Gamma_{\left|\vec{n}\right|}$

$\frac{\partial\left\langle\mathbb{F}\right\rangle}{\partial\bar{t}}=\frac{1}{2}\sum_{\vec{n}}\left|\vec{n}\right|^{2}\left\langle\frac{\partial^{2}\mathbb{F}}{\partial\bar{\xi}_{\vec{n}}\partial\bar{\xi}_{-\vec{n}}}\right\rangle-\sum_{\vec{n}}\Gamma_{\left|\vec{n}\right|}\left\langle\frac{\partial\mathbb{F}}{\partial\bar{\xi}_{\vec{n}}}\bar{\xi}_{\vec{n}}\right\rangle\\\
-\sum_{\vec{n}}\left(g\left|\vec{n}\right|^{4}-\Gamma_{\left|\vec{n}\right|}\right)\left\langle\frac{\partial\mathbb{F}}{\partial\bar{\xi}_{\vec{n}}}\bar{\xi}_{\vec{n}}\right\rangle\\\
-\frac{1}{2}\sum_{\vec{n}}\sum_{\vec{\ell}_{1}\neq\vec{n}}\sum_{\vec{\ell}_{2}}\sum_{\vec{\ell}_{3}}V_{\vec{n},\vec{\ell}_{1},\vec{\ell}_{2},\vec{\ell}_{3}}\left\langle\frac{\partial\mathbb{F}}{\partial\bar{\xi}_{\vec{n}}}\bar{\xi}_{\vec{\ell}_{1}}\bar{\xi}_{\vec{\ell}_{2}}\bar{\xi}_{\vec{\ell}_{3}}\right\rangle\,.$
(19)

One can think of $\Gamma_{\left|\vec{n}\right|}$ as an effective coupling
constant such that a perturbative expansion around the linear theory with
$\Gamma_{\left|\vec{n}\right|}$ is valid. The problem of determining its value
will be deferred to later though due to the isotropic character of our system,
we have already assumed that $\Gamma_{\left|\vec{n}\right|}$ can only depend
on the magnitude of $\vec{n}$ and not its direction. Now if
$\left\langle\mathbb{F}\right\rangle^{\left(m\right)}$ denotes an $m^{th}$
order expansion of $\left\langle\mathbb{F}\right\rangle$, then by assumption,
the latter terms will contribute at a higher order and thus we can write the
iterative relation

$\frac{\partial\left\langle\mathbb{F}\right\rangle^{\left(m\right)}}{\partial\bar{t}}=\frac{1}{2}\sum_{\vec{n}}\left|\vec{n}\right|^{2}\left\langle\frac{\partial^{2}\mathbb{F}}{\partial\bar{\xi}_{\vec{n}}\partial\bar{\xi}_{-\vec{n}}}\right\rangle^{\left(m\right)}\\\
-\sum_{\vec{n}}\Gamma_{\left|\vec{n}\right|}\left\langle\frac{\partial\mathbb{F}}{\partial\bar{\xi}_{\vec{n}}}\bar{\xi}_{\vec{n}}\right\rangle^{\left(m\right)}-\sum_{\vec{n}}\left(g\left|\vec{n}\right|^{4}-\Gamma_{\left|\vec{n}\right|}\right)\left\langle\frac{\partial\mathbb{F}}{\partial\bar{\xi}_{\vec{n}}}\bar{\xi}_{\vec{n}}\right\rangle^{\left(m-1\right)}\\\
-\frac{1}{2}\sum_{\vec{n}}\sum_{\vec{\ell}_{1}\neq\vec{n}}\sum_{\vec{\ell}_{2}}\sum_{\vec{\ell}_{3}}V_{\vec{n},\vec{\ell}_{1},\vec{\ell}_{2},\vec{\ell}_{3}}\left\langle\frac{\partial\mathbb{F}}{\partial\bar{\xi}_{\vec{n}}}\bar{\xi}_{\vec{\ell}_{1}}\bar{\xi}_{\vec{\ell}_{2}}\bar{\xi}_{\vec{\ell}_{3}}\right\rangle^{\left(m-1\right)}\,,$
(20)

supplemented with the convention that for $m=0$ the $m-1$ terms drop out.

This equation can now be used to obtain any moment up to any order. For
instance, in [26], it was shown that subbing in
$\mathbb{F}=\bar{\xi}_{\vec{n}}\bar{\xi}_{\vec{n}^{\prime}}$ or
$\mathbb{F}=\bar{\xi}_{\vec{n}}\bar{\xi}_{\vec{\ell}_{1}}\bar{\xi}_{\vec{\ell}_{2}}\bar{\xi}_{\vec{\ell}_{3}}$
together with $m=0$ directly results in zeroth order expressions for the
static two-point and four-point functions

$\left\langle\bar{\xi}_{\vec{n}}\bar{\xi}_{\vec{n}^{\prime}}\right\rangle^{\left(0\right)}=\frac{\left|\vec{n}\right|^{2}}{2\Gamma_{\left|\vec{n}\right|}}\delta_{\vec{n},-\vec{n}^{\prime}}$
(21)

and

$\left\langle\bar{\xi}_{\vec{n}^{\prime}}\bar{\xi}_{\vec{\ell}_{1}}\bar{\xi}_{\vec{\ell}_{2}}\bar{\xi}_{\vec{\ell}_{3}}\right\rangle^{\left(0\right)}=\left\langle\bar{\xi}_{\vec{n}^{\prime}}\bar{\xi}_{\vec{\ell}_{1}}\right\rangle^{\left(0\right)}\left\langle\bar{\xi}_{\vec{\ell}_{2}}\bar{\xi}_{\vec{\ell}_{3}}\right\rangle^{\left(0\right)}\\\
+\left\langle\bar{\xi}_{\vec{n}^{\prime}}\bar{\xi}_{\vec{\ell}_{2}}\right\rangle^{\left(0\right)}\left\langle\bar{\xi}_{\vec{\ell}_{1}}\bar{\xi}_{\vec{\ell}_{3}}\right\rangle^{\left(0\right)}+\left\langle\bar{\xi}_{\vec{n}^{\prime}}\bar{\xi}_{\vec{\ell}_{3}}\right\rangle^{\left(0\right)}\left\langle\bar{\xi}_{\vec{\ell}_{1}}\bar{\xi}_{\vec{\ell}_{2}}\right\rangle^{\left(0\right)}\,.$
(22)

Upon further substitution of
$\mathbb{F}=\bar{\xi}_{\vec{n}}\bar{\xi}_{\vec{n}^{\prime}}$ and $m=1$, an
expression for the first order static two-point function
$\left\langle\bar{\xi}_{\vec{n}}\bar{\xi}_{\vec{n}^{\prime}}\right\rangle^{\left(1\right)}$
was obtained in terms of the zeroth order static two-point and four-point
functions. Here, we will determine what equation (20) has to say about dynamic
time-dependent quantities.

###  III.2 Time-Dependent SCE

Unlike the static quantities described in [26], subbing time-dependent
quantities into equation (20) does not result in simple expressions for the
moments under consideration. Rather, the time derivative on the left-hand side
of equation (20) ensures that time-dependent quantities are given by ODEs. For
instance, subbing
$\mathbb{F}\left(\left\\{\bar{\xi}_{\vec{n}}\left(\bar{t}\,\right)\right\\}\right)=\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{n}^{\prime}}\left(\bar{t}\,\right)$
and $m=0$ into equation (20) gives the homogeneous ODE

$\frac{\partial\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{n}^{\prime}}\left(\bar{t}\,\right)\right\rangle^{\left(0\right)}}{\partial\bar{t}}=-\Gamma_{\left|\vec{n}^{\prime}\right|}\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{n}^{\prime}}\left(\bar{t}\,\right)\right\rangle^{\left(0\right)}$
(23)

whose solution is simply

$\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{n}^{\prime}}\left(\bar{t}\,\right)\right\rangle^{\left(0\right)}=\left\langle\bar{\xi}_{\vec{n}}\bar{\xi}_{\vec{n}^{\prime}}\right\rangle^{\left(0\right)}e^{-\Gamma_{\left|\vec{n}^{\prime}\right|}\bar{t}}$
(24)

where
$\left\langle\bar{\xi}_{\vec{n}}\bar{\xi}_{\vec{n}^{\prime}}\right\rangle^{\left(0\right)}$
is given by equation (21).

Similarly, subbing
$\mathbb{F}=\bar{\xi}_{\vec{n}^{\prime}}\left(0\right)\bar{\xi}_{\vec{\ell}_{1}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{2}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{3}}\left(\bar{t}\,\right)$
and $m=0$ into equation (20) gives the ODE

$\frac{\partial\left\langle\bar{\xi}_{\vec{n}^{\prime}}\left(0\right)\bar{\xi}_{\vec{\ell}_{1}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{2}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{3}}\left(\bar{t}\,\right)\right\rangle^{\left(0\right)}}{\partial\bar{t}}=\\\
-\sum_{\vec{n}}\Gamma_{\left|\vec{n}\right|}\left\langle\bar{\xi}_{\vec{n}^{\prime}}\left(0\right)\frac{\partial\left[\bar{\xi}_{\vec{\ell}_{1}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{2}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{3}}\left(\bar{t}\,\right)\right]}{\partial\bar{\xi}_{\vec{n}}\left(\bar{t}\,\right)}\bar{\xi}_{\vec{n}}\left(\bar{t}\,\right)\right\rangle^{\left(0\right)}\\\
+\frac{1}{2}\sum_{\vec{n}}\left|\vec{n}\right|^{2}\left\langle\bar{\xi}_{\vec{n}^{\prime}}\left(0\right)\frac{\partial^{2}\left[\bar{\xi}_{\vec{\ell}_{1}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{2}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{3}}\left(\bar{t}\,\right)\right]}{\partial\bar{\xi}_{\vec{n}}\left(\bar{t}\,\right)\partial\bar{\xi}_{-\vec{n}}\left(\bar{t}\,\right)}\right\rangle^{\left(0\right)}\,.$
(25)

Carrying out the derivatives explicitly and computing the sums, the first term
on the right-hand side is simply proportional to the dynamic four-point
function

$\sum_{\vec{n}}\Gamma_{\left|\vec{n}\right|}\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\frac{\partial\left(\bar{\xi}_{\vec{\ell}_{1}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{2}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{3}}\left(\bar{t}\,\right)\right)}{\partial\bar{\xi}_{\vec{n}}\left(\bar{t}\,\right)}\bar{\xi}_{\vec{n}}\left(\bar{t}\,\right)\right\rangle^{\left(0\right)}=\\\
\left(\Gamma_{\left|\vec{\ell}_{1}\right|}+\Gamma_{\left|\vec{\ell}_{2}\right|}+\Gamma_{\left|\vec{\ell}_{3}\right|}\right)\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{\ell}_{1}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{2}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{3}}\left(\bar{t}\,\right)\right\rangle^{\left(0\right)}\,,$
(26)

while the second term is composed of a sum of zeroth order dynamic two-point
functions

$\sum_{\vec{n}}\left|\vec{n}\right|^{2}\left\langle\bar{\xi}_{\vec{n}^{\prime}}\left(0\right)\frac{\partial^{2}\left(\bar{\xi}_{\vec{\ell}_{1}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{2}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{3}}\left(\bar{t}\,\right)\right)}{\partial\bar{\xi}_{\vec{n}}\left(\bar{t}\,\right)\partial\bar{\xi}_{-\vec{n}}\left(\bar{t}\,\right)}\right\rangle^{\left(0\right)}=\\\
\delta_{\vec{\ell}_{1},-\vec{\ell}_{2}}\left(\left|\vec{\ell}_{1}\right|^{2}+\left|\vec{\ell}_{2}\right|^{2}\right)\left\langle\bar{\xi}_{\vec{n}^{\prime}}\left(0\right)\bar{\xi}_{\vec{\ell}_{3}}\left(\bar{t}\,\right)\right\rangle^{\left(0\right)}\\\
+\delta_{\vec{\ell}_{1},-\vec{\ell}_{3}}\left(\left|\vec{\ell}_{1}\right|^{2}+\left|\vec{\ell}_{3}\right|^{2}\right)\left\langle\bar{\xi}_{\vec{n}^{\prime}}\left(0\right)\bar{\xi}_{\vec{\ell}_{2}}\left(\bar{t}\,\right)\right\rangle^{\left(0\right)}\\\
+\delta_{\vec{\ell}_{2},-\vec{\ell}_{3}}\left(\left|\vec{\ell}_{2}\right|^{2}+\left|\vec{\ell}_{3}\right|^{2}\right)\left\langle\bar{\xi}_{\vec{n}^{\prime}}\left(0\right)\bar{\xi}_{\vec{\ell}_{1}}\left(\bar{t}\,\right)\right\rangle^{\left(0\right)}\,.$
(27)

Since we have already computed the zeroth order dynamic two-point function in
equation (24) and can rearrange equation (21) to relate
$\left|\vec{n}\right|^{2}\delta_{\vec{n},-\vec{n}^{\prime}}=2\Gamma_{\left|\vec{n}\right|}\left\langle\bar{\xi}_{\vec{n}}\bar{\xi}_{\vec{n}^{\prime}}\right\rangle^{\left(0\right)}$,
we can observe that

$\delta_{\vec{\ell}_{1},-\vec{\ell}_{2}}\left(\left|\vec{\ell}_{1}\right|^{2}+\left|\vec{\ell}_{2}\right|^{2}\right)\left\langle\bar{\xi}_{\vec{n}^{\prime}}\left(0\right)\bar{\xi}_{\vec{\ell}_{3}}\left(\bar{t}\,\right)\right\rangle^{\left(0\right)}=\\\
2\left(\Gamma_{\left|\vec{\ell}_{1}\right|}+\Gamma_{\left|\vec{\ell}_{2}\right|}+\Gamma_{\left|\vec{\ell}_{3}\right|}-\Gamma_{\left|\vec{n}^{\prime}\right|}\right)\times\\\
\times\left\langle\bar{\xi}_{\vec{\ell}_{1}}\bar{\xi}_{\vec{\ell}_{2}}\right\rangle^{\left(0\right)}\left\langle\bar{\xi}_{\vec{\ell}_{3}}\bar{\xi}_{\vec{n}^{\prime}}\right\rangle^{\left(0\right)}e^{-\Gamma_{\left|\vec{n}^{\prime}\right|}\bar{t}}\,,$
(28)

where we have been able to add and subtract
$\Gamma_{\left|\vec{\ell}_{3}\right|}$ and
$\Gamma_{\left|\vec{n}^{\prime}\right|}$ since the factor
$\left\langle\bar{\xi}_{\vec{\ell}_{3}}\bar{\xi}_{\vec{n}^{\prime}}\right\rangle^{\left(0\right)}\propto\delta_{\vec{\ell}_{3},-\vec{n}^{\prime}}$
ensures that
$\Gamma_{\left|\vec{\ell}_{3}\right|}-\Gamma_{\left|\vec{n}^{\prime}\right|}=0$.
Simplifying each term of equation (27) in this manner, we obtain

$\sum_{\vec{n}}\left|\vec{n}\right|^{2}\left\langle\bar{\xi}_{\vec{n}^{\prime}}\left(0\right)\frac{\partial^{2}\left(\bar{\xi}_{\vec{\ell}_{1}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{2}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{3}}\left(\bar{t}\,\right)\right)}{\partial\bar{\xi}_{\vec{n}}\left(\bar{t}\,\right)\partial\bar{\xi}_{-\vec{n}}\left(\bar{t}\,\right)}\right\rangle^{\left(0\right)}=\\\
2\left(\Gamma_{\left|\vec{\ell}_{1}\right|}+\Gamma_{\left|\vec{\ell}_{2}\right|}+\Gamma_{\left|\vec{\ell}_{3}\right|}-\Gamma_{\left|\vec{n}^{\prime}\right|}\right)\times\\\
\times\left\langle\bar{\xi}_{\vec{n}^{\prime}}\bar{\xi}_{\vec{\ell}_{1}}\bar{\xi}_{\vec{\ell}_{2}}\bar{\xi}_{\vec{\ell}_{3}}\right\rangle^{\left(0\right)}e^{-\Gamma_{\left|\vec{n}^{\prime}\right|}\bar{t}}\,,$
(29)

where
$\left\langle\bar{\xi}_{\vec{n}^{\prime}}\bar{\xi}_{\vec{\ell}_{1}}\bar{\xi}_{\vec{\ell}_{2}}\bar{\xi}_{\vec{\ell}_{3}}\right\rangle^{\left(0\right)}$
is the static four-point function given by equation (22).

Finally, substituting equations (26) and (29) into equation (25), we obtain
the following non-homogeneous ODE for the dynamic four-point function at
zeroth order

$\frac{\partial\left\langle\bar{\xi}_{\vec{n}^{\prime}}\left(0\right)\bar{\xi}_{\vec{\ell}_{1}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{2}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{3}}\left(\bar{t}\,\right)\right\rangle^{\left(0\right)}}{\partial\bar{t}}=\\\
-\left(\Gamma_{\left|\vec{\ell}_{1}\right|}+\Gamma_{\left|\vec{\ell}_{2}\right|}+\Gamma_{\left|\vec{\ell}_{3}\right|}\right)\left\langle\bar{\xi}_{\vec{n}^{\prime}}\left(0\right)\bar{\xi}_{\vec{\ell}_{1}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{2}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{3}}\left(\bar{t}\,\right)\right\rangle^{\left(0\right)}\\\
+\left(\Gamma_{\left|\vec{\ell}_{1}\right|}+\Gamma_{\left|\vec{\ell}_{2}\right|}+\Gamma_{\left|\vec{\ell}_{3}\right|}-\Gamma_{\left|\vec{n}^{\prime}\right|}\right)\left\langle\bar{\xi}_{\vec{n}^{\prime}}\bar{\xi}_{\vec{\ell}_{1}}\bar{\xi}_{\vec{\ell}_{2}}\bar{\xi}_{\vec{\ell}_{3}}\right\rangle^{\left(0\right)}e^{-\Gamma_{\left|\vec{n}^{\prime}\right|}\bar{t}}$
(30)

and it is straightforward to check, though perhaps unsurprising, that this is
simply solved by

$\left\langle\bar{\xi}_{\vec{n}^{\prime}}\left(0\right)\bar{\xi}_{\vec{\ell}_{1}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{2}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{3}}\left(\bar{t}\,\right)\right\rangle^{\left(0\right)}=\\\
\left\langle\bar{\xi}_{\vec{n}^{\prime}}\bar{\xi}_{\vec{\ell}_{1}}\bar{\xi}_{\vec{\ell}_{2}}\bar{\xi}_{\vec{\ell}_{3}}\right\rangle^{\left(0\right)}e^{-\Gamma_{\left|\vec{n}^{\prime}\right|}\bar{t}}\,.$
(31)

Now to study the effect of the nonlinearity, we proceed to higher orders.
Subbing
$\mathbb{F}\left(\left\\{\bar{\xi}_{\vec{n}}\left(\bar{t}\,\right)\right\\}\right)=\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{n}^{\prime}}\left(\bar{t}\,\right)$
and $m=1$ into equation (20) gives after some tedious algebra

$\frac{\partial\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{n}^{\prime}}\left(\bar{t}\,\right)\right\rangle^{\left(1\right)}}{\partial\bar{t}}=-\Gamma_{\left|\vec{n}^{\prime}\right|}\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{n}^{\prime}}\left(\bar{t}\,\right)\right\rangle^{\left(1\right)}\\\
-\left(g\left|\vec{n}^{\prime}\right|^{4}-\Gamma_{\left|\vec{n}^{\prime}\right|}\right)\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{n}^{\prime}}\left(\bar{t}\,\right)\right\rangle^{\left(0\right)}\\\
-\frac{1}{2}\sum_{\vec{\ell}_{1}\neq\vec{n}^{\prime}}\sum_{\vec{\ell}_{2}}\sum_{\vec{\ell}_{3}}V_{\vec{n}^{\prime},\vec{\ell}_{1},\vec{\ell}_{2},\vec{\ell}_{3}}\times\\\
\times\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{\ell}_{1}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{2}}\left(\bar{t}\,\right)\bar{\xi}_{\vec{\ell}_{3}}\left(\bar{t}\,\right)\right\rangle^{\left(0\right)}\,,$
(32)

or after subbing in our expressions for the zeroth order time-dependent two-
point and four-point functions from Eqs. (24) and (31) respectively

$\frac{\partial\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{n}^{\prime}}\left(\bar{t}\,\right)\right\rangle^{\left(1\right)}}{\partial\bar{t}}=-\Gamma_{\left|\vec{n}^{\prime}\right|}\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{n}^{\prime}}\left(\bar{t}\,\right)\right\rangle^{\left(1\right)}\\\
-\Biggl{[}\left(g\left|\vec{n}^{\prime}\right|^{4}-\Gamma_{\left|\vec{n}^{\prime}\right|}\right)\left\langle\bar{\xi}_{\vec{n}}\bar{\xi}_{\vec{n}^{\prime}}\right\rangle^{\left(0\right)}\\\
+\frac{1}{2}\sum_{\vec{\ell}_{1}\neq\vec{n}^{\prime}}\sum_{\vec{\ell}_{2}}\sum_{\vec{\ell}_{3}}V_{\vec{n}^{\prime},\vec{\ell}_{1},\vec{\ell}_{2},\vec{\ell}_{3}}\left\langle\bar{\xi}_{\vec{n}}\bar{\xi}_{\vec{\ell}_{1}}\bar{\xi}_{\vec{\ell}_{2}}\bar{\xi}_{\vec{\ell}_{3}}\right\rangle^{\left(0\right)}\Biggr{]}e^{-\Gamma_{\left|\vec{n}^{\prime}\right|}\bar{t}}\,.$
(33)

It is worth noting that being precise, the exponential decay associated with
the four-point function in this expression should decay with rate
$\Gamma_{\left|\vec{n}\right|}$ instead of
$\Gamma_{\left|\vec{n}^{\prime}\right|}$. A careful analysis of the sum over
the kernel $V_{\vec{n}^{\prime},\vec{\ell}_{1},\vec{\ell}_{2},\vec{\ell}_{3}}$
with the static four-point function reveals however that this term vanishes
unless $\vec{n}=-\vec{n}^{\prime}$ and thus no harm is done by replacing
$\Gamma_{\left|\vec{n}\right|}$ with $\Gamma_{\left|\vec{n}^{\prime}\right|}$.
It is now apparent that the non-homogeneous term in this equation decays at
the natural decay rate $\Gamma_{\left|\vec{n}^{\prime}\right|}$ of the ODE and
thus its general solution will contain a non-physical secular term of the form
$\bar{t}e^{-\Gamma_{\left|\vec{n}^{\prime}\right|}\bar{t}}$. As in the
Poincaré-Lindstedt method for perturbatively solving non-linear ODEs [76, 77,
78], this situation can be avoided by setting
$\Gamma_{\left|\vec{n}^{\prime}\right|}$ such that the coefficient of the non-
homogeneous term vanishes, ie.

$0=\left(g\left|\vec{n}^{\prime}\right|^{4}-\Gamma_{\left|\vec{n}^{\prime}\right|}\right)\left\langle\bar{\xi}_{\vec{n}}\bar{\xi}_{\vec{n}^{\prime}}\right\rangle^{\left(0\right)}\\\
+\frac{1}{2}\sum_{\vec{\ell}_{1}\neq\vec{n}^{\prime}}\sum_{\vec{\ell}_{2}}\sum_{\vec{\ell}_{3}}V_{\vec{n}^{\prime},\vec{\ell}_{1},\vec{\ell}_{2},\vec{\ell}_{3}}\left\langle\bar{\xi}_{\vec{n}}\bar{\xi}_{\vec{\ell}_{1}}\bar{\xi}_{\vec{\ell}_{2}}\bar{\xi}_{\vec{\ell}_{3}}\right\rangle^{\left(0\right)}\,.$
(34)

Using this idea to determine the characteristic decay rate is novel in the
context of the SCE method and might be useful in other problems as well. After
subbing in the static two-point and four-point functions and carrying out the
sums in the last equation, this ultimately simplifies to the following
discrete integral equation for $\Gamma_{\left|\vec{n}\right|}$

$\Gamma_{\left|\vec{n}\right|}=g\left|\vec{n}\right|^{4}+\frac{1}{2}\sum_{\vec{\ell}\neq\vec{n}}\frac{\left|\vec{\ell}\,\right|^{2}\left|\vec{n}\times\vec{\ell}\,\right|^{4}}{\Gamma_{\left|\vec{\ell}\,\right|}\left|\vec{n}-\vec{\ell}\,\right|^{4}}\,.$
(35)

Surprisingly, the above argument for preventing secular terms occurring in our
expression for
$\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{n}^{\prime}}\left(\bar{t}\,\right)\right\rangle^{\left(1\right)}$
is equivalent to simply imposing that the first order approximation for the
dynamic structure factor equals its zeroth order approximation, ie.

$\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{n}^{\prime}}\left(\bar{t}\,\right)\right\rangle^{\left(1\right)}=\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{n}^{\prime}}\left(\bar{t}\,\right)\right\rangle^{\left(0\right)}\,,$
(36)

or in other words, we select $\Gamma_{\left|\vec{n}\right|}$ such that the
zeroth order approximation is exact up to first order. In [26], a static
version of this self-consistent argument was made by setting the first order
approximation for the static structure factor equal to its zeroth order
approximation, ie.

$\left\langle\bar{\xi}_{\vec{n}}\bar{\xi}_{\vec{n}^{\prime}}\right\rangle^{\left(1\right)}=\left\langle\bar{\xi}_{\vec{n}}\bar{\xi}_{\vec{n}^{\prime}}\right\rangle^{\left(0\right)}\,,$
(37)

and indeed, the same discrete integral equation is obtained from both
approaches. It is important to appreciate that these two self-consistent
arguments giving the same discrete integral equation for
$\Gamma_{\left|\vec{n}\right|}$ was by no means anticipated nor trivial and in
fact, it is known that this does not occur for the ordinary unadorned
$\phi^{4}$-model. In such instances, the fact that the two arguments conflict
suggests that the effective decay rate $\Gamma_{\left|\vec{n}\right|}$ also
needs to be appropriately expanded in a manner analogous to the Poincaré-
Lindstedt method [76, 77, 78] if we wish our results to have meaning.
Conversely, the situation where the two arguments result in the same discrete
integral equation implies some degree of quasi-linearity and is indicative
that our approach has self-consistently captured a true feature of the system.

Equation (35) has been solved in the appendix of [26]. Here we simply bring
the final result

$\Gamma_{\left|\vec{n}\right|}\simeq
n^{4}\sqrt{\frac{3\pi}{4}\left[A\left(g\right)-\ln\left(\frac{n}{n_{\textrm{max}}}\right)\right]}\times\\\
\times\left[1+B\left(g\right)\frac{n}{n_{\textrm{max}}}\right]\,,$ (38)

where $n_{\textrm{max}}$ is an upper-frequency cutoff which must be imposed on
the system and $A\left(g\right)$ and $B\left(g\right)$ are constants which
only depend on $g$ and have the following small $g$ expansions

$\displaystyle A$ $\displaystyle\simeq$ $\displaystyle
0.137+0.336g+0.243g^{2}+0.112g^{3}+O\left(g^{4}\right),$ (39) $\displaystyle
B$ $\displaystyle\simeq$
$\displaystyle-0.265+0.360g-0.395g^{2}+0.311g^{3}+O\left(g^{4}\right).\qquad$
(40)

It is worth noting that $B\left(g\right)$ primarily determines the behaviour
of $\Gamma_{\left|\vec{n}\right|}$ for large $\vec{n}$, ie. when
$\left|\vec{n}\right|\rightarrow n_{\textrm{max}}$, and can be neglected when
$\left|\vec{n}\right|$ is small. Accordingly, for small $\vec{n}$, we have
obtained that the decay rate $\Gamma_{\left|\vec{n}\right|}$ grows like a
logarithmically corrected power law $\sim n^{4}$.

To summarise, we have found that at first order, the dynamic structure factor
is given by

$\displaystyle\left\langle\bar{\xi}_{\vec{n}}\left(0\right)\bar{\xi}_{\vec{n}^{\prime}}\left(\bar{t}\,\right)\right\rangle^{\left(1\right)}$
$\displaystyle=\left\langle\bar{\xi}_{\vec{n}}\bar{\xi}_{\vec{n}^{\prime}}\right\rangle^{\left(0\right)}e^{-\Gamma_{\left|\vec{n}\right|}\bar{t}}$
(41)
$\displaystyle=\delta_{\vec{n},-\vec{n}^{\prime}}\frac{n^{2}}{2\Gamma_{\left|\vec{n}\right|}}e^{-\Gamma_{\left|\vec{n}\right|}\bar{t}}$
(42)

where $\Gamma_{\left|\vec{n}\right|}$ is given by equation (38) and can be
seen to simultaneously play the roles of effective coupling constant and
effective decay rate.

##  IV Comparison with Simulations

As in [26], our predictions can be validated by numerical integration of
equation (7) over a square lattice. Since we investigate here the dynamic
properties of the simulation rather than its static properties, the
simulations must be run for long enough to obtain precise averages of the
time-dependent two-point function $S_{\vec{n}}\left(\bar{t}\,\right)$ and thus
they must be run for substantially longer than in [26]. The lattice size
imposes a finite maximum frequency $n_{\textrm{max}}$ but since our theory
necessitates the existence of an upper-frequency cut-off, this in itself is
fine. More pressingly, for increasing maximum frequency $n_{\textrm{max}}$,
the maximum size of the time step $\delta\bar{t}$ that can be used shrinks if
the simulation is to remain stable. This presents a trade-off between the
maximum resolution in time vs the maximum resolution in space and since the
dynamic structure factor can only be extracted from long simulation runs, the
size of $n_{\textrm{max}}$ is sharply constrained by practical considerations.
In practice, our simulations were run with a time-step $\delta\bar{t}=10^{-6}$
over a $41\times 41$ lattice corresponding to a maximum frequency
$n_{\textrm{max}}=20\sqrt{2}\approx 28.3$, though unlike [26] which only used
$10^{6}$ time steps per simulation, these simulations were run for $10^{7}$
time steps each. The consequence of these extended simulations is that an
enormous amount of data is generated though simulation states which are close
to each other in time are practically indistinguishable except at the very
largest modes (an expectation based on Eq. (38)). Since these modes
equilibrate far more rapidly than the small modes, they are far less
interesting in the current context and thus in the interest of keeping memory
resources manageable, simulation data was only saved every $10^{3}$ time
steps. Accordingly, the decay rate of modes which decay faster than
$\Delta\bar{t}=10^{3}\times\delta\bar{t}=10^{-3}$ is not presented here.
Finally, it is worth noting that as in [26], efficient calculation of the
quartic interaction term of equation (7) is non-trivial and as there, was
achieved by implementing the pseudo-spectral method described in [52] in which
the quartic interaction is calculated as the Fourier transform of its real-
space counterpart though this imposes periodic boundary conditions on the
simulation. To achieve precise results, each simulation was run $10$ times for
various values of $0\leq g\leq 1$ and the results averaged.

Figure 1: The dynamic structure factor $S_{\vec{n}}\left(\bar{t}\,\right)$ as
a function of the time difference $\bar{t}$ for $g=0.1$. Each data set
corresponds to a different frequency (see legend) and the solid lines are
linear fits to the logarithm of the data. The fact that these fits are
excellent confirms that $S_{\vec{n}}\left(\bar{t}\,\right)$ indeed decays
exponentially.

Figure 2: $\Gamma_{\left|\vec{n}\right|}^{\textrm{dyn}}$ (top) and
$\Gamma_{\left|\vec{n}\right|}^{\textrm{dyn}}/\left|\vec{n}\right|^{4}$
(bottom) as a function of $n$ for various values of $g$. The solid and dotted
lines are fits of equation (43) to the $g=0.1$ data (green triangles) and
$g=1$ data (purple squares) respectively. The dashed line in the top plot is a
guideline proportional to $n^{4}$. The fact that the fits are excellent over
the entire range of frequencies confirms that our theory correctly predicts
the functional form of the dynamic structure factor
$S_{\vec{n}}\left(\bar{t}\,\right)$ and the appreciable deviation between the
fits and the guideline shows that the logarithmic correction of equation (43)
is non-negligible. This is further emphasised in the bottom plot by the fact
that $\Gamma_{\left|\vec{n}\right|}^{\textrm{dyn}}/\left|\vec{n}\right|^{4}$
is indeed seen to be non-constant and is well fitted by the theory.

Fig. 1 shows how the dynamic structure factor
$S_{\vec{n}}\left(\bar{t}\,\right)$ decays for the first few modes as a
function of the time difference $\bar{t}$. These results are for the
simulation performed with $g=0.1$ though very similar results are obtained for
the other values of $g$. The solid black lines are linear fits to the
logarithm of the simulation data and clearly show that the dynamic structure
factor $S_{\vec{n}}\left(\bar{t}\,\right)$ indeed decays exponentially. By
performing such fits for many frequencies, we are able to numerically
determine how the decay rate, which we will denote
$\Gamma_{\left|\vec{n}\right|}^{\textrm{dyn}}$, varies as a function of $n$.
Fig. 2 shows $\Gamma_{\left|\vec{n}\right|}^{\textrm{dyn}}$ and
$\Gamma_{\left|\vec{n}\right|}^{\textrm{dyn}}/\left|\vec{n}\right|^{4}$ as a
function of $n$ up to $n_{\textrm{max}}/4$ for the various values of $g$ we
used. The solid and dotted lines are fits of the equation

$\frac{\Gamma_{\left|\vec{n}\right|}^{\textrm{dyn}}}{n_{\textrm{max}}^{4}}\simeq
C\left(\frac{n}{n_{\textrm{max}}}\right)^{4}\sqrt{A\left(g\right)-\ln\left(\frac{n}{n_{\textrm{max}}}\right)}$
(43)

to the numerical results of $g=0.1$ (green triangles) and $g=1$ (purple
squares) respectively. Here, $A\left(g\right)$ was simply taken from equation
(39) while the scaling parameter $C$ was fitted and, as can be seen, these
fits are excellent across the entire range of frequencies. As described
towards the end of section III, the constant $B\left(g\right)$ in equation
(38) primarily modifies $\Gamma_{\left|\vec{n}\right|}$ for large values of
$n$ and thus we have not attempted to determine it from our data. Fits for
$g=0.01$ (red circles) and $g=0$ (blue pluses) can also be carried out but the
resulting fits are so similar to the $g=0.1$ case that little insight is
gained by showing them. The dashed line in the top plot of Fig. 2 is a
guideline proportional to $n^{4}$ and together with the bottom plot of Fig. 2
clearly emphasises that the logarithmic correction is non-negligible.

Figure 3: Comparison between the numerical coupling constant
$\Gamma_{\left|\vec{n}\right|}^{\textrm{stat}}=\frac{1}{2}n^{2}/\left\langle\bar{\xi}_{\vec{n}}\bar{\xi}_{-\vec{n}}\right\rangle$
(blue pluses) and the numerical decay rate
$\Gamma_{\left|\vec{n}\right|}^{\textrm{dyn}}$ of the time-dependent two-point
function
$S_{\vec{n}}\left(\bar{t}\,\right)=S_{\vec{n}}\left(0\right)e^{-\Gamma_{\left|\vec{n}\right|}^{\textrm{dyn}}\bar{t}}$
(red circles), for $g=0$. These two quantities are seen to agree over all
orders of magnitude.

The fact that these fits of equation (43) so beautifully capture the behaviour
of $\Gamma_{\left|\vec{n}\right|}^{\textrm{dyn}}$ confirms that our theory has
indeed accurately predicted the functional form of the dynamic structure
$S_{\vec{n}}\left(\bar{t}\,\right)$. It is worth mentioning that in [26], it
was observed that the periodic square lattice introduces an anisotropy into
the simulation and this required each direction to be treated separately, an
aspect which we have not observed in the temporal decay data. We explain this
distinction by observing that in [26], the anisotropy was primarily accounted
for by scaling the parameter $B\left(g\right)$ in equation (38) by some
appropriate function $f\left(\theta\right)$ and thus is only observable for
large frequencies. Since we do not present here the large frequency decay
rate, we have been unable to observe this anisotropy in this data though we
presume it too exists.

Figure 4:
$\Gamma_{\left|\vec{n}\right|}^{\textrm{stat}}/\Gamma_{\left|\vec{n}\right|}^{\textrm{dyn}}$
as a function of $n$. For all values of $g$, this ratio is seen to be constant
and roughly equal to $1$ over all frequency scales.

Finally, it is worth comparing $\Gamma_{\left|\vec{n}\right|}^{\textrm{dyn}}$,
the decay rate of the time-dependent structure factor
$S_{\vec{n}}\left(\bar{t}\,\right)$, with the effective coupling constant
which can be extracted from the relationship

$S_{\vec{n}}\left(0\right)=\left\langle\bar{\xi}_{\vec{n}}\bar{\xi}_{-\vec{n}}\right\rangle=\frac{n^{2}}{2\Gamma_{\left|\vec{n}\right|}^{\textrm{stat}}}\,.$
(44)

Here, we denote the coupling constant by
$\Gamma_{\left|\vec{n}\right|}^{\textrm{stat}}$ as it is obtained by only
considering static quantities. According to the theory developed above and in
[26], $\Gamma_{\left|\vec{n}\right|}^{\textrm{stat}}$ and
$\Gamma_{\left|\vec{n}\right|}^{\textrm{dyn}}$ should be identical and Fig. 3
and Fig. 4 shows that these two methods of obtaining
$\Gamma_{\left|\vec{n}\right|}$ are in fact extremely close. Fig. 3 compares
$\Gamma_{\left|\vec{n}\right|}^{\textrm{stat}}$ and
$\Gamma_{\left|\vec{n}\right|}^{\textrm{dyn}}$ for the case of $g=0$ and shows
that even with maximal non-linear coupling,
$\Gamma_{\left|\vec{n}\right|}^{\textrm{stat}}$ and
$\Gamma_{\left|\vec{n}\right|}^{\textrm{dyn}}$ are practically
indistinguishable. The same result can be obtained for the other values of $g$
and indeed Fig. 4 shows that the ratio
$\Gamma_{\left|\vec{n}\right|}^{\textrm{stat}}/\Gamma_{\left|\vec{n}\right|}^{\textrm{dyn}}\approx
1$ for all values of $g$ over all frequency scales.

##  V Discussion

In this work we have extended a previous paper [26] to study the dynamics of a
vibrating thin sheet, governed by the overdamped Föppl-von Kármán equations.
Specifically, we applied the self-consistent expansion to obtain predictions
which agree with the results of numerical simulations. Surprisingly, the
effective coupling constant $\Gamma_{\left|\vec{n}\right|}$, which determines
the static structure factor of the sheet coincides with the decay rate of the
dynamic structure factor. This observation is reminiscent of linear systems,
where these two quantities are indeed the same, yet it is not obvious that the
Föppl-von Kármán equations, which are strongly nonlinear, would exhibit such a
phenomena. A way to make sense of this situation is by using a recent
correlation-response inequality in dynamical systems [79, 80]. This inequality
was formulated in the context of Langevin dynamics and shows that whenever the
equations of motion can be derived from a Hamiltonian and hence obey a certain
Fluctuation-Dissipation relation (thus belonging to class I using that
classification), and are Galilean invariant (that is, the zero-mode does not
affect the dynamics of the higher modes thus belonging to class II in that
classification), the exponent inequalities become an equality implying quasi-
linear dynamics. The quasi-linearity refers to the fact that the scaling of
the decay rate of the dynamic correlation function is related to the static
exponents in a simple manner. While the overdamped Föppl-von Kármán equations
used here are definitely Galilean invariant since the center of mass does not
drift as a result of the dynamics, the way they are derived from a Hamiltonian
deviates from the models considered in [79, 80], as they belong to neither
model A nor model B in the classical classification of Hohenberg and Halperin
[81]. Nevertheless, they obey the quasi-linear property, which may be the
result of a modified Fluctuation-Dissipation relation. This makes the
overdamped Föppl-von Kármán equations another interesting example of a
physically motivated model that obeys quasi-linearity.

From a methodological point of view, imposing the condition that secular terms
should vanish, inspired by the Poincaré-Lindstedt method, is a major
achievement in the context of the SCE method. In the past this method was only
able to yield approximate results for the dynamical structure factor in
certain cases, usually based on solving another dedicated approximate integral
equation. In this paper, we gained the insight that the resonant/secular terms
that are being generated are actually non-physical, and just like in the
Poincaré-Linstedt method, should be equated to zero. This insight might become
useful in other contexts where the SCE method is applied. The big surprise
here is that, unlike the Poincaré-Linstedt method, we do not need to solve a
new equation for the dynamic decay rate, since the new equation is identical
to the integral equation from the static case.

The results reported here have experimental implications. Both the structure
of thin sheets, as characterised by the static structure factor and the
characteristic decay rate could be measured directly. There has already been
an attempt to measure the structure of crumpled sheets using its optical
signature [31], and it is reasonable that with improved imaging techniques
leading to faster analysis of dynamic laser speckle patterns [82], the full
dynamic structure factor could be effectively sampled. The coincidence of the
effective coupling constant and the decay rate due to the quasi-linearity
should be checked experimentally and if confirmed could be used to enhance the
resolution and accuracy of the measurements by cross validating these two
independent sources.

Thin sheets also exhibit a very distinctive acoustic footprint, often known as
crackling noise [27, 28, 29, 30]. Clearly, the dynamic structure factor of the
vibrating sheet should be correlated with its acoustic emissions however the
coupling between the two is known to be nontrivial. Accordingly, a verified
analytical form of the dynamic structure factor could provide a solid staring
point that may allow progress in that vein.

Last, since the expression for the dynamic structure factor depends on the
elastic properties of the sheet and in particular on its elastic moduli, a
precise time-resolved measurement of the dynamics of the sheet could provide
an indirect measurement of the elastic constants of the material. This might
be advantageous when a more direct mechanical test may be destructive or might
simply modify these properties, while vibrating the sheet introduces only a
mild perturbation.

An interesting extension of this work would be to consider the regime where
inertia is important, which is of major interest in the context of wave
turbulence in thin sheets [44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 42]. In
particular, it would be interesting to see whether the growing knowledge on
the energy cascade could be qualitatively and quantitatively connected to the
structure and dynamics of vibrating sheets. Such a step however would require
a major technical advance, namely a nontrivial extension of the self-
consistent expansion beyond the overdamped regime, or otherwise the
development of an alternative methodology to tackle this problem.

###### Acknowledgements.

The authors wish to thank Arezki Boudaoud for useful discussions. This work
was supported by the Israel Science Foundation Grant No. 1682/18.

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# Depth in a pathological case

Dorin Popescu  Dorin Popescu, Simion Stoilow Institute of Mathematics of
Romanian Academy, Research unit 5, P.O.Box 1-764, Bucharest 014700, Romania
<EMAIL_ADDRESS>
###### Abstract.

Let $I$ be a squarefree monomial ideal of a polynomial algebra over a field
minimally generated by $f_{1},\ldots,f_{r}$ of degree $d\geq 1$, and a set $E$
of monomials of degree $\geq d+1$. Let $J\subsetneq I$ be a squarefree
monomial ideal generated in degree $\geq d+1$. Suppose that all squarefree
monomials of $I\setminus(J\cup E)$ of degree $d+1$ are some least common
multiples of $f_{i}$. If $J$ contains all least common multiples of two of
$(f_{i})$ of degree $d+2$ then $\operatorname{depth}_{S}I/J\leq d+1$ and
Stanley’s Conjecture holds for $I/J$.

Key words  : Monomial Ideals, Depth, Stanley depth.
2010 Mathematics Subject Classification: Primary 13C15, Secondary 13F20,
13F55, 13P10.

The support from grant ID-PCE-2011-1023 of Romanian Ministry of Education,
Research and Innovation is gratefully acknowledged.

## Introduction

Let $K$ be a field and $S=K[x_{1},\ldots,x_{n}]$ be the polynomial $K$-algebra
in $n$ variables. Let $I\supsetneq J$ be two monomial ideals of $S$ and
suppose that $I$ is generated by some monomials of degrees $\geq d$ for some
positive integer $d$. After a multigraded isomorphism we may assume either
that $J=0$, or $J$ is generated in degrees $\geq d+1$.

Suppose that $I\subset S$ is minimally generated by some monomials
$f_{1},\ldots,f_{r}$ of degrees $d$, and a set $E$ of monomials of degree
$\geq d+1$. Let $B$ (resp. $C$) be the set of squarefree monomials of degrees
$d+1$ (resp. $d+2$) of $I\setminus J$. Let $w_{ij}$ be the least common
multiple of $f_{i}$ and $f_{j}$, $i<j$ and set $W$ to be the set of all
$w_{ij}$. By [4, Proposition 3.1] (see [7, Lemma 1.1]) we have
$\operatorname{depth}_{S}I/J\geq d$. It is easy to see that if $d=1$,
$E=\emptyset$ and $B\subset W$ then $\operatorname{depth}_{S}I/J=d$ (see for
instance [7, Lemma 1.8] and [6, Lemma 3]). Attempts to extend this result were
made in [10, Proposition 1.3], [6, Lemma 4]. However [6, Example 1] (see here
Example 1) shows that for $d=2$, $E=\emptyset$ and $B\subset W$ it holds
$\operatorname{depth}_{S}I/J=d+1=3$.

If $B\cap(f_{1},\ldots,f_{r})\subset W$ we call $I/J$ a pathological case. It
is the purpose of this paper to show the following theorem.

######  Theorem 1.

If $B\cap(f_{1},\ldots,f_{r})\subset W$ and $C\cap W=\emptyset$ then
$\operatorname{depth}_{S}I/J\leq d+1$.

In particular, if $C\cap W=\emptyset$ then the so called Stanley’s Conjecture
holds in the pathological case (see Corollary 1). But why is important this
pathological case? The methods used in [11], [6], [9] to show a weak form of
Stanley’s Conjecture when $r\leq 4$ (see [9, Conjecture 0.1]) could be applied
only when $B\cap(f_{1},\ldots,f_{r})\not\subset W$, that is when $I/J$ is not
pathological. Thus the above theorem solves partially one of the obstructions
to prove this weak form. We believe that the condition $C\cap W=\emptyset$
could be removed from the above theorem. The proof of Theorem 1 relies on
Lemmas 2, 3 and Examples 2, 3, 5 found after many computations with the
Computer Algebra System SINGULAR [3].

The above theorem hints a possible positive answer to the following question.

######  Question 1.

Let $i\in[r-1]$. Suppose that $E=\emptyset$ and every squarefree monomial from
$I\setminus J$ of degree $d+i$ is a least common multiple of $i+1$ monomials
$f_{j}$. Then is it $\operatorname{depth}_{S}I/J\leq d+i$?

We owe thanks to A. Zarojanu and a Referee, who noticed some mistakes and a
gap in Section 2 of some previous versions of this paper.

##  1\. Depth and Stanley depth

Suppose that $I$ is minimally generated by some squarefree monomials
$f_{1},\ldots,f_{r}$ of degree $d$ for some $d\in{\mathbb{N}}$ and a set $E$
of some squarefree monomials of degree $\geq d+1$. Let $C_{3}$ be the set of
all $c\in C\cap(f_{1},\ldots,f_{r})$ having all degree $(d+1)$ divisors from
$B\setminus E$ in $W$. In particular each monomial of $C_{3}$ is the least
common multiple of at least three of the $f_{i}$.

Next lemma is closed to [10, Lemma 1.1].

######  Lemma 1.

Suppose that $E=\emptyset$ and $\operatorname{depth}_{S}I/(J,b)=d$ for some
$b\in B$. Then $\operatorname{depth}_{S}I/J\leq d+1$.

###### Proof.

If there exists no $c\in C$ such that $b|c$ then we have
$\operatorname{depth}_{S}I/J\leq d+1$ by [10, Lemma 1.5]. Otherwise, in the
exact sequence

$0\to(b)/J\cap(b)\to I/J\to I/(J,b)\to 0$

the first term has depth $\geq d+2$ because for a multiple $c\in C$ of $b$ all
the variables of $c$ form a regular system. By hypothesis the last term has
depth $\geq d$ and so the middle one has depth $d$ too using the Depth Lemma.

We recall the following example from [6].

######  Example 1.

Let $n=5$, $r=5$, $d=2$, $f_{1}=x_{1}x_{2}$, $f_{2}=x_{1}x_{3}$,
$f_{3}=x_{1}x_{4}$, $f_{4}=x_{2}x_{3}$, $f_{5}=x_{3}x_{5}$ and
$I=(f_{1},\ldots,f_{5})$,
$J=(x_{1}x_{2}x_{5},x_{1}x_{4}x_{5},x_{2}x_{3}x_{4},x_{3}x_{4}x_{5}).$ It
follows that
$B=\\{x_{1}x_{2}x_{3},x_{1}x_{2}x_{4},x_{1}x_{3}x_{4},x_{1}x_{3}x_{5},x_{2}x_{3}x_{5}\\}$
and so $s=|B|=r=5$. Note that $B\subset W$. A computation with SINGULAR when
char $K=0$ gives $\operatorname{depth}_{S}I/J=\operatorname{depth}_{S}S/J=3$
and $\operatorname{depth}_{S}S/I=2$. Since depth depends on the characteristic
of the field it follows in general only that $\operatorname{depth}_{S}S/J\leq
3$, $\operatorname{depth}_{S}S/I\leq 2$ using [1, Lemma 2.4]. In fact
$\operatorname{depth}_{S}I/J\leq d+1=3$ using [12, Proposition 2.4] because
$q=|C|=2<r=5$. Note that choosing any $b\in B$ we have
$\operatorname{depth}_{S}I/(J,b)=2$ because the corresponding $s^{\prime}<r$
and we may apply [7, Theorem 2.2]. But then $\operatorname{depth}_{S}I/J\leq
3$ by Lemma 1.

######  Example 2.

In the above example set $I^{\prime}=(f_{1},\ldots,f_{4})$, $J^{\prime}=J\cap
I^{\prime}=$
$(x_{1}x_{2}x_{5},x_{1}x_{4}x_{5},x_{2}x_{3}x_{4})$. Note that we have an
injection $I^{\prime}/J^{\prime}\to I/J$ and so
$\operatorname{depth}_{S}I^{\prime}/J^{\prime}>2$ because otherwise we get
$\operatorname{depth}_{S}I/J=2$ which is impossible. Given
$B^{\prime},W^{\prime}$ for $I^{\prime}/J^{\prime}$ we see that
$B^{\prime}\not\subset W^{\prime}$ since $x_{2}x_{3}x_{5}\in
B^{\prime}\setminus W^{\prime}$, that is $I^{\prime}/J^{\prime}$ is not
anymore in the pathological case even this was the case of $I/J$. We have
$I/(J,f_{5})\cong I^{\prime}/(J^{\prime},x_{1}x_{3}x_{5},x_{2}x_{3}x_{5})$.
Using SINGULAR when char $K=0$ we see that
$\operatorname{depth}_{S}I^{\prime}/(J^{\prime},x_{1}x_{3}x_{5})=\operatorname{depth}_{S}I^{\prime}/(J^{\prime},x_{1}x_{3}x_{5},x_{2}x_{3}x_{5})=2$,
$\operatorname{depth}_{S}I^{\prime}/(J^{\prime},x_{2}x_{3}x_{5})=3$. It
follows that always
$\operatorname{depth}_{S}I^{\prime}/(J^{\prime},x_{1}x_{3}x_{5}),\operatorname{depth}_{S}I^{\prime}/(J^{\prime},x_{1}x_{3}x_{5},x_{2}x_{3}x_{5})\leq
2$ using [1, Lemma 2.4]. These inequalities are in fact equalities because
$I^{\prime}$ is generated in degree $2$. Thus we cannot apply Lemma 1 for
$I^{\prime},J^{\prime}$, $b=x_{2}x_{3}x_{5}$ but we may apply this lemma for
$I^{\prime},J^{\prime}$, $b^{\prime}=x_{1}x_{3}x_{5}$ to get
$\operatorname{depth}_{S}I^{\prime}/J^{\prime}\leq 3$.

######  Lemma 2.

Suppose that $r>1$ and $\operatorname{depth}_{S}I/(J,f_{r})=d$. Then
$\operatorname{depth}_{S}I/J\leq d+1$.

###### Proof.

Let $B\cap(f_{r})=\\{b_{1},\ldots,b_{p}\\}$. As $I/(J,f_{r})$ has a
squarefree, multigraded free resolution we see that only the components of
squarefree degrees of

$\operatorname{Tor}_{S}^{n-d}(K,I/(J,f_{r}))\cong H_{n-d}(x;I/(J,f_{r}))$

are nonzero, the last module being the Koszul homology of $I/(J,f_{r})$. Thus
we may find

$z=\sum^{r-1}_{i=1}y_{i}f_{i}e_{[n]\setminus\operatorname{supp}f_{i}}\in
K_{n-d}(x;I/(J,f_{r})),$

$y_{i}\in K$ inducing a nonzero element in $H_{n-d}(x;I/(J,f_{r}))$. Here we
set $\operatorname{supp}f_{i}=\\{t\in[n]:x_{t}|f_{i}\\}$ and
$e_{A}=\wedge_{j\in A}e_{j}$ for a subset $A\subset[n]$. We have

$\partial z=\sum_{b\in
B}P_{b}(y)be_{[n]\setminus\operatorname{supp}b}=\sum_{i=1}^{p}P_{b_{i}}(y)b_{i}e_{[n]\setminus\operatorname{supp}b_{i}},$

where $P_{b}$ are linear homogeneous polynomials in $y$. Note that
$P_{b}(y)=0$ for all $b\not\in\\{b_{1},\ldots,b_{p}\\}$. Choose
$j\in\operatorname{supp}f_{r}$ and consider

$z_{j}=\sum^{r-1}_{i=1,f_{i}\not\in(x_{j})}y_{i}f_{i}e_{[n]\setminus(\\{x_{j}\\}\cup\
\operatorname{supp}f_{i})}\in K_{n-d-1}(x;I/J).$

In $\partial z_{j}$ appear only terms of type $ue_{A}$, $j\not\in A$ with
$|A|=n-d-1$ and $u=\Pi_{i\in[n]\setminus(A\cup\\{j\\})}x_{i}$. Thus terms of
type $b_{i}e_{[n]\setminus\operatorname{supp}b_{i}}$, $i\in[p]$ are not
present in $\partial z_{j}$ because $b_{i}\in(x_{j})$. It follows that
$\partial z_{j}=0$ and so $z_{j}$ is a cycle. Note that a cycle of
$K_{n-d-1}(x;I/J)$ could contain also terms of type $ve_{A^{\prime}}$ with
$|A^{\prime}|=n-d-1$ and $v=\Pi_{i\in[n]\setminus A^{\prime}}x_{i}\in B$, but
$z_{j}$ is just a particular cycle.

Remains to show that we may find $j$ such that $z_{j}$ is a nonzero cycle.
Suppose that $y_{m}\not=0$ for $m\in[r-1]$ and choose
$j\in\operatorname{supp}f_{r}\setminus\operatorname{supp}f_{m}$. It follows
that $z_{j}$ is a nonzero cycle because
$y_{m}f_{m}e_{[n]\setminus(\\{x_{j}\\}\cup\ \operatorname{supp}f_{m})}$ is
present in $z_{j}$. Thus $\operatorname{depth}_{S}I/J\leq d+1$ by [2, Theorem
1.6.17].

######  Remark 1.

Applying the above lemma to Example 2 we see that
$\operatorname{depth}_{S}I/J\leq 3$ because
$\operatorname{depth}_{S}I/(J,f_{5})=2$.

Let $P_{I\setminus J}$ be the poset of all squarefree monomials of $I\setminus
J$ with the order given by the divisibility. Let $P$ be a partition of
$P_{I\setminus J}$ in intervals $[u,v]=\\{w\in P_{I\setminus J}:u|w,w|v\\}$,
let us say $P_{I\setminus J}=\cup_{i}[u_{i},v_{i}]$, the union being disjoint.
Define $\operatorname{sdepth}P=\operatorname{min}_{i}\operatorname{deg}v_{i}$
and the Stanley depth of $I/J$ given by
$\operatorname{sdepth}_{S}I/J=\operatorname{max}_{P}\operatorname{sdepth}P$,
where $P$ runs in the set of all partitions of $P_{I\setminus J}$ (see [4],
[13]). Stanley’s Conjecture says that
$\operatorname{sdepth}_{S}I/J\geq\operatorname{depth}_{S}I/J$.

In Example 1 we have $s=5<q+3=7$ and so it follows that
$\operatorname{depth}_{S}I/J\leq d+1$ by [8, Theorem 1.3] (see also [6,
Theorem 2]). Next example follows [9, Example 1.6] and has $s=q+r$,
$\operatorname{sdepth}_{S}I/J=d+2$ but $\operatorname{depth}_{S}I/J=d$.

######  Example 3.

Let $n=12$, $r=11$, $f_{1}=x_{12}x_{1}$, $f_{2}=x_{12}x_{2}$,
$f_{3}=x_{12}x_{3}$, $f_{4}=x_{12}x_{4}$, $f_{5}=x_{12}x_{5}$,
$f_{6}=x_{12}x_{6}$, $f_{7}=x_{6}x_{7}$, $f_{8}=x_{6}x_{8}$,
$f_{9}=x_{6}x_{9}$, $f_{10}=x_{6}x_{10}$, $f_{11}=x_{6}x_{11}$,
$J=(x_{7},\ldots,x_{11})(f_{1},\ldots,f_{5})+(x_{1},\ldots,x_{5})(f_{7},\ldots,f_{11})+f_{6}(x_{9},\ldots,x_{11})$,
$I=(f_{1},\ldots,f_{11})$. We have $B=\\{w_{ij}:1\leq i<j\leq
5\\}\cup\\{w_{kt}:6<k<t\leq 11\\}\cup\\{w_{i6}:i\in[8],i\not=6\\}$, that is
$s=|B|=27$. Let $c_{1}=x_{6}w_{12}$, $c_{2}=x_{6}w_{23}$, $c_{3}=x_{6}w_{34}$,
$c_{4}=x_{6}w_{45}$, $c_{5}=x_{6}w_{15}$, $c_{6}=x_{8}w_{67}$,
$c_{7}=x_{9}w_{78}$, $c_{8}=x_{10}w_{89}$, $c_{9}=x_{11}w_{9,10}$,
$c_{10}=x_{7}w_{10,11}$, $c_{11}=x_{7}w_{8,11}$,
$c^{\prime}_{13}=x_{4}w_{13}$, $c^{\prime}_{14}=x_{5}w_{14}$,
$c^{\prime}_{24}=x_{6}w_{24}$, $c^{\prime}_{25}=x_{3}w_{25}$,
$c^{\prime}_{35}=x_{6}w_{35}$. These are all monomials of $C$, that is
$q=|C|=16$ and so $s=q+r$. The intervals $[f_{i},c_{i}]$, $i\in[11]$ and
$[w_{13},c^{\prime}_{13}]$, $[w_{14},c^{\prime}_{14}]$,
$[w_{24},c^{\prime}_{24}]$, $[w_{25},c^{\prime}_{25}]$,
$[w_{35},c^{\prime}_{35}]$ induce a partition $P$ on $I/J$ with sdepth $4$.

We claim that $\operatorname{depth}_{S}S/J=2$. Indeed, let
$J^{\prime}=(x_{7},\ldots,x_{11})(f_{1},\ldots,f_{5})+(x_{1},\ldots,x_{5})(f_{7},\ldots,f_{11})=(x_{12},x_{6})(x_{7},\ldots,x_{11})(x_{1},\ldots,x_{5})$.
By [5, Theorem 1.4] we get $\operatorname{depth}_{S}S/J^{\prime}=2=d$. Set
$J_{1}=J^{\prime}+(x_{12}x_{6}x_{9})$, $J_{2}=J_{1}+(x_{12}x_{6}x_{10})$. We
have $J=J_{2}+(x_{12}x_{6}x_{11})$. In the exact sequences

$0\to(x_{12}x_{6}x_{9})/(x_{12}x_{6}x_{9})\cap J^{\prime}\to S/J^{\prime}\to
S/J_{1}\to 0,$ $0\to(x_{12}x_{6}x_{10})/(x_{12}x_{6}x_{10})\cap J_{1}\to
S/J_{1}\to S/J_{2}\to 0,$ $0\to(x_{12}x_{6}x_{11})/(x_{12}x_{6}x_{11})\cap
J_{2}\to S/J_{2}\to S/J\to 0$

the first terms have depth $\geq 5$. Applying the Depth Lemma by recurrence we
get our claim.

Now we see that $\operatorname{depth}_{S}S/I=6$. Set
$I_{j}=(f_{1},\ldots,f_{j})$ for $6\leq j\leq 11$. We have $I=I_{11}$,
$I_{6}=x_{12}(x_{1},\ldots,x_{6})$ and $\operatorname{depth}_{S}S/I_{6}=6$. In
the exact sequences

$0\to(f_{j+1})/(f_{j+1})\cap I_{j}\to S/J_{j}\to S/I_{j+1}\to 0,$

$6\leq j<11$ we have $(f_{j+1})\cap I_{j}=f_{j+1}(x_{12},x_{7},\ldots,x_{j})$
and so $\operatorname{depth}_{S}(f_{j+1})/(f_{j+1})\cap I_{j}=12-(j-5)\geq 7$
for $6\leq j<11$. Applying the Depth Lemma by recurrence we get
$\operatorname{depth}_{S}S/I_{j+1}=6$ for $6\leq j<11$ which is enough.

Finally using the Depth Lemma in the exact sequence

$0\to I/J\to S/J\to S/I\to 0$

it follows $\operatorname{depth}_{S}I/J=2=d$.

The following lemma is the key in the proof of Theorem 1 and its proof is
given in the next section.

######  Lemma 3.

Suppose that $E=\emptyset$, $C\subset C_{3}$, $C\cap W=\emptyset$ and Theorem
1 holds for $r^{\prime}<r$. Then $\operatorname{depth}_{S}I/J\leq d+1$.

######  Proposition 1.

Suppose that $C\cap(f_{1},\ldots,f_{r})\subset C_{3}$, $C\cap W=\emptyset$ and
Theorem 1 holds for $r^{\prime}<r$. Then $\operatorname{depth}_{S}I/J\leq
d+1$.

###### Proof.

Suppose that $E\not=\emptyset$, otherwise apply Lemma 3. Set
$I^{\prime}=(f_{1},\ldots,f_{r})$, $J^{\prime}=J\cap I^{\prime}$. In the exact
sequence

$0\to I^{\prime}/J^{\prime}\to I/J\to I/(I^{\prime},J)\to 0$

the last term is isomorphic to something generated by $E$ and so its depth is
$\geq d+1$. The first term satisfies the conditions of Lemma 3 which gives
$\operatorname{depth}_{S}I^{\prime}/J^{\prime}\leq d+1$. By the Depth Lemma we
get $\operatorname{depth}_{S}I/J\leq d+1$ too.

Proof of Theorem 1

Apply induction on $r$. If $r<5$ then $B\cap(f_{1},\ldots,f_{r})\subset W$
implies $|B\cap(f_{1},\ldots,f_{r})|<2r$ and so
$\operatorname{sdepth}_{S}I/J\leq d+1$ and even
$\operatorname{depth}_{S}I/J\leq d+1$ by [12, Proposition 2.4] (we may also
apply [9, Theorem 0.3]). Suppose that $r\geq 5$. Since all divisors of a
monomial $c\in C\cap(f_{1},\ldots,f_{r})$ of degrees $d+1$ are in $B$, they
are also in $W$ by our hypothesis. Thus $C\cap(f_{1},\ldots,f_{r})\subset
C_{3}$ and we may apply Proposition 1 under induction hypothesis. $\hfill\
\square$

######  Corollary 1.

Suppose that $B\cap(f_{1},\ldots,f_{r})\subset W$ and $C\cap W=\emptyset$.
Then $\operatorname{depth}_{S}I/J\leq\operatorname{sdepth}_{S}I/J$, that is
the Stanley Conjecture holds for $I/J$.

###### Proof.

If $\operatorname{sdepth}_{S}I/J=d$ then apply [7, Theorem 4.3], otherwise
apply Theorem 1.

##  2\. Proof of Lemma 3.

We may suppose that $B\subset W$ because each monomial of $B$ must divide a
monomial of $C$, otherwise we get $\operatorname{depth}_{S}I/J\leq d+1$ by
[10, Lemma 1.5]. Then we may suppose that
$B\subset\cup_{i}\operatorname{supp}f_{i}$ and we may reduce to the case when
$[n]=\cup_{i}\operatorname{supp}f_{i}$ because then
$\operatorname{depth}_{S}I/J=\operatorname{depth}_{\tilde{S}}(I\cap{\tilde{S}})/(J\cap{\tilde{S}})$
for ${\tilde{S}}=K[\\{x_{t}:t\in\cup_{i}\operatorname{supp}f_{i}\\}]$.

On the other hand, we may suppose that for each $i\in[r]$ there exists $c\in
C$ such that $f_{i}|c$, otherwise we may apply again [10, Lemma 1.5]. Since
$c\in C_{3}$, let us say $c$ is the least common multiple of
$f_{1},f_{2},f_{3}$ we see that at least,let us say, $w_{12}\in B$. Then
$f_{i}\in(u_{1})$, $i\in[2]$ for some monomial $u_{1}=(f_{1}f_{2})/w_{12}$ of
degree $d-1$.

We may assume that $f_{i}\in(u_{1})$ if and only if $i\in[k_{1}]$ for some
$2\leq k_{1}\leq r$. Set $U_{1}=\\{f_{1},\ldots,f_{k_{1}}\\}$. We also assume
that

$\\{u_{i}:i\in[e]\\}=\\{u:u=gcd(f_{i},f_{j}),\operatorname{deg}u=d-1,i\not=j\in[r]\\},$

and define

$U_{i}=\\{f_{j}:f_{j}\in(u_{i}),j\in[r]\\}$

for each $i\in[e]$. Since each $f_{t}\in U_{i}$ divides a certain $c\in C$ we
see from our construction that there exist $f_{p},f_{l}\in U_{i}$ such that
$w_{tp},w_{tl}\in B$. Note that if $|U_{i}\cap U_{j}|\geq 2$ then we get
$u_{i}=u_{j}$ and so $i=j$. Thus $|U_{i}\cap U_{j}|\leq 1$ for all
$i,j\in[e]$, $i\not=j$.

Suppose that $w_{ij}\in J$ for all $i\in[k_{1}]$ and for all $j>k_{1}$, let us
say $f_{i}=u_{1}x_{i}$ for $i\in[k_{1}]$. Set
$I^{\prime}=(f_{1},\ldots,f_{k_{1}})$, $J^{\prime}=I^{\prime}\cap J$ and
$\hat{S}=K[\\{x_{i}:i\in[k_{1}]\cup\operatorname{supp}u_{1}\\}]$. Then
$\operatorname{depth}_{S}I^{\prime}/J^{\prime}=\operatorname{depth}_{\hat{S}}I^{\prime}\cap\hat{S}/J^{\prime}\cap\hat{S}=\operatorname{deg}u_{1}+\operatorname{depth}_{\hat{S}}(x_{1},\ldots,x_{k_{1}})\hat{S}/(J^{\prime}:u_{1})\cap\hat{S}=d$.
Thus $\operatorname{depth}_{S}I/J=d$ by the Depth Lemma applied to the exact
sequence

$0\to I^{\prime}/J^{\prime}\to I/J\to I/(I^{\prime},J)\to 0,$

since the last term has depth $\geq d$ being generated by squarefree monomials
of degree $\geq d$. In particular, $\operatorname{depth}_{S}I/J=d$ if $e=1$.
If $e>1$ we may assume that for each $i\in[e]$ there exists $j\in[e]$ with
$U_{i}\cap U_{j}\not=\emptyset$.

######  Example 4.

Back to Example 3 note that we may take $u_{1}=x_{12}$, $u_{2}=x_{6}$ and
$U_{1}=\\{f_{1},\ldots,f_{6}\\}$, $U_{2}=\\{f_{6},\ldots,f_{11}\\}$.

######  Lemma 4.

Suppose that $e\geq 2$ and $f_{r}\in U_{e}$. Let $I^{\prime}$ be the ideal
generated by all $f_{k}\in U_{e}\setminus\\{f_{r}\\}$. If
$\operatorname{depth}_{S}I/(J,I^{\prime},f_{r})\geq d+1$ and there exists $t$
with $f_{t}\in(\cup_{i=1}^{e-1}U_{i})\setminus U_{e}$ such that $w_{rt}\in
B\setminus I^{\prime}$ then $\operatorname{depth}_{S}I/(J,I^{\prime})=d+1$.

###### Proof.

Using the Depth Lemma applied to the exact sequence

$0\to(f_{r})/(f_{r})\cap(J,I^{\prime})\to I/(J,I^{\prime})\to
I/(J,I^{\prime},f_{r})\to 0$

we see that it is enough to show that
$\operatorname{depth}_{S}(f_{r})/(f_{r})\cap(J,I^{\prime})=d+1$. By our
hypothesis the squarefree monomials from $(f_{r})\setminus(J,I^{\prime})$ have
the form $w_{rt^{\prime}}$ for some $t^{\prime}$ with
$f_{t^{\prime}}\in(\cup_{i=1}^{e-1}U_{i})\setminus U_{e}$ and
$w_{rt^{\prime}}\in B\setminus I^{\prime}$.

Next we will describe the above set of monomials. If there exists no $U_{l}$
containing $f_{t}$, $f_{t^{\prime}}$ then their contributions to
$(f_{r})/(f_{r})\cap(J,I^{\prime})$ consist in two different monomials
$w_{rt}$, $w_{rt^{\prime}}$. Otherwise, we must have $f_{r}=x_{k}x_{p}v$,
$f_{t}=x_{k}x_{m}v$ and $f_{t^{\prime}}=x_{p}x_{m}v$ for some different
$k,m,p\in[n]$ and one monomial $v$ of degree $d-2$. Thus
$w_{rt}=w_{rt^{\prime}}$ and the contributions of $f_{t}$, $f_{t^{\prime}}$
consist in just one monomial. Let $A$ be the set of all
$f_{k}\in(\cup_{i=1}^{e-1}U_{i})\setminus U_{e}$ such that $w_{rk}\in
B\setminus I^{\prime}$ and define an equivalence relation on $A$ by $f_{t}\sim
f_{t^{\prime}}$ if $f_{t},f_{t^{\prime}}\in U_{i}$ for some $i\in[e-1]$. For
some $f_{t}$ from an equivalence class of $A/\sim$ we have
$w_{rt}=x_{\gamma_{t}}f_{r}$ for one $\gamma_{t}\in[n]$. Let $\Gamma$ be the
set of all these variables $x_{\gamma_{t}}$ for which
$w_{rt}\not\in(J,I^{\prime})$. For two $x_{\gamma_{t}}$,
$x_{\gamma_{t^{\prime}}}$ corresponding to different classes we have
$w_{tt^{\prime}}=x_{\gamma_{t}}x_{\gamma_{t^{\prime}}}f_{r}$ since $f_{t}$,
$f_{t^{\prime}}$ are not in the same equivalence class. Thus
$w_{tt^{\prime}}\in J$ because otherwise $w_{tt^{\prime}}\in C$ which is
impossible by our hypothesis. Let $Q\subset K[\Gamma]$ be the ideal generated
by all squarefree quadratic monomials. The multiplication by $f_{r}$ gives a
bijection between $K[\Gamma]/Q$ and $(f_{r})/(f_{r})\cap(J,I^{\prime})$
because each squarefree monomial of $(B\cap(f_{r}))\setminus I^{\prime}$ has
the form $w_{rt}=f_{r}x_{\gamma_{t}}$ for some $t$, $x_{\gamma_{t}}$ being in
$\Gamma$. Then
$\operatorname{depth}_{S}(f_{r})/(f_{r})\cap(J,I^{\prime})=d+\operatorname{depth}_{K[\Gamma]}K[\Gamma]/Q=d+1$
since the variables of $f_{r}$ form a regular sequence for
$f_{r})/(f_{r})\cap(J,I^{\prime})$ (in the squarefree frame). Note that if
$A/\sim$ has just one class of equivalence containing some $f_{t}$ with
$w_{tr}\in B$ then $|\Gamma|=1$, $Q=0$ and also it holds
$\operatorname{depth}_{K[\Gamma]}K[\Gamma]/Q=1$.

######  Remark 2.

In the notations of the above lemma suppose that $w_{rt}\in(J,I^{\prime})$ for
all $t$ with $f_{t}\in(\cup_{i=1}^{e-1}U_{i})\setminus U_{e}$. Then there
exists no squarefree monomial of degree $d+1$ in
$(f_{r})\setminus(J,I^{\prime})$ and so
$\operatorname{depth}_{S}(f_{r})/(f_{r})\cap(J,I^{\prime})=d$. It follows that
$\operatorname{depth}_{S}I/(J,I^{\prime})=d$ too.

######  Lemma 5.

Suppose that $e\geq 2$. If $\operatorname{depth}_{S}I/(J,(U_{e}))\leq d+1$
then $\operatorname{depth}_{S}I/J\leq d+1$.

###### Proof.

Suppose that $U_{e}\supset\\{f_{k+1},\ldots,f_{r}\\}$ for some $k\leq r$. Let
$I^{\prime}_{k}$ be the ideal generated by all $f_{t}\in
U_{e}\setminus\\{f_{k+1},\ldots,f_{r}\\}$. We claim that
$\operatorname{depth}_{S}I/(J,I^{\prime}_{k})\leq d+1$. Apply induction on
$r-k$, the case $k=r-1$ being done in Lemma 4 and Remark 2. Assume that
$r-k>1$ and note that $I^{\prime}_{k+1}=(I^{\prime}_{k},f_{k+1})$. By
induction hypothesis we have
$\operatorname{depth}_{S}I/(J,I^{\prime}_{k+1})\leq d+1$. If
$\operatorname{depth}_{S}I/(J,I^{\prime}_{k+1})=d$ then we get
$\operatorname{depth}_{S}I/(J,I^{\prime}_{k})\leq d+1$ by Lemma 2. If
$\operatorname{depth}_{S}I/(J,I^{\prime}_{k+1})=d+1$ we get again
$\operatorname{depth}_{S}I/(J,I^{\prime}_{k})\leq d+1$ by Lemma 4 and Remark
2. This proves our claim.

Now choose $r-k$ maxim, let us say $U_{e}=\\{f_{k+1},\ldots,f_{r}\\}$. Then
$I^{\prime}_{k}=0$ and so we get $\operatorname{depth}_{S}I/J\leq d+1$.

######  Example 5.

Let $n=6$, $r=8$, $d=2$, $f_{1}=x_{1}x_{2}$, $f_{2}=x_{1}x_{3}$,
$f_{3}=x_{1}x_{4}$, $f_{4}=x_{2}x_{3}$, $f_{5}=x_{3}x_{5}$,
$f_{6}=x_{2}x_{6}$, $f_{7}=x_{3}x_{6}$, $f_{8}=x_{4}x_{6}$, and
$I=(f_{1},\ldots,f_{8})$,

$J=(x_{1}x_{2}x_{5},x_{1}x_{3}x_{6},x_{1}x_{4}x_{5},x_{1}x_{4}x_{6},x_{2}x_{3}x_{4},x_{2}x_{5}x_{6},x_{3}x_{4}x_{5},x_{3}x_{5}x_{6},x_{4}x_{5}x_{6}).$

It follows that $U_{1}=\\{f_{1},\ldots,f_{3}\\}$,
$U_{2}=\\{f_{2},f_{4},f_{5},f_{7}\\}$, $U_{3}=\\{f_{6},f_{7},f_{8}\\}$,
$U_{4}=\\{f_{1},f_{4},f_{6}\\}$, $U_{5}=\\{f_{3},f_{8}\\}$, $u_{1}=x_{1}$,
$u_{2}=x_{3}$, $u_{3}=x_{6}$, $u_{4}=x_{2}$, $u_{5}=x_{4}$. Let
${\tilde{S}}=K[x_{1},\ldots,x_{5}]$,
${\tilde{J}}\subset{\tilde{I}}\subset{\tilde{S}}$ be the corresponding ideals
given in Example 1. For $I^{\prime}=(U_{3})$ we have
$I/(J,I^{\prime})\cong{\tilde{I}}S/({\tilde{J}},x_{6}{\tilde{I}})S$. Thus
$\operatorname{depth}_{S}I/(J,I^{\prime})=\operatorname{depth}_{\tilde{S}}{\tilde{I}}/{\tilde{J}}\leq
3=d+1$ by Example 1 and so using Lemma 5 we get
$\operatorname{depth}_{S}I/J\leq 3$ too.

######  Proposition 2.

If $\cap_{i\in[e]}U_{i}\not=\emptyset$ then $\operatorname{depth}_{S}I/J\leq
d+1$.

###### Proof.

As we have seen $\operatorname{depth}_{S}I/J=d$ if $e=1$. Assume that $e>1$
and let us say $f_{r}\in\cap_{i\in[e]}U_{i}$. Set $I^{\prime}=(U_{e})$. In
$I/(J,I^{\prime})$ we have $(e-1)$ disjoint
$U^{\prime}_{i}=U_{i}\setminus\\{f_{r}\\}$, $i\in[e-1]$. It follows that
$\operatorname{depth}_{S}I/(J,I^{\prime})=d$ and so
$\operatorname{depth}_{S}I/J\leq d+1$ by Lemma 5.

Proof of Lemma 3.

If $\cap_{i\in[e]}U_{i}\not=\emptyset$ then apply the above proposition.
Otherwise, suppose that $f_{r}\in U_{j}$ if and only if $1\leq j<k$ for some
$1<k\leq e$. Set $I^{\prime}=(U_{k},\ldots,U_{e})$. Applying again the above
proposition we get $\operatorname{depth}_{S}I/(J,I^{\prime})\leq d+1$. Set
$L_{i}=\cup_{j\geq i}U_{j}$. Since $I^{\prime}=(L_{k})$ and
$\operatorname{depth}_{S}I/(J,L_{k})\leq d+1$ we see that
$\operatorname{depth}_{S}I/(J,L_{k+1})\leq d+1$ by Lemma 5. Using by
recurrence Lemma 5 we get
$\operatorname{depth}_{S}I/J=\operatorname{depth}_{S}I/(J,L_{e+1})\leq d+1$
since $L_{e+1}=\emptyset$. $\hfill\ \square$

It is not necessary to assume in Proposition 2 that $C\cap W=\emptyset$
because anyway this follows as shows the following lemma.

######  Lemma 6.

If $\cap_{i\in[e]}U_{i}\not=\emptyset$ then $C\cap W=\emptyset$.

###### Proof.

Clearly we may suppose that $e>1$. Let $f_{r}\in\cap_{i\in[e]}U_{i}$. Suppose
that $w_{tt^{\prime}}\in C$ for some $t\in U_{i}$, $t^{\prime}\in U_{j}$. Then
$i\not=j$ and let us say $f_{t}=u_{i}x_{1}$, $f_{t^{\prime}}=u_{j}x_{2}$ and
$f_{r}=u_{i}x_{3}=u_{j}x_{4}$. It follows that $u_{i}=x_{4}v$, $u_{j}=x_{3}v$
for some monomial $v$ and so $f_{t}=x_{1}x_{4}v$,
$f_{t^{\prime}}=x_{2}x_{3}v$, $f_{r}=vx_{3}x_{4}$. Note that $f_{t}x_{2}\in B$
and so must be of type $w_{tt^{\prime\prime}}$ for some
$t^{\prime\prime}\in[r]$. It follows that $f_{t},f_{t^{\prime\prime}}\in
U_{k}$ for some $k\in[e]$. By hypothesis $f_{r}\in U_{k}$ and so $k=i$ because
otherwise $|U_{k}\cap U_{i}|>1$ which is impossible. Since $f_{t}x_{2}\in B$
we see that $x_{2}\not\in\operatorname{supp}u_{i}$ and it follows that
$f_{t^{\prime\prime}}=x_{2}x_{4}v$. Therefore,
$w_{t^{\prime}t^{\prime\prime}}=x_{2}x_{3}x_{4}v\in B$ and so
$f_{t^{\prime}},f_{t^{\prime\prime}}\in U_{p}$ for some $p$. This is not
possible because otherwise one of $|U_{p}\cap U_{i}|$, $|U_{p}\cap U_{j}|$,
$|U_{i}\cap U_{j}|$ is $\geq 2$. Contradiction!

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