# Theory of Vibrational Polariton Chemistry in the Collective Coupling Regime

Arkajit Mandal<EMAIL_ADDRESS>Department of Chemistry, University
of Rochester, Rochester, New York, 14627  Xinyang Li  Department of Chemistry,
University of Rochester, Rochester, New York, 14627  Pengfei Huo
<EMAIL_ADDRESS>Department of Chemistry, University of Rochester,
Rochester, New York, 14627  Institute of Optics, Hajim School of Engineering
and Applied Sciences, University of Rochester, Rochester, New York, 14627

###### Abstract

We theoretically demonstrate that chemical reaction rate constant can be
significantly suppressed by coupling molecular vibrations with an optical
cavity, exhibiting both the collective coupling effect and the cavity-
frequency modification of the rate constant. When a reaction coordinate is
strongly coupled to the solvent molecules, the reaction rate constant is
reduced due to the dynamical caging effect. We demonstrate that collectively
coupling the solvent to the cavity can further enhance this dynamical caging
effect, leading to additional suppression of the chemical kinetics. This
effect is further amplified when cavity loss is considered.

Hybridizing molecular vibrations and the photonic excitations inside an
optical cavity [1, 2, 3] forms vibrational polaritons (Fig. 1a). Several
recent experiments [4, 3, 5, 6, 7, 8] have demonstrated that it is possible to
modify ground-state chemical reactions by coupling the cavity radiation mode
with the vibrational degrees of freedom (DOF) of molecules. This new strategy
of vibrational strong coupling (VSC), if feasible, will offer a paradigm-shift
in chemical transformations [3, 5]. Despite recent theoretical works [9, 10,
11, 12, 13, 14, 15], a clear theoretical explanation of such remarkable VSC
effects in ground-state reactivity remains elusive, including explanations of
both (i) the collective effect ($N$-dependent effect where $N$ is the total
number of molecules inside the cavity) on chemical reaction rates and (ii) the
resonant effect where the suppression of the rate is achieved with a
particular cavity photon frequency. In this work, we theoretically demonstrate
that the chemical reaction rate constant can be modified when a set of solvent
DOF is collectively coupled to both a reaction coordinate of a solute molecule
and the cavity radiation mode. Our results demonstrate both the collective
coupling effect and the cavity frequency-dependent modifications of the
reaction rate constant, which purely originate from a change of the
transmission coefficient (recrossing factor) of the rate constant due to the
dynamical caging effects from the cavity [11].

We begin by writing the light-matter interaction Hamiltonian in the minimal
coupling form as follows

$\hat{H}_{\mathrm{C}}=\sum_{j}\frac{1}{2m_{j}}(\hat{\bf p}_{j}-{z}_{j}\hat{\bf
A})^{2}+\hat{V}(\hat{\bf x})+\hat{H}_{\mathrm{ph}},$ (1)

where the sum is performed over all charged particles, including both
electrons and nuclei, $m_{j}$ and $z_{j}$ are mass and charge for particle
$j$, respectively, and $\hat{V}$ represents the Coulomb potential of all
charged particles. The total dipole operator of the matter is
$\hat{\bm{\mu}}=\sum_{j}z_{j}x_{j}$. In addition, $\hat{\bf
x}\equiv\\{\hat{\bf x}_{j}\\}=\\{\hat{\bf R},\hat{\bf r}\\}$ with $\hat{\bf
R}$ and $\hat{\bf r}$ representing the nuclear and electronic coordinates,
respectively, $\hat{\bf p}\equiv\\{\hat{\bf p}_{\bf R},\hat{\bf p}_{\bf
r}\\}\equiv\\{\hat{\bf p}_{j}\\}$ is the canonical momentum operator, such
that $\hat{\bf p}_{j}=-i\hbar{\bm{\nabla}}_{j}$. The cavity photon field
Hamiltonian under the single mode assumption is expressed as
$\hat{H}_{\mathrm{ph}}=\hbar\omega_{\mathrm{c}}\big{(}\hat{a}^{\dagger}\hat{a}+\frac{1}{2}\big{)}=\frac{1}{2}\big{(}\hat{p}_{\mathrm{c}}^{2}+\omega_{\mathrm{c}}^{2}\hat{q}_{\mathrm{c}}^{2}\big{)}$,
where $\omega_{\mathrm{c}}$ is the frequency of the mode in the cavity,
$\hat{a}^{\dagger}$ and $\hat{a}$ are the photonic creation and annihilation
operators, and
$\hat{q}_{\mathrm{c}}=\sqrt{\hbar/2\omega_{\mathrm{c}}}(\hat{a}^{\dagger}+\hat{a})$
and
$\hat{p}_{\mathrm{c}}=i\sqrt{\hbar\omega_{\mathrm{c}}/2}(\hat{a}^{\dagger}-\hat{a})$
are the photonic coordinate and momentum operators, respectively. Choosing the
Coulomb gauge, ${\bm{\nabla}}\cdot\hat{\bf A}=0$, the vector potential becomes
purely transverse as $\hat{\bf A}=\hat{{\bf A}}_{\perp}$. Under the long-
wavelength approximation, $\hat{\bf A}={\bf
A}_{0}\big{(}\hat{a}+\hat{a}^{\dagger}\big{)}={\bf
A}_{0}\sqrt{{2\omega_{\mathrm{c}}}/{\hbar}}~{}\hat{q}_{\mathrm{c}}$ for a
Fabry-Pérot cavity, where ${\bf
A}_{0}=\sqrt{\hbar/2\omega_{\mathrm{c}}\varepsilon_{0}\mathcal{V}}\cdot{\bf
e}$, with $\mathcal{V}$ as the quantization volume inside the cavity,
$\varepsilon_{0}$ as the permittivity, and $\hat{{\bf e}}$ as the unit vector
of the field polarization.

Using the Power-Zienau-Woolley (PZW) [16, 17] gauge transformation operator
$\hat{U}=\exp[-\frac{i}{\hbar}\hat{\bm{\mu}}\cdot{\bf
A}_{0}\big{(}\hat{a}+\hat{a}^{\dagger}\big{)}\big{]}$ as well as an unitary
phase transformation operator
$\hat{U}_{\phi}=\exp[-i\frac{\pi}{2}\hat{a}^{\dagger}\hat{a}]$, the Pauli-
Fierz (PF) non-relativistic QED Hamiltonian [18, 19]
$\hat{H}_{\mathrm{PF}}=\hat{U}_{\phi}\hat{U}\hat{H}_{\mathrm{C}}\hat{U}^{\dagger}\hat{U}^{\dagger}_{\phi}$
is obtained as follows

$\hat{H}_{\mathrm{PF}}=\hat{H}_{\mathrm{M}}+\frac{1}{2}\hat{p}_{\mathrm{c}}^{2}+\frac{1}{2}\omega_{\mathrm{c}}^{2}\big{(}\hat{q}_{\mathrm{c}}+\sqrt{\frac{2}{\hbar\omega_{\mathrm{c}}}}\hat{\bm{\mu}}\cdot{\bf
A}_{0}\big{)}^{2},$ (2)

where the matter Hamiltonian is $\hat{H}_{\mathrm{M}}=\hat{T}_{\bf
R}+\hat{T}_{\bf r}+\hat{V}$, with $\hat{T}_{\bf R}$ and $\hat{T}_{\bf r}$
representing the nuclear and electronic kinetic energy, respectively. The
presence of the dipole self-energy (DSE) term
$\frac{\omega_{\mathrm{c}}}{\hbar}(\hat{\bm{\mu}}\cdot{\bf A}_{0})^{2}$ in Eq.
2 is necessary in order to have a gauge invariant Hamiltonian [18, 19] and it
has shown to be crucial for an accurate description of light-matter
interactions under the dipole gauge [18, 19, 20]. We further assume that the
dipole of the matter is oriented in the field polarization direction, such
that $\hat{\bm{\mu}}\cdot{\bf A}_{0}=\hat{\mu}\cdot{A}_{0}$. We acknowledge
that this will not be the situation of the recent VSC experiments [3], where
the molecules in the solution phase should be isotropically disordered. That
said, it is certainly possible to have chemical reactions in anisotropic
solvents like liquid crystals [21, 22] inside an optical cavity [23] More
detailed discussions are provided in the Supplemental Materials.

We are interested in electronically adiabatic reactions, thus we only consider
the electronic ground state of the system defined as
$(\hat{H}_{\mathrm{M}}-\hat{T}_{\bf R})|\Psi_{g}\rangle=E_{g}({\bf
R})|\Psi_{g}\rangle$, with $E_{g}({\bf R})$ as the ground adiabatic potential.
Projecting $\hat{H}_{\mathrm{M}}$ and $\hat{\mu}$ in the ground electronic
state with $\hat{\mathcal{P}}=|\Psi_{g}\rangle\langle\Psi_{g}|$, we obtain the
following model Hamiltonian

$\hat{\mathcal{H}}^{g}_{\mathrm{PF}}=\frac{\hat{\bf P}^{2}}{2}+E_{g}({\bf
R})+\frac{\hat{p}_{\mathrm{c}}^{2}}{2}+\frac{1}{2}\omega_{\mathrm{c}}^{2}\Big{(}\hat{q}_{\mathrm{c}}+\sqrt{\frac{2}{\hbar\omega_{\mathrm{c}}}}A_{0}\cdot{\mu}_{g}({\bf
R})\Big{)}^{2},$ (3)

where ${\mu}_{g}({\bf R})=\langle\Psi_{g}|\hat{\mu}|\Psi_{g}\rangle$. Note
that projecting $\hat{\mu}$ inside the dipole self-energy term is the accurate
matter state truncation scheme for the dipole-gauge Hamiltonian [20, 24]
because it ensures that all operators are properly confined in the same
truncated electronic subspace
$\hat{\mathcal{P}}=|\Psi_{g}\rangle\langle\Psi_{g}|$ in order to generate
consistent results compared to the full Hamiltonian 111Indeed, if
$\hat{\mathcal{I}}=\hat{\mathcal{P}}+\hat{\mathcal{Q}}$ represents the
identity of the full electronic Hilbert space, then
$\hat{\mathcal{P}}\hat{\mu}\hat{\mathcal{P}}$ is properly confined in the
subspace $\hat{\mathcal{P}}$, whereas
$\hat{\mathcal{P}}\hat{\mu}^{2}\hat{\mathcal{P}}=\hat{\mathcal{P}}\hat{\mu}(\hat{\mathcal{P}}+\hat{\mathcal{Q}})\hat{\mu}\hat{\mathcal{P}}$
contains the terms outside the subspace $\hat{\mathcal{P}}$. More numerical
evidence can be found in Fig. S2 of the Supplemental Materials in Ref. [24].

Figure 1: (a) Schematic representation of the reactive molecule coupled to a
set of solvent (phonon) modes which are coupled to quantized radiation in a
cavity. (b) Double-well potential (black solid line) representing a chemical
reaction as a function the reaction coordinate $R_{0}$ where the barrier can
approximated as a inverted Harmonic potential (red dash). Potential energy of
a (c) Harmonic phonon mode (solvent) and (c) a bare cavity mode which when
hybridized leads to a Rabi-splitting $\hbar\Omega_{R}$.

In this letter, we consider a model system for $E_{g}(\bf R)$, where ${\bf
R}=\\{R_{0},R_{1},...R_{N}\\}$ include a reactive molecule with a single
reaction coordinate $R_{0}$ as well as $N$ solvent DOFs $R_{i}$ (where
$i\in[1,N]$) that couples to $R_{0}$. Using the typical Caldeira-Leggett [26]
system-bath Hamiltonian, $E_{g}(\bf R)$ is modeled as

$E_{g}({\bf
R})=U_{0}(R_{0})+\sum_{i=1}^{N}\frac{1}{2}\omega^{2}_{i}\big{(}R_{i}+\frac{c_{i}}{\omega^{2}_{i}}R_{0}\big{)}^{2}.$
(4)

The solute molecule is modeled as a double-well potential $U_{0}(R_{0})=a\cdot
R_{0}^{4}-b\cdot R_{0}^{2}$ as shown in Fig. 1(b) (solid black line) to
represent the chemical reaction along $R_{0}$, and the details of the
parameters are provided in the Supplemental Information. At the top of the
barrier $R_{0}=R^{\ddagger}_{0}$,
$U_{0}(R_{0})\approx-\frac{1}{2}\omega_{\ddagger}^{2}(R_{0}-R^{\ddagger}_{0})^{2}$
as indicated by the red dashed line in Fig. 1(b), where
$\omega_{\ddagger}=~{}1048$ cm-1 is the top of the barrier frequency. Further,
the total dipole of the system is ${\mu}_{g}({\bf
R})=\sum_{i=1}^{N}\mu_{i}(R_{i})$, and we assume that $\mu_{0}(R_{0})=0$. Note
that because the coupling strength between the cavity mode to individual
molecules is very weak under the collective coupling regime; thus, the result
of this letter does not change if $\mu_{0}(R_{0})\neq 0$. We further simplify
our consideration of the solute-solvent coupling as $c_{i}=c_{\mathrm{s}}$ and
$\omega_{i}=\omega_{\mathrm{s}}=1400$ cm-1. The amplitude of the solvent-
friction is
$\lambda=\sum_{i=1}^{N}\frac{c^{2}_{i}}{\omega^{2}_{i}}={Nc^{2}_{\mathrm{s}}}/{\omega^{2}_{\mathrm{s}}}$.
In this work, we keep $\lambda$ as a constant throughout. This means that as
we increase $N$, the corresponding $c_{\mathrm{s}}$ will be decreased by
$1/\sqrt{N}$. In particular, we use $\lambda=3.84\times 10^{-4}$ a.u., and for
$N=2500$, which corresponds to the solvent-solute coupling as
$c_{\mathrm{s}}=2.5\times 10^{-6}$ a.u. Further generalization of the solvent-
solute coupling with an arbitrary spectral density is possible [27], with
details provided in the Supplemental Material.

Treating both ${\bf R}$ and $q_{\mathrm{c}}$ in Eq. 3 on an equally classical
footing [28, 14, 29, 11, 9], one can express the reaction rate constant as
follows [30, 31, 32]

$k=\lim_{t\to t_{\mathrm{p}}}\kappa(t)\cdot k_{\mathrm{TST}},$ (5)

where $k_{\mathrm{TST}}$ is the Transition State Theory (TST) rate constant,
$t_{\mathrm{p}}$ refers to the plateau time of the transmission coefficient
$\kappa(t)$. The transmission coefficient captures the dynamical recrossing
effects through the flux-side correlation function formalism [30, 31, 32]

$\kappa(t)=\frac{\langle{\mathcal{F}}(0)\cdot{h}[R_{0}(t)-R_{0}^{\ddagger}]\rangle}{\langle{\mathcal{F}}(0)\cdot{h}[\dot{R}_{0}^{\ddagger}(0)]\rangle},$
(6)

where ${h}[R_{0}-R_{0}^{\ddagger}]$ is the Heaviside function of the reaction
coordinate $R_{0}$, with the dividing surface $R_{0}^{\ddagger}$ that
separates the reactant and the product regions (for the model system studied
here, $R_{0}^{\ddagger}=0$), the flux function
${\mathcal{F}}(t)=\dot{h}(t)=\delta[R_{0}(t)-R_{0}^{\ddagger}]\cdot\dot{R}_{0}(t)$
measures the reactive flux across the dividing surface (with $\delta(R)$ as
the Dirac delta function), $\dot{R}_{\ddagger}(0)$ represents the initial
velocity of the nuclei on the dividing surface, and $\langle...\rangle$
represents the canonical ensemble average with the constrain on the dividing
surface enforced by $\delta[R(t)-R_{0}^{\ddagger}]$ inside ${\mathcal{F}}(t)$.

It has been shown that the classical potential of mean force (free energy
profile) is invariant under the change of light-matter coupling strength or
photon frequency [14]. Other theoretical investigations based on a simple TST
analysis 222When the DSE is explicitly considered, the barrier height
$E^{\ddagger}$ on the CBO surface remains invariant to changes of the light-
matter coupling strength or the photon frequency. This is because the
equilibrium position along the photonic coordinate $q_{\mathrm{c}}$ is
$q^{0}_{\mathrm{c}}({\bf
R})=-\sqrt{\frac{2}{\hbar\omega_{\mathrm{c}}}}A_{0}\cdot{\mu}_{g}({\bf R})$
for all possible ${\bf R}$. Thus, the last term in Eq. 3 is always 0 for the
reactant well or the transition state on the CBO surface. This explains why
one cannot observe any effects from a simple TST analysis when treating
$q_{\mathrm{c}}$ classically. also suggest no significant change of the
reaction rate constant [34, 29]. Thus, it is reasonable to conjecture that the
VSC modification of the rate constant is purely dynamical [11] and completely
dictated by $\kappa(t)$. Based on this, we have demonstrated the cavity
frequency dependence of the VSC modification of $\kappa$ for a single molecule
coupled to the cavity [11]. In this work, we consider the scenario in which
such cavity modification can also be observed in the collective coupling
regime.

We numerically compute $\kappa(t)$ using the flux-side correlation function
formalism in Eq. 6, with the details provided in the Supplemental Materials.
On the other hand, $\kappa=\lim_{t\to t_{\mathrm{p}}}\kappa(t)$ can also be
obtained using the Grote-Hynes (GH) theory [35, 36, 37, 38, 39, 40] through
the multi-dimensional transition-state treatment [37, 41, 42, 43]. The
transmission coefficient using the GH theory is
$\kappa_{\mathrm{GH}}={{\sqrt{-(\Omega^{\ddagger}_{-})^{2}}}/\omega_{\mathrm{\ddagger}}}$,
where ${\Omega^{\ddagger}_{-}}$ is the unstable imaginary normal-mode
frequency at the dividing surface $R_{0}=R^{\ddagger}_{0}$. To obtain
${\Omega^{\ddagger}_{-}}$, we further approximate the dipole of the
$i_{\mathrm{th}}$ solvent molecule as
$\mu_{i}(R_{i})\approx\mu_{0}+\mu^{\prime}R_{i}$, and define the collective
bright mode of the solvent [29] as
$R_{\mathrm{B}}=\frac{1}{\sqrt{N}}\sum_{i=1}^{N}R_{i}$. The QED Hamiltonian
${\mathcal{H}}^{g}_{\mathrm{PF}}$ can be shown to have three coupled modes,
${\bf x}=\\{R_{0},R_{\mathrm{B}},q_{\mathrm{c}}\\}$, whereas the rest of
normal modes (commonly referred to as the dark modes [29]) are completely
decoupled. At the dividing surface $R_{0}=R_{0}^{\ddagger}$, the Hessian
matrix in the 3-mode ${\bf x}$ subspace is

$\displaystyle\mathcal{H}_{\bf
x}\equiv\frac{\partial^{2}\mathcal{H}^{g}_{\mathrm{PF}}}{\partial
x_{i}\partial
x_{j}}=\begin{bmatrix}-\omega^{2}_{\ddagger}+N\frac{c_{s}^{2}}{\omega_{s}^{2}}&\sqrt{N}c_{s}&0\\\
\sqrt{N}c_{s}&\omega^{2}_{s}+N\frac{\mathcal{C}^{2}}{\omega_{c}^{2}}&\sqrt{N}\mathcal{C}\\\
0&\sqrt{N}\mathcal{C}&\omega^{2}_{c}\\\ \end{bmatrix},$ (7)

where $\mathcal{C}=A_{0}\mu^{\prime}\sqrt{\frac{2\omega_{c}^{3}}{\hbar}}$ and
$N$ is number of the solvent DOFs. A detailed derivation of Eq. 7 is provided
in the Supplemental Material. The imaginary frequency
${\Omega^{\ddagger}_{-}}$ can be obtained by diagonalizing $\mathcal{H}_{\bf
x}$ in Eq. 7. The key to the emerging collective VSC effects is the
$\sqrt{N}c_{s}$ term as well as $\sqrt{N}\mathcal{C}$ in the above Hessian
matrix, which does not exist if one ignores dipole self-energies or
considering $N$ solute molecules (with potential $U_{0}$) coupled to the
cavity [29].

While ${\Omega^{\ddagger}_{-}}$ can be easily computed by numerically
diagonalizing Eq. 7, a simple and concise analytical expression for
${\Omega^{\ddagger}_{-}}$ is not readily available. Nonetheless, we find (see
Supplemental Material) an approximate expression of ${\Omega^{\ddagger}_{-}}$
as follows

$\displaystyle\Omega^{\ddagger}_{-}\approx\Bigg{[}\frac{1}{2}(\tilde{\omega}^{2}_{\ddagger}-\omega^{2}_{c})+\frac{1}{2}\sqrt{(\omega^{2}_{c}+\tilde{\omega}^{2}_{\ddagger})^{2}+4N\sin^{2}\Theta_{\ddagger}\mathcal{C}^{2}}\Bigg{]}^{1/2}$
(8)

where
$-\tilde{\omega}^{2}_{\ddagger}=\frac{1}{2}(-\omega^{2}_{\ddagger}+N\frac{c_{s}^{2}}{\omega_{s}^{2}}+\omega^{2}_{s}+N\frac{\mathcal{C}^{2}}{\omega_{c}^{2}})-\frac{1}{2}\sqrt{(\omega^{2}_{\ddagger}-N\frac{c_{s}^{2}}{\omega_{s}^{2}}+\omega^{2}_{s}+N\frac{\mathcal{C}^{2}}{\omega_{c}^{2}})^{2}+4Nc_{s}^{2}}$
and
$\Theta_{\ddagger}=\frac{1}{2}\tan^{-1}[{2\sqrt{N}c_{s}/({-\omega^{2}_{\ddagger}+N\frac{c_{s}^{2}}{\omega_{s}^{2}}-\omega^{2}_{s}-N\frac{\mathcal{C}^{2}}{\omega_{c}^{2}}}})]$.
It is interesting to note that Eq. 8 has a similar (but not identical)
structure as the case of a single molecule coupled to cavity [11]. That said,
the dependence of ${\Omega^{\ddagger}_{-}}$ (or $\kappa_{\mathrm{GH}}$) on
$\omega_{c}$ is complicated as both $\Theta_{\ddagger}$ and
$\tilde{\omega}_{\ddagger}$ depends on $\omega_{c}$ in a non-trivial fashion.
Nevertheless, one can clearly see from Eqs. 7-8 that ${\Omega^{\ddagger}_{-}}$
depends on both $\omega_{\mathrm{c}}$ and $N$, giving rise to the cavity
frequency dependence and the collective coupling effect. Further, both
${\Omega^{\ddagger}_{-}}$ and $\kappa_{\mathrm{GH}}$ is also a function of
$\sqrt{N}\mu^{\prime}$, which is a signature of the collective coupling
effect. This is similar to the collective Rabi-splitting $\hbar\Omega$ , which
depends on $\sqrt{N}\mu^{\prime}$ when hybridizing the solvent modes
$\\{R_{i}\\}$ to the cavity mode $q_{\mathrm{c}}$ (as shown in Fig. 1c-d). In
the Travis-Cummings model, under the resonant condition (when
$\omega_{s}=\omega_{c}$), $\hbar\Omega$ is given as

$\displaystyle\hbar\Omega=\sqrt{N}\mu^{\prime}A_{0}\sqrt{2\hbar\omega_{s}}\equiv
2\eta\cdot\hbar\omega_{\mathrm{c}},$ (9)

where the unitless parameter $\eta$ (defined above) characterizes the
normalized light-matter coupling strength.

Figure 2: (a) Cavity frequency-dependent suppression of transmission
coefficient $\kappa$ ($\kappa_{0}$ represents the outside cavity scenario) as
a function of cavity photon frequency $\omega_{\mathrm{c}}$ with $N=2500$
solvent modes at various collective light-matter coupling strength $\eta$. (b)
$\kappa$ as function of $N$ while keeping collective solvent-solute coupling
$\sqrt{N}c_{s}$ the same.

Fig. 2 demonstrates both (a) the cavity frequency dependence and (b) the
collective effects on modifying the reaction rate constant. Outside cavity,
the transmission coefficient in the absence of light-matter interaction is
$\kappa(\eta=0)=\kappa_{0}\approx 0.317$, which is much lower than 1 due to
solvent molecules coupled to the reaction coordinate $R_{0}$ around the
barrier region.

Fig. 2a presents the results of coupling $N=2500$ solvent DOFs to the cavity
and obtaining $\kappa$ using both the GH theory (solid lines) and the direct
numerical simulations (dots) using Eq. 6. The results from both approaches are
nearly identical. Importantly, we observe a strong dependence of $\kappa$ on
the photon frequency $\hbar\omega_{\mathrm{c}}$, and $\kappa$ is minimized at
a certain photon frequency which we refer to as $\omega_{\mathrm{c}}^{0}$.
While we do not have a simple analytic expression of $\omega_{\mathrm{c}}^{0}$
(which can be in principle obtained from
$\partial\kappa_{\mathrm{GH}}/\partial\omega_{\mathrm{c}}=0$), it is a
function of $\omega_{\mathrm{s}}$, $\omega_{\ddagger}$, $c_{\mathrm{s}}$ and
$\eta$ (see its definition in Eq. 9). As can be seen, increasing $\eta$
results in a significant red-shift in $\omega_{c}^{0}$. Note that $\eta$
signifies collective coupling, and the individual solvent-cavity coupling is
weak ($\eta/\sqrt{N}<0.0093$). We emphasize, that the suppression of the
$\kappa$ is originated from the collective dynamical caging effects, where the
cavity radiation mode is effectively acting as an additional “solvent” DOF
coupled to the collective bright solvent coordinate $R_{\mathrm{B}}$, which in
turn coupled to the reactive coordinate $R_{0}$, such that the presence of the
cavity mode enhance the recrossing of the reaction coordinate and reduces the
transmission coefficient. As a result, with an increasing light-matter
coupling strength, the plateau value of $\kappa(t)$ keeps decreasing, and at
the same time, becoming more oscillatory (see Fig. S3 of the Supplemental
Materials). This phenomenon is well explored in the context of solvent-
mediated dynamical caging effects [44, 45, 36, 37, 46]. More detailed
discussions on the dynamical caging hypothesis of the VSC reaction can be
found in Ref. [11].

Fig. 2b demonstrates the $N$-dependence of the transmission coefficient using
the GH theory while keeping $\omega_{\mathrm{c}}$ constant. Note that to
clearly identify the effect of increasing $N$ on the light-matter
interactions, we have kept the solvent reorganization energy $\lambda$ a
constant. We find that increasing $N$ effectively increasing the light-matter
interaction strength (see Eq. 9), leading to additional suppression of the
chemical kinetics. Interestingly, unlike the $\hbar\Omega$ which linearly
depends on $\sqrt{N}$ (see 9), $\kappa$ has a non-linear monotonic dependence
of $\sqrt{N}$. This theoretical prediction agrees with the recent VSC
experiments by Ebbesen and co-workers (such as Fig. 3d in Ref. [5]).

Up to now, we have considered a perfect micro-cavity setup with no photon
leaking. Despite the recent progress in the development of high-quality factor
Fabry-Pérot cavities, optical micro-cavities are generally leaky. The typical
values of cavity losses for a Fabry-Pérot cavity is in the range of
$\Gamma_{c}=5-100$ meV [47, 48, 3, 49]. In the classical Markovian limit, the
cavity loss (dissipation) can be described with Langevin dynamics of the
cavity radiation mode (see the Supplemental Material), with the equation of
motion

$\displaystyle\ddot{q}_{\mathrm{c}}=-\frac{d\mathcal{H}_{\mathrm{PF}}^{g}}{dq_{\mathrm{c}}}-\Gamma_{\mathrm{c}}p_{\mathrm{c}}+{\bf
F}_{\mathrm{c}}(t),$ (10)

where $\Gamma_{\mathrm{c}}$ is cavity loss rate and ${\bf F}_{\mathrm{c}}(t)$
is a Gaussian random force bounded by the fluctuation-dissipation theorem
through $\langle{\bf F}_{\mathrm{c}}(0){\bf
F}_{\mathrm{c}}(t)\rangle=2\Gamma_{\mathrm{c}}k_{\mathrm{B}}T\delta(t)$. Using
this approach, we can numerically compute $\kappa$ using the flux-side
correlation function expression in Eq. 6. Incorporating such dissipative
dynamics in the GH theory makes it non-trivial to derive a simple analytic
expression for $\kappa_{\mathrm{GH}}$ and therefore, we study the effect of
the cavity loss only through direct numerical simulations.

Fig. 3 demonstrates the effect of cavity loss on $\kappa$. Here, we choose
$\eta=0.28$ and keep $N=2500$ (with per-molecule coupling
$\eta/\sqrt{N}=0.0056$) as a constant while varying the cavity loss rate
$\Gamma_{\mathrm{c}}$. In Fig. 3a we observe that increasing
$\Gamma_{\mathrm{c}}$ results in further suppression of the chemical rate
constant while concurrently blue-shifting the maximum suppression frequency
$\omega_{\mathrm{c}}^{0}$. It can be seen in Fig. 3b that increasing
$\Gamma_{\mathrm{c}}$ shifts $\omega_{\mathrm{c}}^{0}$ gradually towards to
the value of $\omega_{0}$, which is the vibrational frequency of the solvent
molecules. This cavity loss-assisted suppression of chemical kinetics can be
attributed to the fact that the cavity loss dynamics arises when the cavity
mode is coupled to other non-cavity radiation modes which act as additional
dissipative baths [50] (see Supplemental Material). Just like the cavity mode
which acts like a dissipative bath to the collective solvent coordinate
$R_{\mathrm{B}}$ that leads to suppression of chemical kinetics (as shown in
Fig. 2), introducing an additional dissipative environment to the cavity-mode
itself also leads to further suppression of the chemical rate constant. Note
that when explicitly considering cavity loss dynamics, the collective coupling
effects (Fig. 2) still persist, as demonstrated in the Supplementary
Materials.

Figure 3: Transmission coefficient $\kappa$ when explicitly considering the
cavity loss $\Gamma_{\mathrm{c}}$ at $\eta=0.28$. (a) Cavity frequency
dependence of the transmission coefficient $\kappa$ at various
$\Gamma_{\mathrm{c}}$. (b) $\kappa$ as a function of photon frequency and
cavity loss $\Gamma_{\mathrm{c}}$.

In conclusion, we theoretically demonstrate both the collective coupling
effects and the cavity frequency-dependent modifications of the reaction rate
constant in the polaritonic vibrational strong coupling regime. The model
system includes the coupling between a collective solvent coordinate, the
cavity radiation mode, and the reaction coordinate, giving rise to Rabi-
splitting as well as cavity-modified chemical kinetics that both scales as
$\sqrt{N}$, where $N$ is the number of solvent DOFs. We further demonstrate
that the suppression in chemical kinetics is cavity photon-frequency dependent
such that the maximal suppression occurs around a particular photon frequency
$\omega^{0}_{\mathrm{c}}$. Further, we find a red-shift in
$\omega^{0}_{\mathrm{c}}$ when increasing the light-matter coupling strength.
In addition, we demonstrated that cavity loss significantly enhances this
suppression effect and increasing cavity loss, resulting in a blue-shift of
$\omega^{0}_{\mathrm{c}}$. Overall, when a collective solvent mode is strongly
coupled to a reaction coordinate in the non-Markovian limit, the reaction
coordinate becomes dynamically caged near the barrier region. We find that the
cavity radiation mode as well as intrinsic cavity loss assist in dynamically
caging of the reaction coordinate, leading to the suppression of the reaction
rate constant if the solvent dipoles are aligned in the cavity-polarization
direction. This effect operates under the collective coupling regime and
survives when per-molecule (per solvent DOF) light-matter coupling is weak. We
envision that the current theoretical work brings us one step closer to
resolving the mysteries of VSC enabled chemistry demonstrated in recent
experiments [4, 3, 5, 6, 7, 8] by demonstrating both the collective coupling
effect and the cavity frequency dependent modification of the rate constant.
On the other hand, we expect that the present strategy will be applicable in
controlling chemical reactions in anisotropic solvents such as liquid crystals
[21] inside an optical cavity [23].

###### Acknowledgements.

##  I Acknowledgments

This work was supported by the National Science Foundation CAREER Award under
Grant No. CHE-1845747, by a Cottrell Scholar award (a program by Research
Corporation for Science Advancement), as well as by a University Research
Award from the University of Rochester. Computing resources were provided by
the Center for Integrated Research Computing (CIRC) at the University of
Rochester. The authors appreciate valuable comments to the manuscript from
Braden M. Weight.

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# On Minimal Gorenstein Hilbert functions

Lenin Bezerra  **Departamento de Matematica, Universidade Federal de
Pernambuco, Recife PE (Brasil<EMAIL_ADDRESS>,  Rodrigo Gondim
**Departamento de Matematica, Universidade Federal Rural de Pernambuco, Recife
PE (Brasil<EMAIL_ADDRESS>,  Giovanna Ilardi  *Dipartimento
Matematica ed Applicazioni “Renato Caccioppoli”, Università Degli Studi Di
Napoli “Federico II”, Via Cinthia - Complesso Universitario Di Monte S. Angelo
80126 - Napoli - Italia<EMAIL_ADDRESS>and  Giuseppe Zappalà
***Dipartimento di Matematica e Informatica, Università di Catania, Viale A.
Doria 6, 95125 Catania, Italy<EMAIL_ADDRESS>
###### Abstract.

We conjecture that a class of Artinian Gorenstein Hilbert algebras called full
Perazzo algebras always have minimal Hilbert function, fixing codimension and
length. We prove the conjecture in length four and five, in low codimension.
We also prove the conjecture for a particular subclass of algebras that occurs
in every length and certain codimensions. As a consequence of our methods we
give a new proof of part of a known result about the asymptotic behavior of
the minimum entry of a Gorenstein Hilbert function.

###### Key words and phrases:

Hilbert function, Non unimodal h-vector, full Perazzo algebra, Turan algebra.

###### 2000 Mathematics Subject Classification:

Primary 13D10; Secondary 13H10

## Introduction

Gorenstein algebras appear as cohomology rings in several categories. For
instance, real orientable manifolds, projective varieties, Kahler manifolds,
convex polytopes, matroids, Coxeter groups and tropical varieties are examples
of categories for which the ring of cohomology is an Artinian Gorenstein
$\mathbb{K}$-algebra. The fundamental point is that these algebras can be
characterized as algebras satisfying Poincaré duality, see [18].

We deal with standard graded Artinian Gorenstein $\mathbb{K}$-algebras over a
field of characteristic zero. A natural and classical problem consists in
understanding their possible Hilbert function, sometimes also called Hilbert
vector. When the codimension of the algebra is less than or equal to $3$, all
possible Hilbert vectors were characterized in [21]; in particular, they are
unimodal, i.e. they never strictly increase after a strict decrease. While it
is known that non unimodal Gorenstein $h$-vectors exist in every codimension
greater than or equal to $5$ (see [4, 5, 6]), it is open whether non unimodal
Gorenstein $h$-vectors of codimension $4$ exist. For algebras with codimension
$4$ having small initial degree the Hilbert vector is unimodal (see [20, 17]).

Consider the family $\cal{AG}_{\mathbb{K}}(r,d)$ of standard graded Artinian
Gorenstein $\bf{\mathbb{K}}$-algebras of socle degree $d$ and codimension $r$.
By Poncaré duality, the Hilbert function of $A\in\cal{AG}_{\mathbb{K}}(r,d)$
is a symmetric vector $\operatorname{Hilb}(A)=(1,r,h_{2},\ldots,h_{d-2},r,1)$,
that is $h_{k}=h_{d-k}$. There is a natural partial order in this family given
by:

$(1,r,h_{2},\dots,h_{d-2},r,1)\preceq(1,r,\tilde{h}_{2},\dots,\tilde{h}_{d-2},r,1),$

if $h_{i}\leq\tilde{h}_{i}$, for all $i\in\\{2,\ldots,d-2\\}$. The maximal
Hilbert functions are associated to compressed algebras and completely
described in [13]. In fact the Hilbert vector of a compressed Gorenstein
algebra is a maximum in $\cal{AG}_{\mathbb{K}}(r,d)$. On the other hand,
classifying minimal Hilbert functions is a hard problem. We do not know in
general if there is a minimum. Moreover, given two comparable Gorenstein
Hilbert functions, it is not true that any symmetric vector between them is
Gorenstein. Some partial results in this direction were obtained in [23] and
called the interval conjecture.

The first example of a non-unimodal Gorenstein $h$-vector was given by Stanley
(see [21, Example 4.3]). He showed that the $h$-vector $(1,13,12,13,1)$ is
indeed a Gorenstein $h$-vector. In [14] the authors showed that Stanley’s
example is optimal, i.e. if we consider the $h-$vector $(1,12,11,12,1)$, it is
not Gorenstein. We say that a vector is totally non unimodal if

$h_{1}>h_{2}>\ldots>h_{k}\ \text{for}\ k=\lfloor d/2\rfloor.$

A totally non-unimodal Gorenstein Hilbert vector exists for every socle degree
$d\geq 4$ when the codimension $r$ is large enough. It is related to a
conjecture posed by Stanley and proved in [15, 16] and also a consequence of
our Proposition 2.3, see Corollary 2.4.

From Macaulay-Matlis duality, every standard graded Artinian Gorenstein
$\mathbb{K}$-algebra can be presented by a quotient of a ring of differential
operators by a homogeneous ideal that is the annihilator of a single form in
the dual ring of polynomials. full Perazzo algebras are associated with full
Perazzo polynomials, they are the family that we will study in detail. Perazzo
polynomials are related to Gordan and Noether theory of forms with vanishing
Hessian (see [19, Chapter 7] and [10]). In [10] the author introduced the
terminology Perazzo algebras to denote the Artinian Gorenstein algebra
associated to a Perazzo polynomial. In [9, AAdPFIMN] the authors study the
Hilbert vector and the Lefschetz properties for Perazzo algebras in
codimension $5$. In [8] the authors study full Perazzo algebras focusing on
socle degree $4$, showing that they have minimal Hilbert vector in some cases.
In this paper we deal with codimension greater than $13$ and we are more
interested in full Perazzo algebras. In the case of socle degree $4$ we recall
the known results.

We now describe the contents of the paper in more detail. In the first section
we recall the basics on Macaulay-Matlis duality, see 1.1. In the next
subsection we recall the classical bounds for Hilbert functions given by
Macaulay, Gotzman and Green summarized in 1.4.

In the second section we recall the definition of full Perazzo algebras and we
pose the full Perazzo Conjecture (see Conjecture 2.6). A full Perazzo
polynomial of type $m$ and degree $d$ is a bigraded polynomial of bidegree
$(1,d-1)$ given by $f=\sum x_{j}M_{j}$ where
$\\{M_{j}|j=1,\ldots,\binom{m+d-2}{d-1}\\}$ is a basis for
$\mathbb{K}[u_{1},\ldots,u_{m}]_{(d-1)}$. The associated Artinian Gorenstein
algebra is called full Perazzo algebra (see Section $2$ for more details).

######  Conjecture.

Let $H$ be the Hilbert vector of a full Perazzo algebra of type $m\geq 3$ and
socle degree $d\geq 4$ and let $r=r(m,d)$ its codimension. Then $H$ is minimal
in the family of Hilbert vectors of Artinian Gorenstein algebras of
codimension $r$ and socle degree $d$, that is, if $\hat{H}$ is a comparable
Artinian Gorenstein Hilbert vector such that $\hat{H}\preceq H$, then
$\hat{H}=H$.

In the third section we prove special cases of the Conjecture in socle degree
$4$ and we try to fill the gaps in order to classify all possible Hilbert
functions up to codimension $25$ (see Theorem 3.6, Corollary 3.7 and
Proposition 3.8). In socle degree $5$ we prove the Conjecture for
$m\in\\{3,4,5,6,7,8,9,10\\}$ (see Theorem 3.15) and a stronger version of the
conjecture for $m=3$ (see Corollary 3.16).

In the fourth section we prove our main result that the full Perazzo
Conjecture is true for arbitrary socle degree $d\geq 4$ and type $m=3$.

######  Theorem.

Every full Perazzo algebra with socle degree $d\geq 4$ of type $m=3$ has
minimal Hilbert function.

In the last section we give a new proof of part of a result originally proved
in [16], concerning the asymptotic behavior of the minimum entry of a
Gorenstein Hilbert function (see Theorem 5.2).

##  1\. Preliminaries

Most of the background material presented here can be found in [8].

###  1.1. Macaulay-Matlis duality

In this section we recall some basic results from Macaulay-Matlis duality for
Artinian Gorenstein algebras over a field $\mathbb{K}$ of characteristic zero.
We recall that in characteristic zero we can use a differential version of
Macaulay-Matlis duality.

Let $A=\mathbb{K}[X_{1},\ldots,X_{n}]/I=\bigoplus_{i=0}^{d}A_{i}$ be a
standard graded Artinian $\mathbb{K}-$algebra with $A_{d}\neq 0$. The Hilbert
function of $A$ can be described by the vector
$\operatorname{Hilb}(A)=(1,h_{1},\ldots,h_{d})$ where $h_{i}=\dim A_{i}$. We
say that $A$ is Gorenstein if $\dim A_{d}=1$ and for every $i=1,\ldots,d$, the
natural pairing given by multiplication $A_{i}\times A_{d-i}\to
A_{d}\simeq\mathbb{K}$ is perfect. There is an isomorphism $A_{i}^{*}\simeq
A_{d-i}$. In this context, $d$ is the socle degree of the algebra and assuming
$I_{1}=0$, $n$ is the codimension of $A$.

Let us regard the polynomial algebra $R=\mathbb{K}[x_{1},\ldots,x_{n}]$ as a
module over the algebra $Q=\mathbb{K}[X_{1},...,X_{n}]$ via the identification
$X_{i}=\partial/\partial x_{i}.$ If $f\in R$ we set

$\operatorname{Ann}_{Q}(f)=\\{\alpha=p(X_{1},\ldots,X_{n})\in
Q\mid\alpha(f):=p(\partial/\partial x_{1},\ldots,\partial/\partial
x_{n})f=0\\}.$

More generally, given any $Q$ submodule $M$ of $R$ we define the ideal of $Q$:

$\operatorname{Ann}_{Q}(M)=\\{\alpha\in Q\mid\alpha(f)=0\ \text{for all}\ f\in
M\\}.$

On the other side we have the notion of inverse system. Given $I\subset Q$ be
an ideal, we define the inverse system $I^{-1}$ which is a $Q$ submodule of
$R$:

$I^{-1}=\\{f\in R\mid\alpha(f)=0\ \text{for all}\ \alpha\in I\\}.$

By Macaulay-Matlis duality we have a bijection:

$\begin{array}[]{ccc}\\{\text{Homogeneous ideals of}\
Q\\}&\leftrightarrow&\\{\text{Graded}\ Q\ \text{submodules}\ \text{of}\
R\\}\\\ \operatorname{Ann}_{Q}(M)&\leftarrow&M\\\
I&\rightarrow&I^{-1}\end{array}$

From the Theory of Inverse Systems, we get the following characterization of
standard Artinian Gorenstein graded $\mathbb{K}$-algebras. A proof of this
result can be found in [18, Theorem 2.1].

######  Theorem 1.1.

(Double annihilator Theorem of Macaulay)
Let $R=\mathbb{K}[x_{1},\ldots,x_{n}]$ and let
$Q=\mathbb{K}[X_{1},\ldots,X_{n}]$ be the ring of differential operators. Let
$A=\bigoplus_{i=0}^{d}A_{i}=Q/I$ be an Artinian standard graded
$\mathbb{K}$-algebra. Then $A$ is Gorenstein if and only if there exists $f\in
R_{d}$ such that $A\simeq Q/\operatorname{Ann}(f)$.

In the sequel we always assume that $\operatorname{char}(\mathbb{K})=0$,
$A=Q/I$, $I=\operatorname{Ann}_{Q}(f)$ and $I_{1}=0$.

We will deal with standard bigraded Artinian Gorenstein algebras
$A=\bigoplus_{i=0}^{d}A_{i},$ $A_{d}\neq 0$, with
$A_{k}=\bigoplus_{i=0}^{k}A_{(i,k-i)}$, $A_{(d_{1},d_{2})}\neq 0$ for some
$d_{1},d_{2}$ such that $d_{1}+d_{2}=d$, we call $(d_{1},d_{2})$ the socle
bidegree of $A$. Since $A^{*}_{k}\simeq A_{d-k}$ and since duality is
compatible with direct sum, we get $A_{(i,j)}^{*}\simeq
A_{(d_{1}-i,d_{2}-j)}$.
Let $R=\mathbb{K}[x_{1},\ldots,x_{n},u_{1},\ldots,u_{m}]$ be the polynomial
ring viewed as standard bigraded ring in the sets of variables
$\\{x_{1},\ldots,x_{n}\\}$ and $\\{u_{1},\ldots,u_{m}\\}$ and let
$Q=\mathbb{K}[X_{1},\ldots,X_{n},U_{1},\ldots,U_{m}]$ be the associated ring
of differential operators.

We want to stress that the bijection given by Macaulay-Matlis duality
preserves bigrading, that is, there is a bijection:

$\begin{array}[]{ccc}\\{\text{Bihomogeneous ideals of}\
Q\\}&\leftrightarrow&\\{\text{Bigraded}\ Q\ \text{submodules}\ \text{of}\
R\\}\\\ \operatorname{Ann}_{Q}(M)&\leftarrow&M\\\
I&\rightarrow&I^{-1}\end{array}$

If $f\in R_{(d_{1},d_{2})}$ is a bihomogeneous polynomial of total degree
$d=d_{1}+d_{2}$, then $I=\operatorname{Ann}_{Q}(f)\subset Q$ is a
bihomogeneous ideal and $A=Q/I$ is a standard bigraded Artinian Gorenstein
algebra of socle bidegree $(d_{1},d_{2})$ and codimension $r=m+n$ if we
assume, without loss of generality, that $I_{1}=0$.

######  Remark 1.2.

With the previous notation, all bihomogeneous polynomials of bidegree
$(1,d-1)$ can be written in the form

$f=x_{1}g_{1}+\dotsm+x_{n}g_{n},$

where $g_{i}\in\mathbb{K}[u_{1},\ldots,u_{m}]_{d-1}$. The associated algebra,
$A=Q/\operatorname{Ann}_{Q}(f)$, is bigraded, has socle bidegree $(1,d-1)$ and
we assume that $I_{1}=0,$ so $\operatorname{codim}A=m+n$.

###  1.2. Classical Bounds of Hilbert function

We recall some classical bounds for the growth of the Hilbert function of
Artinian $\mathbb{K}-$algebras. The three main results are due to Macaulay,
Gotzmann and Green; before stating them, we need to recall the following
definition:

######  Definition 1.3.

Let $k$ and $i$ be positive integers. The $i-$binomial expansion of $k$,
denoted by $k_{(i)},$ is

(1) $k=k_{(i)}=\binom{k_{i}}{i}+\binom{k_{i-1}}{i-1}+\cdots+\binom{k_{j}}{j}$

where $k_{i}>k_{i-1}>\ldots>k_{j}\geq j\geq 1$.

An expansion of type (1) always exists and is unique (see, e.g., [7, Lemma
4.2.6]). Following [7], we define for any integers $a$ and $b$,

$(k_{(i)})^{b}_{a}=\binom{k_{i}+b}{i+a}+\binom{k_{i-1}+b}{i-1+a}+\cdots+\binom{k_{j}+b}{j+a}$

where we set $\binom{s}{c}=0$ whenever $s<c$ or $c<0$.

######  Theorem 1.4.

Let $A=R/I$ be a standard graded $\mathbb{K}$-algebra, and $L\in A$ a general
linear form (according to the Zariski topology). Denote by $h_{d}$ the degree
$d$ entry of the Hilbert function of $A$ and by $h^{\prime}_{d}$ the degree
$d$ entry of the Hilbert function of $A/(L)$. Then:

(Macaulay):

$h_{d+1}\leq((h_{d})_{(d)})^{+1}_{+1}.$

(Gotzmann):

If $h_{d+1}=((h_{d})_{(d)})^{+1}_{+1}$ and $I$ is generated in degrees $\leq
d+1$, then

$h_{d+s}=((h_{d})_{(d)})^{s}_{s}\mbox{ for all }s\geq 1.$

(Green):

$h^{\prime}_{d}\leq((h_{d})_{(d)})^{-1}_{0}.$

###### Proof.

For _Macaulay_ , see [7, Theorem 4.2.10]. For _Gotzmann_ , see [7, Theorem
4.3.3] or [11]. For _Green_ , see [12, Theorem 1]. ∎

######  Definition 1.5.

A sequence of non-negative integers $h=(1,h_{1},h_{2},\ldots,h_{i},\ldots)$ is
said to be an $\mathcal{O}$-sequence if it satisfies Macaulay’s Theorem (1.4)
for all $i$.

Recall that when A is Artinian and Gorenstein, then its Hilbert function is a
finite, symmetric $\mathcal{O}$-sequence.

##  2\. Minimal Gorenstein Hilbert functions

We recall the construction of full Perazzo algebras, introduced in [8].

######  Definition 2.1.

Let $\mathbb{K}[x_{1},\ldots,x_{n},u_{1},\ldots,u_{m}]$ be the polynomial ring
in the $n$ variables $x_{1},\ldots,x_{n}$ and in the $m$ variables
$u_{1},\ldots,u_{m}.$ A Perazzo polynomial is a reduced bihomogeneous
polynomial $f\in\mathbb{K}[x_{1},\ldots,x_{n},u_{1},\ldots,u_{m}]_{(1,d-1)}$,
of degree $d,$ of the form

(2) $f=\displaystyle{\sum_{i=1}^{n}x_{i}g_{i}}$

with $g_{i}\in\mathbb{K}[u_{1},\ldots,u_{m}]_{d-1}$, for $i=1,\ldots,n$,
linearly independent and algebraically dependent polynomials in the variables
$u_{1},\ldots,u_{m}$. The associated algebra is called a Perazzo algebra, it
has codimension $m+n$ and socle degree $d$.

Now we fix $m\geq 2$ and we consider the $m$ variables $u_{1},\ldots,u_{m}$.
For a multi-index $\alpha=(e_{1},\ldots,e_{m})$ with $e_{1}+\cdots+e_{m}=d-1$,
let

$M_{\alpha}=u_{1}^{e_{1}}\cdots u_{m}^{e_{m}}\in Q_{d-1}$

be a $\mathbb{K}$-linear basis for $Q_{d-1}$ and denote $\tau_{m}=\dim
Q_{d-1}=\binom{m+d-2}{d-1}$.

######  Definition 2.2.

Let $f\in\mathbb{K}[x_{1},\ldots,x_{\tau_{m}},u_{1},\ldots,u_{m}]_{(1,d-1)}$
be a Perazzo polynomial of degree $d$ of form:

(3) $f=\displaystyle{\sum_{j=1}^{\tau_{m}}x_{j}M_{j}}.$

In this case $f$ is called full Perazzo polynomial of type $m$ and degree $d$.
The associated algebra is a full Perazzo algebra of socle degree $d$ and
codimension $m+\tau_{m}$.

######  Proposition 2.3.

Let $A$ be a full Perazzo algebra of type $m\geq 2$ and socle degree $d$. Then
for $k=0,\ldots,\lfloor\frac{d}{2}\rfloor$

$h_{k}=\dim A_{k}=\binom{m+k-1}{k}+\binom{m+d-k-1}{d-k}.$

In particular, its Hilbert function is totally non-unimodal for $r>>0$.

###### Proof.

Using the bigrading of $A$ and considering that the polynomial $f$ has degree
$1$ in the variables $x_{1},\ldots,x_{\tau_{m}}$, fixed
$k=0,\ldots,\lfloor\frac{d}{2}\rfloor$, we have the following decomposition:

$A_{k}=A_{(0,k)}\oplus A_{(1,k-1)}.$

  1. (i)

It is clear that $A_{(0,k)}=Q_{(0,k)}$, hence $\dim A_{(0,k)}=\dim
Q_{(0,k)}=\binom{m+k-1}{k}$.

  2. (ii)

We have $A^{*}_{(1,k-1)}\simeq A_{(0,d-k)}$ and $A_{(0,d-k)}=Q_{(0,d-k)},$
hence $\dim A_{(1,k-1)}=\dim Q_{(0,d-k)}=\binom{m+d-k-1}{d-k}$.

To verify that the Hilbert vector is asymptotically totally non unimodal it is
enough to see that as a function of $m$,
$h_{k}(m)\simeq\frac{1}{(d-k)!}m^{d-k}$ for $k\leq d/2$. ∎

######  Corollary 2.4.

For every $d\geq 4$ there is a positive integer $r_{0}$ such that for all
$r\geq r_{0}$ there is an Artinian Gorenstein algebra with socle degree $d$
and codimension $r$ having a totally non unimodal Hilbert vector.

###### Proof.

Let $m$ be large enough in order to guarantee that the Hilbert vector of the
full Perazzo algebra $A=Q/\operatorname{Ann}(f)$, of type $m$ and socle degree
$d$ has a totally non unimodal Hilbert vector. For every
$r>m+\binom{m+d-2}{d-1}$, let $s=r-[m+\binom{m+d-2}{d-1}]$ and consider the
algebra $A^{\prime}=Q^{\prime}/\operatorname{Ann}(g)$ where
$Q^{\prime}=Q[Y_{1},\ldots,Y_{s}]$ and $g=f+\sum_{i=1}^{s}Y_{i}^{d}$. It is
easy to see that the Hilbert vector of $A^{\prime}$ is given by
$h^{\prime}_{k}=h_{k}+s$ for $k\neq 0,d$, therefore, it is totally non-
unimodal and the result follows. ∎

Let $d\geq 4$, $r\geq 3$. Consider the family $\cal{AG}(r,d)$ of standard
graded artinian Gorenstein $\bf{\mathbb{K}}$-algebras of socle degree $d$ and
codimension $r$. In this section we will consider $\mathbb{K}$, a fixed field
of characteristic $0$. We know that the Hilbert function of
$A\in\cal{AG}(r,d)$ is a symmetric vector
$\operatorname{Hilb}(A)=(1,r,h_{2},\ldots,h_{d-2},r,1)$, with $h_{i}=h_{d-i}$
by Poincaré duality.

Consider the family of length $d$ symmetric vectors of type
$(1,r,h_{2},\ldots,h_{d-2},r,1)$, where $h_{i}=h_{d-i}$. There is a natural
partial order in this family

$(1,r,h_{2},\dots,h_{d-2},r,1)\preceq(1,r,\tilde{h}_{2},\dots,\tilde{h}_{d-2},r,1).$

If $h_{i}\leq\tilde{h}_{i}$, for all $i\in\\{2,\ldots,d-2\\}$. This order can
be restricted to $\cal{AG}(r,d)$ which becomes a poset.

######  Definition 2.5.

Let $r,d$ be fixed positive integers and let $H$ be a length $d+1$ symmetric
vector $(1,r,h_{2},\ldots,h_{d-2},r,1)$. We say that $H$ is a minimal Artinian
Gorenstein Hilbert function of socle degree $d$ and codimension $r$ if there
is an Artinian Gorenstein algebra such that $\operatorname{Hilb}(A)=H$ and $H$
is minimal in $\cal{AG}(r,d)$ with respect to $\preceq$. To be precise, if
$\hat{H}$ is a comparable Artinian Gorenstein Hilbert vector such that
$\hat{H}\preceq H$, then $\hat{H}=H$.

We now present the full Perazzo Conjecture.

######  Conjecture 2.6.

Let $H$ be the Hilbert vector of a full Perazzo algebra of type $m$ and socle
degree $d$. Then $H$ is minimal in $\cal{AG}(r,d)$.

##  3\. Minimal Gorenstein Hilbert functions in low socle degree

In this section we study Gorenstein Hilbert functions of algebras with socle
degree $4$ and $5.$ Part of the results in socle degree $4$ can be found in
[8].

###  3.1. Minimal Gorenstein Hilbert functions in socle degree 4

In socle degree $4,$ a Gorenstein sequence is of the form

$(1,r,h,r,1).$

Let $\mu(r)$ be the integer such that $(1,r,\mu(r),r,1)$ is a Gorenstein
sequence, but $(1,r,\mu(r)-1,r,1)$ is not a Gorenstein sequence. Then
$\mu(r)\leq h\leq\binom{r+1}{2}.$

It is well known that $(1,r,h,r,1)$ is a Gorenstein sequence if and only if
$\mu(r)\leq h\leq\binom{r+1}{2}$ (see [23]). We set $\delta(r)=r-\mu(r).$ This
function was introduced in [15] and also studied in [8]. The function
$\delta(r)$ is not decreasing, so $\delta(r)\leq\delta(r+1),$ for every $r$
(see [15], Proposition 8).

By Remark 5.4 in [8], if $\delta(r-1)<\delta(r)$ then
$\delta(r)=\delta(r-1)+1.$

######  Definition 3.1.

We say that the Gorenstein sequence $(1,r,\mu(r),r,1)$ is minimal. Moreover we
say that the Gorenstein sequence $(1,r,\mu(r),r,1)$ is strongly minimal if
$\delta(r-1)<\delta(r).$

By Remark 5.4 in [8], if $(1,r,\mu(r),r,1)$ is strongly minimal, then
$\delta(r)=\delta(r-1)+1.$

The minimal $r$ such that $(1,r,\mu(r),r,1)$ is not unimodal is $r=13$ ([14]).
So $\delta(r)=0$ for $r\leq 12.$

######  Proposition 3.2.

$\delta(r)=1$ iff $13\leq r\leq 19.$

Consequently the sequence $(1,13,12,13,1)$ is strongly minimal.

###### Proof.

The sequence $(1,13,12,13,1)$ is a Gorenstein sequence. This was originally
proved by Stanley in [21]. This sequence is also the Hilbert Function of the
full Perazzo algebra with $m=3.$ In [14], Proposition 3.1, it was proved that
$(1,12,11,12,1)$ is not a Gorenstein sequence. Consequently $\delta(12)=0,$
therefore $\delta(r)=0$ for every $r\leq 12.$ In [2], Theorem 4.1, was shown
that $(1,19,17,19,1)$ is not a Gorenstein sequence, so $\delta(r)=1$ for
$13\leq r\leq 19.$ In [14], Remark 3.5, was observed that $(1,20,18,20,1)$ is
a Gorenstein sequence, so for $r\geq 20$ we have that $\delta(r)\geq 2.$ ∎

######  Corollary 3.3.

$\delta(20)=2.$ Consequently the sequence $(1,20,18,20,1)$ is strongly
minimal.

###### Proof.

In [14], Remark 3.5, it was observed that $(1,20,18,20,1)$ is a Gorenstein
sequence. So, by Remark 5.4 in [8], $\delta(20)=2.$ ∎

######  Proposition 3.4.

Let $m\geq 3.$ We have that

$\delta\left(m+\binom{m+2}{3}\right)\geq\binom{m}{3}.$

###### Proof.

For $r=m+\binom{m+2}{3}$ there exists the full Perazzo Algebra. It realizes
the Gorenstein sequence

$(1,m+\binom{m+2}{3},m(m+1),m+\binom{m+2}{3},1).$

So
$\delta\left(m+\binom{m+2}{3}\right)\geq\binom{m+2}{3}+m-m(m+1)=\binom{m+2}{3}-m^{2}=\binom{m}{3}.$
∎

######  Lemma 3.5.

Let $(1,r,h,r,1)$ be a Gorenstein sequence. Let $u=r-h,$ with $u\geq 0$. Then

$\big{(}\big{(}(r_{(3)})^{-1}_{0}-u\big{)}_{(2)}\big{)}^{1}_{1}\geq(r_{(3)})^{-1}_{0}.$

###### Proof.

Let $A$ be a Gorenstein algebra with Hilbert function $(1,r,h,r,1)$ and let
$L$ be a general linear form. Using the same argument as in Proposition 3.1 in
[14], we get that the Hilbert function of $A/(L)$ is of the type

$(1,r-1,s-u,s).$

By the theorems of Green and of Macaulay we have $s\leq(r_{(3)})^{-1}_{0}$ and

$\big{(}(s-u)_{(2)}\big{)}^{1}_{1}\geq s.$

Consequently

$\big{(}(s+t-u)_{(2)}\big{)}^{1}_{1}\geq s+t,\,\text{ for every }t\geq 0;$

In particular, for $t=(r_{(3)})^{-1}_{0}-s$ we are done. ∎

######  Theorem 3.6.

$\delta(24)=4$ and $\delta(40)=10.$

###### Proof.

By Proposition 3.4, $\delta(24)\geq\binom{4}{3}=4.$ We have to prove that
$(1,24,19,24,1)$ is not a Gorenstein sequence. Indeed
$24_{(3)}=\binom{6}{3}+\binom{3}{2}+\binom{1}{1},$ so
$(24_{(3)})^{-1}_{0}=11.$ Since $u=5$ we have that

$\big{(}(11-5)_{(2)}\big{)}^{1}_{1}=10<11.$

By Lemma 3.5, $(1,24,19,24,1)$ is not a Gorenstein sequence.

By Proposition 3.4, $\delta(40)\geq\binom{5}{3}=10.$ We have to prove that
$(1,40,29,40,1)$ is not a Gorenstein sequence. Indeed
$40_{(3)}=\binom{7}{3}+\binom{3}{2}+\binom{2}{1},$ so
$(40_{(3)})^{-1}_{0}=22.$ Since $u=11$ we have that

$\big{(}(22-11)_{(2)}\big{)}^{1}_{1}=21<22.$

By Lemma 3.5, $(1,40,29,40,1)$ is not a Gorenstein sequence. ∎

######  Corollary 3.7.

$\delta(25)=4.$

###### Proof.

By Theorem 3.6 and by Theorem 2.5 in [14], $(1,25,21,25,1)$ is a Gorenstein
sequence, so $\delta(25)\geq 4.$ We have to prove that $(1,25,20,25,1)$ is not
a Gorenstein sequence. Indeed
$25_{(3)}=\binom{6}{3}+\binom{3}{2}+\binom{2}{1},$ so
$(25_{(3)})^{-1}_{0}=12.$ Since $u=5$ we have that

$\big{(}(12-5)_{(2)}\big{)}^{1}_{1}=11<12.$

By Lemma 3.5, $(1,25,20,25,1)$ is not a Gorenstein sequence. ∎

######  Proposition 3.8.

$2\leq\delta(21)\leq\delta(22)\leq\delta(23)\leq 4.$

###### Proof.

This follows trivially by the fact that $\delta(20)=2$ and $\delta(24)=4.$ ∎

######  Proposition 3.9.

$20\leq\delta(62)\leq 21.$

###### Proof.

By Proposition 3.4, for $m=6,$ we get that $(1,62,42,62,1)$ is a Gorenstein
sequence.

On the other hand, $(1,62,40,62,1)$ is not a Gorenstein sequence by Lemma 3.5.
Indeed

$(62_{(3)})^{-1}_{0}=\binom{7}{3}+\binom{3}{2}=38$

and

$\big{(}(38-22)_{(2)}\big{)}^{1}_{1}=36<38.$

∎

######  Proposition 3.10.

$\delta(26)=4=\delta(27).$

###### Proof.

For $r=26$, we have to prove that $(1,26,21,26,1)$ is not a Gorenstein
sequence. Indeed, let $A=R/I$ be a Gorenstein algebra with Hilbert function
$(1,26,21,26,1)$, $L$ be a general linear form. We set $J=(I_{\leq 3})$,
$\bar{J}=(J,L)/(L)$ and $S=R/(L)$. By the theorems of Green and of Macaulay
and repeating the above method, $R/\bar{J}$ has Hilbert function
$(1,25,8,13)$. As $R/\bar{J}$ has maximal growth from degree $2$ to degree $3$
and $\bar{J}$ has no new generators in degree $4$, by Gotzmann’s theorem we
get $h_{R/\bar{J}}(t)=\binom{t+2}{2}+t.$ Therefore, $\bar{J}$ is the saturated
ideal, in all degrees $\geq 2$ of the union of a plane and a line in
$\mathbb{P}^{24}$. It follows that, up to saturation, $J$ is the ideal of a
scheme $T$ given by the union in $\mathbb{P}^{25}$ of a $3-$dimensional linear
variety, a plane and $m$ points (possibly embedded). Hence, $50\leq
h_{R/\bar{J}}(4)\leq 45$, which is absurd.

Now, for $r=27$, following the same argument as above, we prove that the
sequence $(1,27,22,27,1)$ is not Gorenstein. In this case we conclude $50\leq
h_{R/\bar{J}}(4)\leq 46.$

∎

###  3.2. Minimal Gorenstein Hilbert functions in socle degree 5

In socle degree $5,$ a Gorenstein sequence is of the form

$(1,r,h,h,r,1).$

Let $\mu(r)$ be the integer such that $(1,r,\mu(r),\mu(r),r,1)$ is a
Gorenstein sequence, but $(1,r,\mu(r)-1,\mu(r)-1,r,1)$ is not a Gorenstein
sequence. Then $\mu(r)\leq h\leq\binom{r+1}{2}.$

It is well known that $(1,r,h,h,r,1)$ is a Gorenstein sequence iff $\mu(r)\leq
h\leq\binom{r+1}{2}$ (see [23]). We set $\delta(r)=r-\mu(r).$

######  Definition 3.11.

We say that the Gorenstein sequence $(1,r,\mu(r),\mu(r),r,1)$ is minimal.
Moreover we say that the Gorenstein sequence $(1,r,\mu(r),\mu(r),r,1)$ is
strongly minimal if $\delta(r-1)<\delta(r).$

######  Proposition 3.12.

In socle degree $5$ we have

$\delta(r)=0$ iff $r\leq 16.$

$\delta(r)=1$ iff $r=17.$

For $18\leq r\leq 25,$ $\delta(r)=2$

###### Proof.

For $18\leq r\leq 25,$ see Theorem 3.4 and Remark 3.5 in [14]. ∎

######  Lemma 3.13.

Let $(1,r,h,h,r,1)$ be a Gorenstein sequence. Let $u=r-h.$ Then

$\big{(}\big{(}(r_{(4)})^{-1}_{0}-u\big{)}_{(2)}\big{)}^{2}_{2}\geq(r_{(4)})^{-1}_{0}.$

###### Proof.

Analogous to Lemma 3.5. ∎

######  Proposition 3.14.

Let $m\geq 3.$ We have that

$\delta\left(m+\binom{m+3}{4}\right)\geq\frac{m+5}{4}\binom{m}{3}.$

###### Proof.

For $r=m+\binom{m+3}{4}$ there exists the full Perazzo Algebra. It realizes
the Gorenstein sequence

$\left(1,m+\binom{m+3}{4},\binom{m+1}{2}+\binom{m+2}{3},\binom{m+1}{2}+\binom{m+2}{3},m+\binom{m+3}{4},1\right).$

So

$\delta\left(m+\binom{m+3}{4}\right)\geq\binom{m+3}{4}+m-\binom{m+1}{2}-\binom{m+2}{3}=\frac{m+5}{4}\binom{m}{3}.$

∎

######  Theorem 3.15.

For $m\in\\{3,4,5,6,7,8,9,10\\},$ we have
$\delta\left(m+\binom{m+3}{4}\right)=\frac{m+5}{4}\binom{m}{3}.$ That is, the
full Perazzo conjecture is true in these cases.

###### Proof.

By Proposition 3.14, we have to prove that for $m\in\\{3,4,5,6,7,8,9,10\\},$
the sequence

$\left(1,m+\binom{m+3}{4},\binom{m+1}{2}+\binom{m+2}{3}-1,\binom{m+1}{2}+\binom{m+2}{3}-1,m+\binom{m+3}{4},1\right)$

is not a Gorenstein sequence.

The case $m=3$ will be dealt with in general in the next section. We can
assume $m\geq 4$.

For $m=4$ we have to prove that the sequence $(1,39,29,29,39,1)$ is not a
Gorenstein sequence. Indeed, using Lemma 3.13, we have:

$(39_{(4)})^{-1}_{0}=16;\,\,u=10;\,\,16-10=6;\,\,(6_{(2)})^{2}_{2}=15<16.$

For $m=5$ we have to prove that the sequence $(1,75,49,49,75,1)$ is not a
Gorenstein sequence. Indeed, by Lemma 3.13, we have:

$(75_{(4)})^{-1}_{0}=36;\,\,u=26;\,\,36-26=10;\,\,(10_{(2)})^{2}_{2}=35<36.$

For $m=6$ we have to prove that the sequence $(1,132,76,76,132,1)$ is not a
Gorenstein sequence. Indeed, using Lemma 3.13, we have:

$(132_{(4)})^{-1}_{0}=71;\,\,u=56;\,\,71-56=15;\,\,(15_{(2)})^{2}_{2}=70<71.$

For $m=7$ we have to prove that the sequence $(1,217,111,111,217,1)$ is not a
Gorenstein sequence. Indeed, by Lemma 3.13, we have:

$(217_{(4)})^{-1}_{0}=128;\,\,u=106;\,\,128-106=22;\,\,(22_{(2)})^{2}_{2}=127<128.$

For $m=8$ we have to prove that the sequence $(1,338,155,155,338,1)$ is not a
Gorenstein sequence. Indeed, using Lemma 3.13, we have:

$(338_{(4)})^{-1}_{0}=212;\,\,u=183;\,\,212-183=29;\,\,(29_{(2)})^{2}_{2}=211<212.$

For $m=9$ we have to prove that the sequence $(1,504,209,209,504,1)$ is not a
Gorenstein sequence. Indeed, using symmetry, Green’s theorem and Macaulay’s
theorem, the following diagram represents the Hilbert functions of $R/I$,
$R/(I:L)$ and $R/(I,L)$

$\begin{array}[]{ccccccccc}1&504&209&209&504&1\\\ &1&171&54&171&1\\\ \hline\cr
1&503&38&155&333&&\end{array}$

By Lemma 3.5 the middle line is not a Gorenstein sequence.

For $m=10$ we have to prove that the sequence $(1,725,274,274,725,1)$ is not a
Gorenstein sequence. Indeed, using symmetry, Green’s theorem and Macaulay’s
theorem, the following diagram represents the Hilbert functions of $R/I$,
$R/(I:L)$ and $R/(I,L)$

$\begin{array}[]{ccccccccc}1&725&274&274&725&1\\\ &1&226&65&226&1\\\ \hline\cr
1&724&48&209&499&&\end{array}$

By Lemma 3.5 the middle line is not a Gorenstein sequence. ∎

######  Corollary 3.16.

The Gorenstein vector

$\left(1,18,16,16,18,1\right)$

is strongly minimal.

###### Proof.

By the previous theorem we know that it is a minimal Gorenstein Hilbert
vector. We have to prove that $(1,17,15,15,17,1)$ is not a Gorenstein
sequence. Indeed, by Proposition 3.12, $\delta(17)=1.$ ∎

##  4\. A family of minimal Gorenstein Hilbert functions

Consider the family of full Perazzo algebras of type $m=3$ and socle degree
$d\geq 4$. Its Hilbert function is given by
$h_{k}=\binom{k+2}{k}+\binom{2+2q-k}{2q-k}$, for $k\leq\lfloor d/2\rfloor$ and
by symmetry we get $h_{d-k}=h_{k}$.

######  Lemma 4.1.

Let $k\leq\lfloor d/2\rfloor$. Then we have:

$\left(\left(\binom{k+1}{2}\right)_{(d-k)}\right)_{0}^{-1}\leq k-2.$

###### Proof.

First of all, consider $d>2k-1$. In this case, $d-k+1>k$, i.e.
$(d-k+1)+(d-k)+\ldots+(d-2k+3)>k+(k-1)+\ldots+2+1=k(k+1)/2=\binom{k+1}{2}$.

We have:
$(\binom{k+1}{2})_{(d-k)}<\binom{d-k+1}{d-k}+\binom{d-k}{d-k-1}+\ldots\binom{d-2k+3}{d-2k+2}$.
Then

$\left(\left(\binom{k+1}{2}\right)_{(d-k)}\right)^{-1}_{0}\leq k-2.$

Now there are only two other cases to consider: 1) $d=2k-1$, 2) $d=2k$.

They are similar, we will do the details for $d=2k$. In this case, we have:

$\left(\left(\binom{k+1}{2}\right)_{(k)}\right)^{-1}_{0}\leq k-2.$

Indeed we know that the $k$-binomial expansion of $\binom{k+1}{2}$ has two
blocks

$(\binom{k+1}{2})_{(k)}=[\binom{k+1}{k}+\binom{k}{k-1}+\ldots+\binom{j+1}{j}]+[\binom{j-1}{j-1}+\ldots+\binom{i}{i}]$.
The first block consists of binomials of type $\binom{s+1}{s}$ and the second
one of type $\binom{s}{s}$.

Therefore:

$\left(\left(\binom{k+1}{2}\right)_{(k)}\right)^{-1}_{0}=k-j+1.$

If $k-j+1>k-2$, then $j\leq 2$, but the cases $j=1$ and $j=2$ are not
possible. In fact, suppose, $j=2$, since:

$\binom{k+1}{k}+\binom{k}{k-1}+\ldots+\binom{3}{2}=(k+4)(k-1)/2\geq\binom{k+1}{2}.$

It is absurd for $k>2$. The case $j=1$ is analogous, the result follows.

∎

######  Theorem 4.2.

Every full Perazzo algebra with socle degree $d\geq 4$ of type $m=3$ has
minimal Hilbert function.

###### Proof.

We want to show that the Hilbert vector of the full Perazzo algebra
$H=(1,h_{1},h_{2},h_{3},\ldots,h_{d-1},h_{d}=1)$ with
$h_{k}=\binom{m+k-1}{k}+\binom{m+d-k-1}{d-k}$ is a minimal Gorenstein Hilbert
vector. Let

$\hat{H}=(1,\hat{h}_{1},\hat{h}_{2},\hat{h}_{3},\ldots,\hat{h}_{d-1},1)$

be a comparable Artinian Gorenstein Hilbert vector $\hat{H}\preceq H$ of
length $d+1$ and $\hat{h}_{1}=h_{1}$. We will proceed in steps to show that
$\hat{H}=H$. Consider, on the contrary, one of the following situations:

  1. (1)

For some $k\in\\{2,\ldots,\lfloor d/2\rfloor-1\\}$, $\hat{h}_{k}<h_{k}$;

  2. (2)

For $d=2q$, suppose that $\hat{h}_{t}=h_{t}$ for all $t<q$ and
$\hat{h}_{q}<h_{q}$;

  3. (3)

For $d=2q+1$, suppose that $\hat{h}_{t}=h_{t}$ for all $t<q$ and
$\hat{h}_{q}<h_{q}$.

We will show that all of these situations give rise to a contradiction.

(1). Let $A=Q/I$ with $I=\operatorname{Ann}(f)$ be a standard graded Artinian
Gorenstein $\mathbb{K}$-algebra such that $H_{A}=\hat{H}$ with
$\hat{h}_{k}=\dim A_{k}<h_{k}=\binom{m+k-1}{k}+\binom{m+d-k-1}{d-k}$ for some
$k\in\\{2,\ldots,\lfloor d/2\rfloor-1\\}$. Suppose that $k$ is minimal
satisfying this property, that is, for $t<k$ we get $\hat{h}_{t}=h_{t}$, by
the comparability hypothesis. Let $L\in A_{1}$ be a generic linear form and
let $S=Q/(L)$. We get the following exact sequence:

$\begin{CD}0@>{}>{}>Q/(I:L)(-1)@>{}>{}>Q/I@>{}>{}>S/\overline{I}@>{}>{}>0\end{CD}$

with $\overline{I}=\dfrac{(I,L)}{L}$ and $(I:L)=Ann(f)$ and $f^{\prime}=L(f)$
denoting the derivative of $f$ with respect to $L\in Q$. Therefore $Q/(I:L)$
is also Gorenstein. We get the following diagram:

$\begin{array}[]{ccccccccc}1&\hat{h}_{1}&\ldots&\hat{h}_{k}&\ldots&\hat{h}_{d-k}&\hat{h}_{d-k+1}&\ldots&1\\\
&1&\ldots&a_{k-1}&\ldots&&a_{k-1}&\ldots&1\\\ \hline\cr
1&\hat{h}_{1}-1&\ldots&h_{k}^{\prime}&\ldots&&h_{d-k+1}^{\prime}&\end{array}$

We have
$\hat{h}_{d-k+1}=\hat{h}_{k-1}=h_{k-1}=h_{d-k+1}=\binom{k+1}{k-1}+\binom{d-k+3}{d-k+1}$.
The $(d-k+1)$-binomial decomposition of $h_{d-k+1}$ is
$(h_{d-k+1})_{(d-k+1)}=\binom{d-k+3}{d-k+1}+(\binom{k+1}{k-1})_{(d-k)}$. By
Green’s theorem we have

$h_{d-k+1}^{\prime}\leq((\hat{h}_{d-k+1})_{(d-k+1)})_{0}^{-1}=((h_{d-k+1})_{(d-k+1)})_{0}^{-1}=\binom{d-k+2}{d-k+1}+\left(\left(\binom{k+1}{k-1}\right)_{(d-k)}\right)^{-1}_{0}.$

By Lemma 4.1, we have

$h_{d-k+1}^{\prime}\leq d-k+2+k-2=d$.

We consider only the case $h^{\prime}_{d-k+1}=d$, the other cases are similar.

We have $a_{k-1}=\hat{h}_{d-k+1}-d$,
$h^{\prime}_{k}=\hat{h}_{k}-(\hat{h}_{d-k+1}-d)$. Since $\hat{h}_{k}\leq
h_{k}-1$ we have $h_{k}^{\prime}\leq h_{k}-h_{d-k+1}+d-1$.

We recall that

$h_{k}=\binom{k+2}{2}+\binom{d-k+2}{2}$,
$h_{d-k+1}=\binom{d-k+3}{2}+\binom{k+1}{2}$.

Therefore

$\displaystyle h_{k}-h_{d-k+1}$ $\displaystyle=$
$\displaystyle\left[\binom{k+2}{2}-\binom{k+1}{2}\right]-\left[\binom{d-k+3}{2}-\binom{d-k+2}{2}\right]$
$\displaystyle=$ $\displaystyle(k+1)-(d-k+2)$ $\displaystyle=$ $\displaystyle
2k-d-1$

We obtain $h^{\prime}_{k}\leq 2k-2$. Thence $h^{\prime}_{k}\leq 2k-2=k+1+k-3$
which implies that
$(h^{\prime}_{k})_{k}\leq\binom{k+1}{k}+\binom{k-1}{k-1}+\cdots+\binom{3}{3}$.

By Macaulay’s theorem applied $d-2k+1$ times we have

$h^{\prime}_{d-k+1}\leq((h^{\prime}_{k})_{k})^{d-2k+1}_{d-2k+1}\leq\binom{k+1+d-2k+1}{k+d-2k+1}+k-3=k+1+d-2k+1+k-3$

therefore $d\leq d-1$,a contradiction.

(2). Case $d=2q$ is even. Suppose that $\hat{h}_{t}=h_{t}$ for all $t<q$ and
$\hat{h}_{q}<h_{q}$. Let $L\in Q$ be a generic linear form and $S=Q/(L)$. We
have the following exact sequence:

$\begin{CD}0@>{}>{}>Q/(I:L)(-1)@>{}>{}>Q/I@>{}>{}>S/\overline{I}@>{}>{}>0\end{CD}$

where $\overline{I}=\dfrac{(I,L)}{L}$ and $(I:L)=Ann(f)$, $f^{\prime}=L(f)$,
that is, $Q/(I:L)$ is also Gorenstein. In the middle we get the following
diagram:

$\begin{array}[]{cccccccc}1&h_{1}&\ldots&h_{q-1}&\hat{h}_{q}&h_{q+1}&\ldots&1\\\
\\\ &1&\ldots&a_{q-2}&a_{q-1}&a_{q-1}&\ldots&1\\\ \hline\cr\\\
1&h_{1}-1&\ldots&&h_{q}^{\prime}&h_{q+1}^{\prime}\end{array}$

Since $(h_{q+1})_{(q+1)}=(\binom{q+3}{q+1}+(\binom{q+1}{2})_{q}$, from Green’s
theorem

$h_{q+1}^{\prime}\leq((h_{q+1})_{q+1})^{-1}_{0}=\binom{q+2}{q+1}+((\binom{q+1}{2})_{q})^{-1}_{0}$.

By Lemma 4.1 we have

$((h_{q+1})_{q+1})^{-1}_{0}=\binom{q+2}{q+1}+((\binom{q+1}{2})_{q})^{-1}_{0}\leq
q+2+q-2=2q$.

We study the case $h_{q+1}^{\prime}=2q$, the other cases are similar.

We have $a_{q-1}^{\prime}=h_{q+1}-2q$, $h^{\prime}_{q}=\hat{h}_{q}-a_{q-1}\leq
h_{q}-h_{q+1}+2q-1$, then $h^{\prime}_{q}\leq 2q-2=(q+1)+(q-3)$.

Therefore

$(h^{\prime}_{q})_{q}\leq\binom{q+1}{q}+\binom{q-1}{q-1}+\binom{q-2}{q-2}+\cdots+\binom{3}{3}$,

with $\binom{q-1}{q-1}+\binom{q-2}{q-2}+\cdots+\binom{3}{3}$ being counted
$q-3$ times.

From Macaulay’s theorem we have
$h^{\prime}_{q+1}\leq((h^{\prime}_{q})_{(q)})^{+1}_{+1}$, hence

$\displaystyle 2q$ $\displaystyle\leq$
$\displaystyle\binom{q+2}{q+1}+\binom{q}{q}+\binom{q-1}{q-1}+\cdots+\binom{4}{4}$
$\displaystyle\leq$ $\displaystyle q+2+q-3=2q-1.$

It is a contradiction.

(3). If $d=2q+1$ is odd. Suppose that $\hat{h}_{t}=h_{t}$ for all $t<q$ and
$\hat{h}_{q}<h_{q}$. By the same argument:

$\begin{array}[]{cccccccc}1&h_{1}&\ldots&\hat{h}_{q}&\hat{h}_{q+1}&h_{q+2}&\ldots&1\\\
\\\ &1&\ldots&a_{q-1}&a_{q}&a_{q-1}&\ldots&1\\\ \hline\cr\\\
1&h_{1}-1&\ldots&h_{q}^{\prime}&h_{q+1}^{\prime}&h_{q+2}^{\prime}\end{array}$

Since $h_{q+2}=\binom{q+4}{q+2}+\binom{q+1}{2}$ and
$(h_{q+2})_{q+2}=\binom{q+4}{q+2}+(\binom{q+1}{2})_{q+1}$, by Green’s theorem

$h_{q+2}^{\prime}\leq((h_{q+2})_{(q+2)})^{-1}_{0}=\binom{q+3}{q+2}+(\binom{q+1}{2})_{(q+1)})^{-1}_{0}\leq
q+3+q-2=2q+1.$

We consider only the case $h^{\prime}_{q+2}=2q+1$. We have
$a_{q-1}=h_{q+2}-(2q+1)$, $h^{\prime}_{q}=\hat{h}_{q}-a_{q-1}$. Then
$h^{\prime}_{q}\leq h_{q}-1-a_{q-1}$, $h^{\prime}_{q}\leq
h_{q}-1-(h_{q+2}-(2q+1))$, thence $h_{q}^{\prime}\leq 2q-2$. We have

$h_{q}-h_{q+2}=\binom{q+2}{2}-\binom{q+1}{2}+\binom{q+3}{2}-\binom{q+4}{2}=-2$.

Therefore

$\displaystyle h^{\prime}_{q}$ $\displaystyle\leq$ $\displaystyle(q+1)+(q-3)$
$\displaystyle\leq$
$\displaystyle\binom{q+1}{q}+\binom{q-1}{q-1}+\cdots+\binom{3}{3}$

where the terms $\binom{q-1}{q-1}+\cdots+\binom{3}{3}$ are $q-3$.

By Macaulay’s theorem we have

$h^{\prime}_{q+1}\leq((h^{\prime}_{q})_{q})^{+1}_{+1}=\binom{q+2}{q+1}+\binom{q}{q}+\cdots+\binom{4}{4}$,

the last terms are $q-3$.

By Macaulay’s theorem we have

$h^{\prime}_{q+2}\leq\binom{q+3}{q+2}+\binom{q+1}{q+1}+\dots+\binom{5}{5}=q+3+q-3=2q$,

then $2q+1\leq 2q$.

It is a contradiction. The result follows.

∎

##  5\. Asymptotic behavior of the minimum

In this section we give a new proof of part of Theorem $3.6$ in [16].

Let $P_{m}=m+\binom{m+d-2}{d-1}$ be the codimension of a full Perazzo algebra
of type $m$. Denote by $\mu_{d,k}(r)$ the minimal entry in degree $k$ of a
Gorenstein $h$-vector with codimension $r$ and socle degree $d$.

######  Lemma 5.1.

$\mu_{d,k}(P_{m})\geq\binom{m+d-k-1}{d-k}$.

###### Proof.

We proceed by induction on $k$.

For $k=1$, we have
$\mu_{d,1}(P_{m})=P_{m}=m+\binom{m+d-2}{d-1}>\binom{m+d-2}{d-1}$.

Now, suppose that

$\mu_{d,k-1}(P_{m})>\binom{m+d-k}{d-k+1}.$

From Theorem $2.4$ in [16], we get

$\mu_{d,k}(P_{m})\geq\left(\left(\mu_{d,k-1}(P_{m})\right)_{(d-k+1)}\right)_{-1}^{-1}+\left(\left(\mu_{d,k-1}(P_{m})\right)_{(d-k+1)}\right)_{-d+2k+1}^{-d+2k}.$

By inductive hypothesis, and by basic properties of binomial expansions, we
have:
$\left(\left(\mu_{d,k-1}(P_{m})\right)_{(d-k+1)}\right)_{-1}^{-1}>\binom{m+d-k-1}{d-k}$
and
$\left(\left(\mu_{d,k-1}(P_{m})\right)_{(d-k+1)}\right)_{-d+2k+1}^{-d+2k}>\binom{m+k}{k+2}.$

So,

$\mu_{d,k}(P_{m})\geq\binom{m+d-k-1}{d-k}+\binom{m+k}{k+2}>\binom{m+d-k-1}{d-k}.$

as we wanted.

∎

######  Theorem 5.2 (Migliore-Nagel-Zanello [16]).

Let $A$ be a Gorenstein algebra of codimension $r$ and socle degree $d$. Then,
for all $k<\lfloor d/2\rfloor$

$\lim_{r\rightarrow\infty}\dfrac{\mu_{d,k}(r)}{r^{\frac{d-k}{d-1}}}=\dfrac{((d-1)!)^{\frac{d-k}{d-1}}}{(d-k)!}.$

###### Proof.

For any integer $r>>0$ there is a unique integer $P_{m}=m+\binom{m+d-2}{d-1}$
such that

$P_{m}\leq r\leq P_{m+1}.$

Applying the function $\mu_{d,k}$ we have

$\mu_{d,k}(P_{m})\leq\mu_{d,k}(r)\leq\mu_{d,k}(P_{m+1}).$

By Lemma 5.1

$\binom{m+d-k-1}{d-k}\leq\mu_{d,k}(r)\leq\binom{m+d-k}{d-k}+\binom{m+k}{k}.$

Therefore

$\frac{m^{d-k}}{(d-k)!}+o(m^{d-k-1})\leq\mu_{d,k}(r)\leq\frac{m^{d-k}}{(d-k)!}+o(m^{d-k-1})$

where $o(m^{s})$ denote all terms of degree less than $s$.

On other hand, since $P_{m}\leq r\leq P_{m+1}$, then

$\displaystyle\frac{m^{d-1}}{(d-1)!}+o(m^{d-2})\leq$ $\displaystyle r$
$\displaystyle\leq\frac{m^{d-1}}{(d-1)!}+o(m^{d-2})$
$\displaystyle\frac{m^{d-k}}{((d-1)!)^{\frac{d-k}{d-1}}}+o(m^{d-k-1})\leq$
$\displaystyle r^{\frac{d-k}{d-1}}$
$\displaystyle\leq\frac{m^{d-k}}{((d-1)!)^{\frac{d-k}{d-1}}}+o(m^{d-k-1})$
$\displaystyle\dfrac{1}{\frac{m^{d-k}}{((d-1)!)^{\frac{d-k}{d-1}}}+o(m^{d-k-1})}\leq$
$\displaystyle\dfrac{1}{r^{\frac{d-k}{d-1}}}$
$\displaystyle\leq\dfrac{1}{\frac{m^{d-k}}{((d-1)!)^{\frac{d-k}{d-1}}}+o(m^{d-k-1})}$

Multiplying, we get

$\dfrac{\frac{m^{d-k}}{(d-k)!}+o(m^{d-k-1})}{\frac{m^{d-k}}{((d-1)!)^{\frac{d-k}{d-1}}}+o(m^{d-k-1})}\leq\dfrac{\mu_{d,k}(r)}{r^{\frac{d-k}{d-1}}}\leq\dfrac{\frac{m^{d-k}}{(d-k)!}+o(m^{d-k-1})}{\frac{m^{d-k}}{((d-1)!)^{\frac{d-k}{d-1}}}+o(m^{d-k-1})}.$

Since in both sides the limit exists and are the same, the result follows.

∎

Acknowledgments. We would like to thank the anonymous referee for the careful
reading, suggestions and corrections of a previous version of the manuscript.
A CNPq Research Fellowship partially supported the second named author (Proc.
309094/2020-8). The first author was financed in part by the Cordenação de
Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code
001.

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