\section{Introduction}\label{Introduction}

A matroid is a simplicial complex with the independence augmentation property.
Facets of a matroid are called bases for the matroid.
They satisfy a property of ``symmetricity'', so called the basis exchange property.
For a matroid $M$, define the polynomial
\begin{align*}
F_{M}=\sum_{F}\prod_{i\in F}x_{i}
\end{align*}
where $F$ runs the collection of facets of $M$.
The polynomials are studied from various viewpoints.
In \cite{ALOV2018, MR4003314, MR3899575, BH2018, MR4172622},
it was shown that the polynomials are log-concave on the positive orthant.
For a polynomial $F$, to study log-concavity and to study the Hessian matrix are equivalent.
More precisely, $F$ is log-concave on the positive orthant if and only if the Hessian matrix $H_{F}(\bm{a})$ of $F$ has exactly one positive eigenvalue for $\bm{a}$ in the positive orthant.
In \cite{MNY2020}, it was shown that the polynomials are strictly log-concave on the positive orthant,
equivalently, the Hessian matrices have exactly one positive eigenvalue and are not degenerate.
For an Artinian Gorenstein algebra associated to the polynomial, it was also shown that the strong Lefschetz property and Hodge--Riemann relation at degree one.

We introduce the homogeneous polynomial $F_{\mathcal{P}}$ for a regular polyhedron $\mathcal{P}$ as an analogue of above polynomials.
For a regular polyhedra $\mathcal{P}$, we define a homogeneous polynomial by
\begin{align*}
F_{\mathcal{P}}=\sum_{F\in F(\mathcal{P})}\prod_{v\in F}x_{v},
\end{align*}
where $F(\mathcal{P})$ is the collection of the facets, and $v$ is a $0$-dimensional face.
Then we consider the Artinian Gorenstein algebra
\begin{align*}
A_{\mathcal{P}}=\mathbb{R}[\partial_{v}|v\in V(\mathcal{P})]/\operatorname{Ann}(F_{\mathcal{P}}),
\end{align*}
where $\partial_{v}$ is the partial derivative operator of $x_{v}$, and $V(\mathcal{P})$ is the collection of $0$-dimensional faces.
We discuss the strong Lefschetz property of $A_{\mathcal{P}}$ and Hodge--Riemann relation with respect to the Poincar\'e duality
\begin{align*}
P_{F_{\mathcal{P}}}^{k}: A_{k}\times A_{s-k}\to \mathbb{R}, && (f,g)\mapsto fgF_{\mathcal{P}}.
\end{align*}

In this paper, we consider the Platonic solids.
And, we show the following, and study the strong Lefschetz elements.
\begin{Theorem}[cf.\ Theorems \ref{n=4}, \ref{n=6}, \ref{n=8}, \ref{n=12} and \ref{n=20}]
The following hold for the Platonic solids:
\begin{itemize}
\item The Artinian Gorenstein algebras have the strong Lefschetz property.
\item For the regular tetrahedron and octahedron, the algebra satisfy the Hodge--Riemann relation on the positive orthant.
For the others, the algebras do not satisfy the Hodge--Riemann relation with respect to some strong Lefschetz elements.
\end{itemize}
\end{Theorem}

This paper is organized as follows:
In Section \ref{SLP}, we recall the strong Lefschetz property and Hodge--Riemann relation.
Then we see the relation between the strong Lefschetz property and the Hessian matrices, and between the Hodge--Riemann relation and the Hessian matrices.
In Section \ref{Main results}, we discuss the strong Lefschetz property and Hodge--Riemann relation for the Artinian Gorenstein algebras defined by the Platonic solids.
In Section \ref{Hessian matrices}, we calculate the Hessian matrices of the homogeneous polynomials of the Platonic solids.

\begin{acknowledgment}
This work was supported by the Sasakawa Scientific Research Grant from The Japan Science Society.
\end{acknowledgment}

\section{Strong Lefchetz property, Hodge--Riemann relation and Hessian matrices}\label{SLP}

We recall some properties for a graded algebra, the strong Lefschetz property and Hodge--Riemann relation.

Let $A=\bigoplus_{k=0}^{s} A_{k}$, $A_{s}\neq \boldsymbol{0}$, be a graded Artinian algebra with a symmetric bilinear map $P^{k}$ from $A_{k}\times A_{s-k}$ to $\mathbb{R}$.
For a graded algebra $A=\bigoplus_{k=0}^{s} A_{k}$,
define $h_{k}$ by the dimension of $k$-th homogeneous component of $A$.
We call the sequence $(h_{0}, h_{1}, \ldots, h_{s})$ the \emph{Hilbert series} of $A$.

We say that $A$ has the \emph{strong Lefschetz property}
if there exists an element $\ell \in A_{1}$ such that the linear map
\begin{align}\label{slp}
\times \ell^{s-2k}\colon A_{k}\to A_{s-k}, && f\mapsto \ell^{s-2k}f
\end{align}
is bijective for each nonnegative integer $k\leq \frac{s}{2}$.
We call $\ell \in A_{1}$ with this property a \emph{strong Lefschetz element}.
If $A$ has the strong Lefschetz property, then the Hilbert series of $A$ is palindromic.

We say that $A$ satisfies the \emph{Hodge--Riemann relation} with respect to $\ell\in A_{1}$
if the symmetric bilinear form
\begin{align}\label{hrr}
Q^{k}_{\ell}\colon A_{k}\times A_{k}\to\mathbb{R}, && (f,g)\mapsto (-1)^{k}P^{k}(f, \ell^{s-2k}g)
\end{align}
is positive definite on the kernel $\times \ell^{s-2k+1}\colon A_{k}\to A_{s-k+1}$ for each nonnegative integer $k\leq \frac{s}{2}$.

The strong Lefschetz property and Hodge--Riemann relation are defined on a general graded algebra with a symmetric bilinear form, but we consider those properties on a graded Artinian Gorenstin algebra with the Poincar\'e duality in this paper.

Let $\partial_{i}=\frac{\partial}{\partial x_{i}}$ be the partial derivative operator of $x_{i}$.
The polynomial ring $\mathbb{R}[\partial_{1}, \partial_{2}, \ldots, \partial_{n}]$ acts on $\mathbb{R}[x_{1}, x_{2}, \ldots, x_{n}]$ in the usual manner.
For a homogeneous polynomial $F\in \mathbb{R}[x_{1}, x_{2}, \ldots, x_{n}]$, we define the annihilator $\operatorname{Ann}(F)$ by
\begin{align*}
\operatorname{Ann}(F)=\Set{f\in \mathbb{R}[\partial_{1}, \ldots, \partial_{n}]| fF=0}.
\end{align*}
Then $\operatorname{Ann}(F)$ is a homogeneous ideal of $\mathbb{R}[\partial_{1}, \ldots, \partial_{n}]$.
Let $A=\mathbb{R}[\partial_{1}, \ldots, \partial_{n}]/\operatorname{Ann}(F)$.
Since $\operatorname{Ann}(F)$ is homogeneous, the algebra $A$ is graded.
Furthermore $A$ is an Artinian Gorenstein algebra.
Conversely, a graded Artinian Gorenstein algebra $A$ has the presentation
\begin{align*}
A=\mathbb{R}[\partial_{1}, \ldots, \partial_{n}]/\operatorname{Ann}(F)
\end{align*}
for some homogeneous polynomial $F\in \mathbb{R}[x_{1}, x_{2}, \ldots, x_{n}]$.
The socle degree $s$ of $A$ is the degree of $F$.
Thus the maps
\begin{align*}
P_{F}^{k}: A_{k}\times A_{s-k}\to \mathbb{R}, && (f,g)\mapsto fgF
\end{align*}
are bilinear maps.
We call $P_{F}=\bigoplus_{k}P_{F}^{k}$ the \emph{Poincar\'e duality} of $A$.
The Hilbert series is palindromic for every graded Artinian Gorenstein algebra.

Let $A=\mathbb{R}[\partial_{1}, \ldots, \partial_{n}]/\operatorname{Ann}(F)=\bigoplus_{k=0}^{s}A_{k}$ be a graded Artinian Gorenstein algebra,
and $\Lambda_{k}$ the basis for $A_{k}$.
We define the matrix $H_{F}^{k}$ by
\begin{align*}
H_{F}^{k}=
\left(
e_{i
e_{j
F
\right)_{e_{i},e_{j}\in \Lambda_{k}}.
\end{align*}
The matrix $H_{F}^{k}$ is called the \emph{$k$-th Hessian matrix} and $\det H_{F}^{k}$ is called the $k$th \emph{Hessian} of $F$ with respect to the basis $\Lambda_{k}$.
We define the $0$th Hessian of $F$ to be $F$.
If $\Lambda_{1}=\Set{\partial_{1}, \partial_{2}, \ldots, \partial_{n}}$, then $H_{F}^{1}$ coincides with the usual Hessian matrix of $F$.

\begin{Theorem}[Watanabe \cite{W2000}, Maeno--Watanabe \cite{MR2594646}]\label{SLP and Hess}
Let $\bm{a}=(a_{1}, a_{2}, \ldots, a_{n})\in\mathbb{R}^{n}$, and $\ell_{\bm{a}}=a_{1}\partial_{1}+a_{2}\partial_{2}+\cdots+a_{n}\partial_{n}$.
The multiplication map
$\times \ell_{\bm{a}}^{s-2k}\colon A_{k}\to A_{s-k}$ is bijective
if and only if $\det H_{F}^{k}(\bm{a})\neq 0$.
\end{Theorem}

\begin{Remark}
By Theorem \ref{SLP and Hess}, a strong Lefschetz element comes from an open dense space where the determinants do not vanish.
Thus, if the $k$-th Hessian does not vanish as a polynomial for all $k$, then the Artinian Gorenstein algebra $A$ has the strong Lefschetz property.
\end{Remark}

\begin{Proposition}\label{HRR and Hess}
Assume that $F(\bm{a})>0$ for $\bm{a}=(a_{1}, a_{2}, \ldots, a_{n})\in \mathbb{R}^{n}$.
Let $\ell_{\bm{a}}=a_{1}\partial_{1}+a_{2}\partial_{2}+\cdots+a_{n}\partial_{n}$.
The algebra $A$ satisfies the condition (\ref{hrr}) when $k=1$ with respect to $\ell_{\bm{a}}$
if and only if
the first Hessian matrix $H_{F}^{1}(\bm{a})$ has $n-1$ negative eigenvalues and one positive eigenvalue.
\end{Proposition}

\begin{proof}
Since the map
\begin{align*}
\times \ell_{\bm{a}}^{s}:A_0 \xrightarrow{\times \ell_{\bm{a}}} A_1 \xrightarrow{\times \ell_{\bm{a}}^{s-1}} A_s
\end{align*}
is bijective, we have $A_1=\mathbb{R} \ell_{\bm{a}}\oplus \operatorname{Ker}(\times \ell_{\bm{a}}^{s-1})$.
And we have
\begin{align*}
Q_{\ell_{\bm{a}}}^{1}(\ell_{\bm{a}}, \ell_{\bm{a}})&=-P_{F}^{1}(\ell_{\bm{a}}, \ell_{\bm{a}}^{s-2}\ell_{\bm{a}}) \\
&=-P_{F}^{1}(\ell_{\bm{a}}, \ell_{\bm{a}}^{s-1}) \\
&=-\ell_{\bm{a}}^{s}F \\
&=-s!F(\bm{a}) <0.
\end{align*}
Hence,
the algebra $A$ satisfies the condition (\ref{hrr}) when $k=1$ with respect to $\ell_{\bm{a}}$
if and only if
$Q_{\ell_{\bm{a}}}^{1}$ has $n-1$ positive eigenvalues and one negative eigenvalue.

Furthermore, the representing matrix associated to $Q_{\ell_{\bm{a}}}^{1}$ with respect to a basis $\Lambda_{1}$ for $A_1$ is given by the first Hessian matrix $-H_{F}^{1}(\bm{a})$.
In fact, for $e_{i}, e_{j}\in \Lambda_{1}$,
\begin{align*}
Q_{\ell_{\bm{a}}}^{1}(e_{i}, e_{j})&=-P_{F}^{1}(e_{i}, \ell_{\bm{a}}^{s-2}e_{j}) \\
&=-\ell_{\bm{a}}^{s-2}e_{i}e_{j}F \\
&=-(a_{1}\partial_{1}+a_{2}\partial_{2}+\cdots+a_{n}\partial_{n})^{s-2}e_{i}e_{j}F \\
&=-(s-2)!(e_{i}e_{j}F)(\bm{a}) \\
&=-(s-2)!\left(H_{F}^{1}(\bm{a})\right)_{i,j}.
\end{align*}
\end{proof}

\section{Main results}\label{Main results}

In this section, we discuss the strong Lefschetz property and Hodge--Riemann relation for the Artinian Gorenstein algebras defined by the regular polyhedra.

For a polyhedron $\mathcal{P}$, we call a $0$-dimensional face a \emph{vertex} of $\mathcal{P}$, an $1$-dimensional face an \emph{edge} of $\mathcal{P}$, and a $2$-dimensional face, simply, a \emph{face} of $\mathcal{P}$.
Let $V(\mathcal{P})$, $E(\mathcal{P})$ and $F(\mathcal{P})$ denote the collection of vertices, edges and faces of $\mathcal{P}$, respectively.
We focus a combinatorial data, the face poset of $\mathcal{P}$.
The face poset of $\mathcal{P}$ is the set $V(\mathcal{P})\cup E(\mathcal{P})\cup F(\mathcal{P})$ ordered by inclusion.
We regard edges and faces of $\mathcal{P}$ as subsets of $V(\mathcal{P})$.

In this paper, we are only interested in the regular polyhedra.
For a regular polyhedra $\mathcal{P}$, we define a homogeneous polynomial by
\begin{align*}
F_{\mathcal{P}}=\sum_{F\in F(\mathcal{P})}\prod_{v\in F}x_{v}.
\end{align*}
The number of vertices appears as the number of variables, and the number of faces appears as the numbers of terms.
The shape of faces appears as the degree.
For example, if faces of $\mathcal{P}$ are $d$-gons, then $F_{\mathcal{P}}$ is a homogeneous of degree $d$.

For a polyhedron $\mathcal{P}$, $A_{\mathcal{P}}=\bigoplus_{k=0}^{s}A_{k}$ denotes the Artinian Gorenstein algebra defined by the homogeneous polynomial $F_{\mathcal{P}}$.
We call $A_{\mathcal{P}}$ the algebra associated to $\mathcal{P}$.
For the algebra $A_{\mathcal{P}}$, we consider the Poincar\'e duality $P_{F_{\mathcal{P}}}=\bigoplus_{k}P_{F_{\mathcal{P}}}^{k}$, where
\begin{align*}
P_{F_{\mathcal{P}}}^{k}: A_{k}\times A_{s-k}\to \mathbb{R}, && (f,g)\mapsto fgF_{\mathcal{P}}.
\end{align*}
We see the strong Lefschetz property of $A_{\mathcal{P}}$ and Hodge--Riemann relation with respect to the Poincar\'e duality  $P_{F_{\mathcal{P}}}=\bigoplus_{k}P^{k}_{F_{\mathcal{P}}}$ of $A_{\mathcal{P}}$.

\begin{Remark}
As mentioned in Section \ref{SLP}, if $\Set{\partial_{i}}_{i\in V(\mathcal{P})}$ is a basis for $A_{1}$, then the first Hessian matrix $H^{1}_{F_{\mathcal{P}}}$ coincides with the usual Hessian matrix of $F_{\mathcal{P}}$.
Unless noted, we calculate $H^{1}_{F_{\mathcal{P}}}$ as the Hessian matrix of $F_{\mathcal{P}}$.
In other words, we calculate $H^{1}_{F_{\mathcal{P}}}$ with respect to $\Set{\partial_{i}}_{i\in V(\mathcal{P})}\subset A_{1}$.
\end{Remark}

\subsection{Regular tetrahedron}\label{n=4}

Let us consider the regular tetrahedron $\mathcal{P}$.
The regular tetrahedron has $4$ vertices, $6$ edges and $4$ faces.
We assign the number $1,2,3,4$ to the vertices.
Let $$F(\mathcal{P})=\Set{\Set{1,2,3}, \Set{1,2,4}, \Set{1,3,4}, \Set{2,3,4}}. $$
See Figure \ref{4}.
\begin{figure}
\centering
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.98\textwidth]{4.png}
\end{minipage}
\caption{Tetrahedron}\label{4}
\end{figure}
Then
\begin{align*}
F_{\mathcal{P}}=x_{1}x_{2}x_{3}+x_{1}x_{2}x_{4}+x_{1}x_{3}x_{4}+x_{2}x_{3}x_{4}.
\end{align*}
The Hilbert series of the algebra associated to $\mathcal{P}$ is $(1,4,4,1)$.

The homogeneous polynomial $F_{\mathcal{P}}$ is equal to the elementary symmetric polynomial $e_{3}(x_{1}, x_{2}, x_{3}, x_{4})$ of degree $3$ in $4$ variables.
In \cite{MR3566530}, Maeno and Numata show that the algebra defined by the annihilator of the elementary symmetric polynomial $e_{d}(x_{1}, x_{2}, \ldots, x_{n})$ of degree $d$ in $n$ variables satisfies the strong Lefschetz property for every $d$ and $n$.

Furthermore, $F_{\mathcal{P}}$ is equal to the Kirchhoff polynomial of the cycle graph $C_{4}$ with $4$ vertices.
In \cite{NY2019}, Nagaoka and Yazawa show that the algebra defined by a Kirchhoff polynomial, a homogeneous polynomial defined by a graph, satisfies the strong Lefschetz property and Hodge--Riemann relation ``at degree one'' (it will be explained in Remark \ref{rem deg1}) on the positive orthant.
More generally, in \cite{MNY2020}, Murai, Nagaoka and Yazawa show that the algebra defined by the basis generating polynomial, a generalization of a Kirchhoff polynomial, satisfies the strong Lefschetz property and Hodge--Riemann relation at degree one on the positive orthant.

\begin{Remark}\label{rem deg1}
Let $A=\bigoplus_{k=0}^{s} A_{k}$ be a graded algebra with a symmetric bilinear map $P^{k}$ from $A_{k}\times A_{s-k}$ to $\mathbb{R}$.
We say that $A$ satisfies the strong Lefschetz property at degree one if the condition (\ref{slp}) holds when $k=1$, and that $A$ satisfies the Hodge--Riemann relation with respect to $\ell\in A_1$ if the condition (\ref{hrr}) holds when $k=1$.
\end{Remark}

To summarize of this section, we obtain the following.

\begin{Theorem}\label{n=4
The algebra associated to the regular tetrahedron satisfies the strong Lefschetz property and Hodge--Riemann relation on the positive orthant.
\end{Theorem}

\subsection{Regular hexahedron}\label{n=6}

Let us consider the regular hexahedron $\mathcal{P}$.
The regular hexahedron has $8$ vertices, $12$ edges and $6$ faces.
We assign the number $1,2,\ldots, 8$ to the vertices.
Let $$F(\mathcal{P})=\Set{\Set{1,2,3,4}, \Set{2,3,6,7}, \Set{3,4,7,8}, \Set{1,4,5,8}, \Set{1,2,5,6}, \Set{5,6,7,8}}. $$
See Figure \ref{6}.
\begin{figure}
\centering
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.98\textwidth]{6.png}
\end{minipage}
\caption{Hexahedron}\label{6}
\end{figure}
Then
\begin{align*}
F_{\mathcal{P}}=x_{1}x_{2}x_{3}x_{4}+x_{2}x_{3}x_{6}x_{7}+x_{3}x_{4}x_{7}x_{8}+x_{1}x_{4}x_{5}x_{8}+x_{1}x_{2}x_{5}x_{6}+x_{5}x_{6}x_{7}x_{8}.
\end{align*}

\begin{Proposition}\label{eigen n=6}
The eigenvalues of the matrix $H_{F_\mathcal{P}}^{1}(1,1,\ldots, 1)$ are $9, 1, -3$ with the dimensions of the eigenspaces $1,3,4$, respectively.
\end{Proposition}
The matrix $H_{F_\mathcal{P}}^{1}(1,1,\ldots, 1)$ is in Section \ref{Hessian matrices}.
Also the proof of Proposition \ref{eigen n=6} is in Section \ref{Hessian matrices}.

From Proposition \ref{eigen n=6}, Theorem \ref{SLP and Hess} and Proposition \ref{HRR and Hess}, we obtain the following.

\begin{Theorem}\label{n=6}
The algebra associated to the regular hexahedron satisfies the strong Lefschetz property with a strong Lefschetz element $\sum_{i=1}^{8}\partial_{i}$,
but does not satisfy the Hodge--Riemann relation with respect to $\sum_{i=1}^{8}\partial_{i}$.
\end{Theorem}

\begin{Remark}
If $\Set{v, v'}$ does not a subset of a face of $\mathcal{P}$, then $\partial_{v}\partial_{v'}$ is in $\operatorname{Ann}(F_{\mathcal{P}})$.
In this case,
\begin{align*}
\partial_{1}\partial_{7}, \partial_{2}\partial_{8}, \partial_{3}\partial_{5}, \partial_{4}\partial_{6}\in \operatorname{Ann}(F_{\mathcal{P}}).
\end{align*}
Further, monomials of degree $2$ of $A_{\mathcal{P}}$ have the following relations:
\begin{align}\label{eqs}
\begin{array}{ccc}
\partial_{1}\partial_{2}=\partial_{7}\partial_{8}, & \partial_{1}\partial_{4}=\partial_{6}\partial_{7}, & \partial_{1}\partial_{5}=\partial_{3}\partial_{7}, \\
\partial_{2}\partial_{3}=\partial_{5}\partial_{8}, & \partial_{2}\partial_{6}=\partial_{4}\partial_{8}, & \partial_{3}\partial_{4}=\partial_{5}\partial_{6}.
\end{array}
\end{align}
The other monic monomials of degree two and monomials in left hand sides of the equations \eqref{eqs} generate $A_{2}$.
Furthermore, since the second Hessian with respect to them does not vanish, they form basis for $A_{2}$.
Thus the Hilbert series of the algebra associated to $\mathcal{P}$ is $(1,8,18,8,1)$.
The second Hessian matrix $H_{F_\mathcal{P}}^{2}$ is in Section \ref{Hessian matrices}.
\end{Remark}

\subsection{Regular octahedron}\label{n=8}

Let us consider the regular octahedron $\mathcal{P}$.
The regular octahedron has $6$ vertices, $12$ edges and $8$ faces.
We assign the number $1,2,\ldots, 6$ to the vertices.
Let $$F(\mathcal{P})=\Set{\Set{1,2,3}, \Set{1,3,4}, \Set{1,4,5}, \Set{1,2,5}, \Set{2,3,6}, \Set{3,4,6}, \Set{4,5,6}, \Set{2,5,6}}. $$
See Figure \ref{8}.
\begin{figure}
\centering
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.98\textwidth]{8.png}
\end{minipage}
\caption{Octahedron}\label{8}
\end{figure}
Then
\begin{align*}
F_{\mathcal{P}}=x_{1}x_{2}x_{3}+x_{1}x_{3}x_{4}+x_{1}x_{4}x_{5}+x_{1}x_{2}x_{5}+x_{2}x_{3}x_{6}+x_{3}x_{4}x_{6}+x_{4}x_{5}x_{6}+x_{2}x_{5}x_{6}.
\end{align*}
The algebra $A_{\mathcal{P}}$ has $6$ variables.

We can reduce the number of variables.
\begin{Proposition}\label{prop reduce}
For the regular octahedron $\mathcal{P}$, we have
\begin{align*}
A_{\mathcal{P}}\cong \mathbb{R}[t_{1}, t_{2}, t_{3}]/(t_{1}^{2}, t_{2}^{2}, t_{3}^{2}).
\end{align*}
\end{Proposition}

\begin{proof}
Let
\begin{align*}
\Phi: \mathbb{R}[\partial_{1}, \partial_{2}, \ldots, \partial_{6}] &\to \mathbb{R}[t_{1}, t_{2}, t_{3}]/(t_{1}^{2}, t_{2}^{2}, t_{3}^{2}) \\
\partial_{1}, \partial_{6} &\mapsto t_{1}, \\
\partial_{2}, \partial_{4} &\mapsto t_{2}, \\
\partial_{3}, \partial_{5} &\mapsto t_{3}.
\end{align*}
We calculate $\operatorname{Ker}(\Phi)$.
In our situation, we have a way to calculate the kernel:
For sets $X=\Set{x_{1}, x_{2}, \ldots, x_{n}}$, $Y=\Set{y_{1}, y_{2}, \ldots, y_{m}}$ of variables, and
a subset $D$ of the polynomial ring $\mathbb{R}[Y]$, consider
\begin{align*}
\varphi&: \mathbb{R}[X] \to \mathbb{R}[Y]/\langle D\rangle, \\
G&=\Set{x_{i}-\varphi(x_{i})|i\in\Set{1,2,\ldots, n}}\cup D.
\end{align*}
Then we have $\operatorname{Ker}(\varphi)=\langle G \rangle_{\mathbb{R}[X, Y]}\cap \mathbb{R}[X]$.

For $\Phi$, the map defined above, we consider $D=\Set{t_{1}^{2}, t_{2}^{2}, t_{3}^{2}}$.
Then we have
\begin{align*}
G=\Set{\partial_{1}-t_{1}, \partial_{6}-t_{1}, \partial_{2}-t_{2}, \partial_{4}-t_{2}, \partial_{3}-t_{3}, \partial_{5}-t_{3}, t_{1}^{2}, t_{2}^{2}, t_{3}^{2}}.
\end{align*}
A Gr\"{o}bner basis for $\langle G \rangle_{\mathbb{R}[X, Y]}$ with respect to the lexicographical order with
$t_{3}>t_{2}>t_{1}>\partial_{6}>\partial_{5}>\partial_{4}>\partial_{3}>\partial_{2}>\partial_{1}$ is
\begin{align*}
\Set{
\partial_{1}-t_{1}, \partial_{2}-t_{2}, \partial_{3}-t_{3},
\partial_{1}-\partial_{6}, \partial_{2}-\partial_{4}, \partial_{3}-\partial_{5},
\partial_{1}^{2}, \partial_{2}^{2}, \partial_{3}^{2}
}.
\end{align*}
Hence,
$\operatorname{Ker}(\varphi)=\langle \partial_{1}-\partial_{6}, \partial_{2}-\partial_{4}, \partial_{3}-\partial_{5}, \partial_{1}^{2}, \partial_{2}^{2}, \partial_{3}^{2} \rangle_{\mathbb{R}[X]}$.

The Hilbert series of $A_{\mathcal{P}}=\mathbb{R}[\partial_{1}, \partial_{1}, \ldots, \partial_{6}]/\operatorname{Ann}(F_{\mathcal{P}})$ is $(1,3,3,1)$
since the degree of $F_{\mathcal{P}}$ is $3$, and the first Hessian $H^{1}_{F_{\mathcal{P}}}$ with respect to the $\Set{\partial_{1}, \partial_{2}, \partial_{3}}$ does not vanish.

It is obvious that $\operatorname{Ker}(\varphi)\subset \operatorname{Ann}(F)$, and the Hilbert series of $\mathbb{R}[\partial_{1}, \partial_{1}, \ldots, \partial_{6}]/\operatorname{Ker}(\varphi)$ is $(1,3,3,1)$, so we obtain
\begin{align*}
A_{\mathcal{P}}\cong \mathbb{R}[\partial_{1}, \partial_{1}, \ldots, \partial_{6}]/\operatorname{Ker}(\varphi) \cong \mathbb{R}[t_{1}, t_{2}, t_{3}]/(t_{1}^{2}, t_{2}^{2}, t_{3}^{2}).
\end{align*}
\end{proof}

In general, the algebra $\mathbb{R}[\partial_{1}, \partial_{2}, \ldots, \partial_{n}]/(\partial_{1}^{a_{1}+1}, \partial_{2}^{a_{2}+1}, \ldots,  \partial_{n}^{a_{n}+1})$, where $a_{i}\geq 0$ for all $i$ satisfies the strong Lefschetz property with a strong Lefschetz element $\sum_{i=1}^{n}\partial_{i}$ (\cite{MR3112920}).
If $F=x_{1}^{a_{1}}x_{2}^{a_{2}}\cdots x_{n}^{a_{n}}$, then we have $A_{F}=\mathbb{R}[\partial_{1}, \partial_{2}, \ldots, \partial_{n}]/(\partial_{1}^{a_{1}+1}, \partial_{2}^{a_{2}+1}, \ldots,  \partial_{n}^{a_{n}+1})$.
In \cite{NY2019} and \cite{MNY2020}, when $a_{1}=a_{2}=\cdots=a_{n}=1$, namely $F=x_{1}x_{2}\cdots x_{n}$,
they show that the algebra $\mathbb{R}[\partial_{1}, \partial_{2}, \ldots, \partial_{n}]/(\partial_{1}^{2}, \partial_{2}^{2}, \ldots,  \partial_{n}^{2})$ satisfies the strong Lefschetz property and Hodge--Riemann relation at degree one on the positive orthant.
In fact, the homogeneous polynomial $F=x_{1}x_{2}\cdots x_{n}$ coincides with the Kirchhoff polynomial a tree graph with $n+1$ vertices.

To summarize of this section, we obtain the following.

\begin{Theorem}\label{n=8
The algebra associated to the regular octahedron satisfies the strong Lefschetz property and Hodge--Riemann relation on the positive orthant.
\end{Theorem}

\subsection{Regular dodecahedron}\label{n=12}

Let us consider the regular dodecahedron $\mathcal{P}$.
The regular dodecahedron has $20$ vertices, $30$ edges and $12$ faces.
We assign the number $1,2,\ldots, 20$ to the vertices.
Let
\begin{align*}
F(\mathcal{P})=\left\{
\begin{array}{cccc}
\Set{1,2,3,4,5},
\Set{1,2,6,7,19},
\Set{2,3,7,8,20},
\Set{3,4,8,9,16}, \\
\Set{4,5,9,10,17},
\Set{1,5,6,10,18},
\Set{11,12,13,14,15}, \\
\Set{11,12,16,17,9},
\Set{12,13,17,18,10},
\Set{13,14,18,19,6}, \\
\Set{14,15,19,20,7},
\Set{11,15,16,20,8}
\end{array}
\right\}.
\end{align*}
See Figure \ref{12}.
\begin{figure}
\centering
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.98\textwidth]{12.png}
\end{minipage}
\caption{Dodecahedron}\label{12}
\end{figure}
Then
\begin{align*}
F_{\mathcal{P}}=
&x_{1}x_{2}x_{3}x_{4}x_{5}+x_{1}x_{2}x_{6}x_{7}x_{19}+x_{2}x_{3}x_{7}x_{8}x_{20}+x_{3}x_{4}x_{8}x_{9}x_{16}\\
+&x_{4}x_{5}x_{9}x_{10}x_{17}+x_{1}x_{5}x_{6}x_{10}x_{18}+x_{11}x_{12}x_{13}x_{14}x_{15}+x_{11}x_{12}x_{16}x_{17}x_{9}\\
+&x_{12}x_{13}x_{17}x_{18}x_{10}+x_{13}x_{14}x_{18}x_{19}x_{6}+x_{14}x_{15}x_{19}x_{20}x_{7}+x_{11}x_{15}x_{16}x_{20}x_{8}.
\end{align*}

The socle degree of $A_{\mathcal{P}}$ is $5$, in other words, $A_{\mathcal{P}}=\bigoplus_{k=0}^{5}A_{k}$.
We calculate the first and second Hessian matrices of $F_{\mathcal{P}}$.

\begin{Proposition}\label{eigen n=12 Hess1}
Let $\bm{a}=(a_{i})_{i}\in\mathbb{R}^{20}$ with $a_{i}=1$ for $i\not\in\Set{6,7,8,9,10}$ and $a_{i}=0$ for $i\in\Set{6,7,8,9,10}$.
The matrix $H_{F_\mathcal{P}}^{1}(\bm{a})$ is non-degenerate.
Moreover, the number of the positive eigenvalues is more than two.
\end{Proposition}

\begin{Proposition}\label{eigen n=12 Hess2}
Let $\bm{a}=(a_{i})_{i}\in\mathbb{R}^{20}$ with $a_{1}=0$ and $a_{i}=1$ for $i\neq 1$.
The matrix $H_{F_\mathcal{P}}^{2}(\bm{a})$ with some basis for $A_2$ is not degenerate.
\end{Proposition}

The matrices  $H_{F_\mathcal{P}}^{1}(\bm{a})$ and $H_{F_\mathcal{P}}^{2}(\bm{a})$ are in Section \ref{Hessian matrices}.
Also the proof of Propositions \ref{eigen n=12 Hess1} and \ref{eigen n=12 Hess2} are in Section \ref{Hessian matrices}.
To show Proposition \ref{eigen n=12 Hess2}, we use CoCalc \cite{CoCalc}.

To summarize of this section, we obtain the following.

\begin{Theorem}\label{n=12}
The algebra associated to the regular dodecahedron satisfies the strong Lefschetz property,
but does not satisfy the Hodge--Riemann relation with respect to $\sum_{i\neq 6,7,8,9,10}\partial_{i}$.
\end{Theorem}

\begin{Remark}
Let
$\bm{a}=(a_{i})_{i}\in\mathbb{R}^{20}$ with $a_{i}=1$ for all $i$;
$\bm{b}=(b_{i})_{i}\in\mathbb{R}^{20}$ with $b_{i}=1$ for $i\not\in\Set{6,7,8,9,10}$ and $b_{i}=0$ for $i\in\Set{6,7,8,9,10}$;
and
$\bm{c}=(c_{i})_{i}\in\mathbb{R}^{20}$ with $c_{1}=0$ and $c_{i}=1$ for $i\neq 1$.
We have
\begin{itemize}
\item $\det H^{1}_{F_{\mathcal{P}}}(\bm{a})=0$ and $\det H^{2}_{F_{\mathcal{P}}}(\bm{a})=0$,
\item $\det H^{1}_{F_{\mathcal{P}}}(\bm{b})\neq 0$ and $\det H^{2}_{F_{\mathcal{P}}}(\bm{b})=0$,
\item $\det H^{1}_{F_{\mathcal{P}}}(\bm{c})\neq 0$ and $\det H^{2}_{F_{\mathcal{P}}}(\bm{c})\neq 0$.
\end{itemize}
Hence, $\ell_{\bm{a}}=\sum_{i=1}^{20}a_{i}\partial_{i}$ and $\ell_{\bm{b}}=\sum_{i=1}^{20}b_{i}\partial_{i}$ are not strong Lefschetz elements.
The linear element $\ell_{\bm{c}}=\sum_{i=1}^{20}c_{i}\partial_{i}$ is a strong Lefschetz element.
We use Cocalc for calculation the determinant $\det H^{1}_{F_{\mathcal{P}}}(\bm{c})$.
\end{Remark}

\begin{Remark}
The Hilbert series of $A_{\mathcal{P}}$ is $(1,20,90,20,1)$.
\end{Remark}

\subsection{Regular icosahedron}\label{n=20}

Let us consider the regular icosahedron $\mathcal{P}$.
The regular icosahedron has $12$ vertices, $30$ edges and $20$ faces.
We assign the number $1,2,\ldots, 12$ to the vertices.
Let
\begin{align*}
F(\mathcal{P})=
\left\{
\begin{array}{cccc}
\Set{1,2,3}, \Set{1,3,4}, \Set{1,4,5}, \Set{3,4,9}, \Set{1,5,6}, \\
\Set{1,2,6}, \Set{2,3,8}, \Set{3,8,9}, \Set{2,6,7}, \Set{2,7,8}, \\
\Set{10,11,12}, \Set{4,5,10}, \Set{4,9,10}, \Set{5,6,11}, \Set{5,10,11}, \\
\Set{9,10,12}, \Set{8,9,12}, \Set{6,7,11}, \Set{7,11,12}, \Set{7,8,12}
\end{array}
\right\}.
\end{align*}
See Figure \ref{20}.
\begin{figure}
\centering
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.98\textwidth]{20.png}
\end{minipage}
\caption{Icosahedron}\label{20}
\end{figure}
Then
\begin{align*}
F_{\mathcal{P}}=
&x_{1}x_{2}x_{3}+x_{1}x_{3}x_{4}+x_{1}x_{4}x_{5}+x_{3}x_{4}x_{9}+x_{1}x_{5}x_{6}\\
+&x_{1}x_{2}x_{6}+x_{2}x_{3}x_{8}+x_{3}x_{8}x_{9}+x_{2}x_{6}x_{7}+x_{2}x_{7}x_{8}\\
+&x_{10}x_{11}x_{12}+x_{4}x_{5}x_{10}+x_{4}x_{9}x_{10}+x_{5}x_{6}x_{11}+x_{5}x_{10}x_{11}\\
+&x_{9}x_{10}x_{12}+x_{8}x_{9}x_{12}+x_{6}x_{7}x_{11}+x_{7}x_{11}x_{12}+x_{7}x_{8}x_{12}.
\end{align*}

\begin{Proposition}\label{eigen n=20}
The eigenvalues of the matrix $H_{F_\mathcal{P}}^{1}(1,1,\ldots, 1)$ of the homogeneous polynomial $F_{\mathcal{P}}$ are $10,-2,2\sqrt{5},-2\sqrt{5}$ with the dimensions of the eigenspaces $1,5,3,3$, respectively.

\end{Proposition}
The matrix $H_{F_\mathcal{P}}^{1}(1,1,\ldots, 1)$ is in Section \ref{Hessian matrices}.
Also the proof of Proposition \ref{eigen n=20} is in Section \ref{Hessian matrices}.

To summarize of this section, we obtain the following.

\begin{Theorem}\label{n=20}
The algebra associated to the regular icosahedron satisfies the strong Lefschetz property with a strong Lefschetz element $\sum_{i=1}^{20}\partial_{i}$,
but does not satisfy the Hodge--Riemann relation with respect to $\sum_{i=1}^{20}\partial_{i}$.
\end{Theorem}

\section{Hessian matrices}\label{Hessian matrices}

Here, we calculate the Hessian matrices of the homogeneous polynomials of the regular polyhedra.

\subsection{Regular tetrahedron}\label{appen n=4}

Let $\mathcal{P}$ be the regular tetrahedron.
The matrix $H_{F_\mathcal{P}}^{1}(1,1,1,1)$ is
\begin{align*}
\begin{pmatrix}
0&2&2&2 \\
2&0&2&2 \\
2&2&0&2 \\
2&2&2&0 \\
\end{pmatrix}.
\end{align*}
The eigenvalues of the matrix are $6, -2, -2, -2$.

\subsection{Regular hexahedron}\label{appen n=6}

Let $\mathcal{P}$ be the regular hexahedron.
The matrix $H_{F_\mathcal{P}}^{1}(1,1,\ldots, 1)$ is the following, and has the following block decomposition:
\begin{align*}
\left(
\begin{array}{cccc|cccc}
0&2&1&2&2&1&0&1 \\
2&0&2&1&1&2&1&0 \\
1&2&0&2&0&1&2&1 \\
2&1&2&0&1&0&1&2 \\ \hline
2&1&0&1&0&2&1&2 \\
1&2&1&0&2&0&2&1 \\
0&1&2&1&1&2&0&2 \\
1&0&1&2&2&1&2&0 \\
\end{array}
\right).
\end{align*}
By \cite[Section 2]{Y2018}, the eigenvalues of the matrix come from the eigenvalues of the following four matrices $M_0, M_1, M_2, M_3$:
\begin{align*}
M_0=
\begin{pmatrix}
5&4 \\
4&5
\end{pmatrix}, &&
M_1=M_3=
\begin{pmatrix}
-1&2 \\
2&-1
\end{pmatrix}, &&
M_2=
\begin{pmatrix}
-3&0 \\
0&-3
\end{pmatrix}.
\end{align*}
The eigenvalues of $M_0$ are $9,1$.
The eigenvalues of $M_1=M_3$ are $1,-3$.
The eigenvalues of $M_2$ are $-3,-3$.
Hence we have Proposition \ref{eigen n=6}.

By Section \ref{Main results}, we know that
\begin{align*}
\Lambda_{2}=
\Set{
\partial_{1}\partial_{3}, \partial_{2}\partial_{4}, \partial_{1}\partial_{8}, \partial_{4}\partial_{5}, \partial_{2}\partial_{5},
\partial_{1}\partial_{6}, \partial_{3}\partial_{6}, \partial_{2}\partial_{7}, \partial_{4}\partial_{7}, \partial_{3}\partial_{8},
\partial_{6}\partial_{8}, \partial_{5}\partial_{7}, \partial_{1}\partial_{2}, \partial_{1}\partial_{4}, \partial_{1}\partial_{5},
\partial_{2}\partial_{6}, \partial_{2}\partial_{3}, \partial_{3}\partial_{4}
}
\end{align*}
generates $A_{2}$.
Let us calculate the second Hessian matrix with respect to $\Lambda_{2}$ is
\begin{align*}
\begin{pmatrix}
A&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0} \\
\boldsymbol{0}&A&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0} \\
\boldsymbol{0}&\boldsymbol{0}&A&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0} \\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&A&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0} \\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&A&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0} \\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&A&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0} \\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&A&\boldsymbol{0}&\boldsymbol{0} \\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&A&\boldsymbol{0} \\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&A \\
\end{pmatrix},
\end{align*}
where
\begin{align*}
A=
\begin{pmatrix}
0&1 \\
1&0
\end{pmatrix},
\boldsymbol{0}=
\begin{pmatrix}
0&0 \\
0&0
\end{pmatrix}.
\end{align*}
The eigenvalues are $1$ and $-1$.
The multiplicity of both are $9$.
Since the second Hessian with respect to $\Lambda_{2}$ does not vanish, $\Lambda_{2}$ is basis for $A_{2}$.

\subsection{Regular octahedron}\label{appen n=8}

Let $\mathcal{P}$ be the regular octahedron.
By Proposition \ref{prop reduce}, the algebra $A_{\mathcal{P}}$ is isomorphic to the graded Artinian Gorenstein algebra $\mathbb{R}[\partial_{1}, \partial_{2}, \partial_{3}]/(\partial_{1}^{2}, \partial_{2}^{2}, \partial_{3}^{2})$.
If $F=x_{1}x_{2}x_{3}$, then $\mathbb{R}[\partial_{1}, \partial_{2}, \partial_{3}]/(\partial_{1}^{2}, \partial_{2}^{2}, \partial_{3}^{2})=\mathbb{R}[\partial_{1}, \partial_{2}, \partial_{3}]/\operatorname{Ann}(F)$.

The first Hessian matrix of $F$ evaluated $1$ is the following.
\begin{align*}
\begin{pmatrix}
0&1&1 \\
1&0&1 \\
1&1&0
\end{pmatrix}.
\end{align*}
The eigenvalues of the matrix are $2, -1, -1$.

\subsection{Regular dodecahedron}\label{appen n=12}

Let $\mathcal{P}$ be the regular dodecahedron.
The socle degree of $A_{\mathcal{P}}$ is $5$, in other words, $A_{\mathcal{P}}=\bigoplus_{k=0}^{5}A_{k}$.
We calculate the first and second Hessian matrices of $F_{\mathcal{P}}$.

At first, we consider the first Hessian.
The matrix $H_{F_\mathcal{P}}^{1}(1,1,\ldots, 1)$ is degenerate.
In fact, if we set
\begin{align*}
A=
\begin{pmatrix}
0&2&1&1&2 \\
2&0&2&1&1 \\
1&2&0&2&1 \\
1&1&2&0&2 \\
2&1&1&2&0 \\
\end{pmatrix}, &&
B=
\begin{pmatrix}
2&1&0&0&1 \\
1&2&1&0&0 \\
0&1&2&1&0 \\
0&0&1&2&1 \\
1&0&0&1&2 \\
\end{pmatrix}, &&
C=
\begin{pmatrix}
0&0&1&1&0 \\
0&0&0&1&1 \\
1&0&0&0&1 \\
1&1&0&0&0 \\
0&1&1&0&0 \\
\end{pmatrix}, \\
D=
\begin{pmatrix}
0&1&0&0&1 \\
1&0&1&0&0 \\
0&1&0&1&0 \\
0&0&1&0&1 \\
1&0&0&1&0 \\
\end{pmatrix}, &&
\boldsymbol{0}=
\begin{pmatrix}
0&0&0&0&0 \\
0&0&0&0&0 \\
0&0&0&0&0 \\
0&0&0&0&0 \\
0&0&0&0&0 \\
\end{pmatrix}, &&
A'=
\begin{pmatrix}
0&1&1&1&1 \\
1&0&1&1&1 \\
1&1&0&1&1 \\
1&1&1&0&1 \\
1&1&1&1&0 \\
\end{pmatrix},
\end{align*}
then the matrix $H_{F_\mathcal{P}}^{1}(1,1,\ldots, 1)$ is
\begin{align*}
\begin{pmatrix}
A&B&\boldsymbol{0}&C \\
B&D&C&2C \\
\boldsymbol{0}&C&A&B \\
C&2C&B&D \\
\end{pmatrix}.
\end{align*}
One can check that the matrix $H_{F_\mathcal{P}}^{1}(1,1,\ldots, 1)$ is degenerate.
By Theorem \ref{SLP and Hess}, the linear element $\sum_{i=1}^{20}\partial_{i}$ is not a strong Lefschetz element of $A_{\mathcal{P}}$.
Instead of the matrix, we consider another matrix $H'$ evaluated the following:
\begin{align*}
\text{$x_{i}=1$ \ for $i \not\in \Set{6,7,8,9,10}$}, \\
\text{$x_{i}=0$ \ for $i \in \Set{6,7,8,9,10}$}.
\end{align*}
The matrix $H'$ is the following, and has the following block decomposition
\begin{align*}
\left(
\begin{array}{c|ccc}
A'&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0} \\ \hline
\boldsymbol{0}&D&C&C \\
\boldsymbol{0}&C&A'&\boldsymbol{0} \\
\boldsymbol{0}&C&\boldsymbol{0}&\boldsymbol{0} \\
\end{array}
\right).
\end{align*}
The eigenvalues of $A'$ is $4$, $-1$.
The multiplicities are $1$ and $3$, respectively.
The matrix
\begin{align*}
\left(
\begin{array}{ccc}
D&C&C \\
C&A'&\boldsymbol{0} \\
C&\boldsymbol{0}&\boldsymbol{0} \\
\end{array}
\right)
\end{align*}
is non-degenerate since $A'$ and $C$ are non-degenerate.
Moreover, since the trace is zero, the matrix has at least one positive eigenvalue.
Therefore, $H'$ has at least two positive eigenvalues.

Next, we consider the second Hessian.
The second Hessian matrix of $F_{\mathcal{P}}$ is too large to calculate by hands, so we use CoCalc \cite{CoCalc} to calculation.
\begin{lstlisting}[basicstyle=\ttfamily\footnotesize, frame=single, numbers=left]
xx=[var("

Facet=[
[0, 1, 2, 3, 4],
[0, 1, 5, 6, 18],
[1, 2, 6, 7, 19],
[2, 3, 7, 8, 15],
[3, 4, 8, 9, 16],
[0, 4, 5, 9, 17],
[10, 11, 12, 13, 14],
[8, 10, 11, 15, 16],
[9, 11, 12, 16, 17],
[5, 12, 13, 17, 18],
[6, 13, 14, 18, 19],
[7, 10, 14, 15, 19]
]

F=sum(prod(xx[i] for i in f) for f in Facet)

A2=Combinations(xx,2).list()
Basis=[]
for i in A2:
    if (F.derivative(i[0])).derivative(i[1])==0:
        pass
    else:
        Basis=Basis+[i]

ss={}
for xi in xx:
    ss[xi]=1
ss[x0]=0

h1=[F.derivative(i[0]).derivative(i[1]) for i in Basis]
h=[[hi.derivative(j[0]).derivative(j[1]) for j in Basis] for hi in h1]
H=Matrix([[ZZ(hij.substitute(ss)) for hij in hi] for hi in h])
det(H)
\end{lstlisting}

By this program, you obtain $\det H^{2}_{F_{\mathcal{P}}}(\bm{a})=342456532992$, where $\bm{a}=(a_{i})_{i}\in\mathbb{R}^{20}$ with $a_{1}=0$ and $a_{i}=1$ for $i\neq 1$.

\subsection{Regular icosahedron}\label{appen n=20}

The first Hessian matrix $H_{F_\mathcal{P}}^{1}(1,1,\ldots, 1)$ of the homogeneous polynomial of the regular icosahedron is the following, and has the following block decomposition:

\begin{align*}
\left(
\begin{array}{ccccc|ccccc|c|c}
0&2&0&0&2&2&0&0&0&2&2&0 \\
2&0&2&0&0&2&2&0&0&0&2&0 \\
0&2&0&2&0&0&2&2&0&0&2&0 \\
0&0&2&0&2&0&0&2&2&0&2&0 \\
2&0&0&2&0&0&0&0&2&2&2&0 \\ \hline
2&2&0&0&0&0&2&0&0&2&0&2 \\
0&2&2&0&0&2&0&2&0&0&0&2 \\
0&0&2&2&0&0&2&0&2&0&0&2 \\
0&0&0&2&2&0&0&2&0&2&0&2 \\
2&0&0&0&2&2&0&0&2&0&0&2 \\ \hline
2&2&2&2&2&0&0&0&0&0&0&0 \\ \hline
0&0&0&0&0&2&2&2&2&2&0&0 \\
\end{array}
\right).
\end{align*}

By \cite[Section 2]{Y2018}, the eigenvalues of the matrix are come from the eigenvalues of the following five matrices $M_0, M_1, M_2, M_3, M_4$:
Let $\zeta_{5}$ be the primitive $5$th root of unity and $k\in\Set{1,2,3,4}$.
Then we define the matrices $M_0, M_1, M_2, M_3, M_4$ by
\begin{align*}
M_0=
\left(
\begin{array}{cc|cc}
4&4&2&0 \\
4&4&0&2 \\ \hline
10&0&0&0 \\
0&10&0&0 \\
\end{array}
\right),
&&
M_k=
\begin{pmatrix}
2(\zeta_{5}^{k}+\zeta_{5}^{-k})&2(1+\zeta_{5}^{-k})&2&0 \\
2(1+\zeta_{5}^{k})&2(\zeta_{5}^{k}+\zeta_{5}^{-k})&0&2 \\
0&0&0&0 \\
0&0&0&0 \\
\end{pmatrix}.
\end{align*}
Note that $M_1=M_4$ and $M_2=M_3$.
Moreover, the eigenvalues of $M_0$ come from the eigenvalues of the following two matrices $M_{0}', M_{0}''$, and
the eigenvalues of $M_{k}$ come from some eigenvalues of the following two matrices $M_{k}', M_{k}''$:
\begin{align*}
M_{0}'&=
\begin{pmatrix}
8&2 \\
10&0 \\
\end{pmatrix}, &
M_{0}''&=
\begin{pmatrix}
0&2 \\
10&0 \\
\end{pmatrix}, \\
M_{k}'&=
\begin{pmatrix}
2(\zeta_{5}^{k}+\zeta_{5}^{-k})&2(1+\zeta_{5}^{-k}) \\
2(1+\zeta_{5}^{k})&2(\zeta_{5}^{k}+\zeta_{5}^{-k}) \\
\end{pmatrix}, &
M_{k}''&=
\begin{pmatrix}
0&0 \\
0&0 \\
\end{pmatrix}.
\end{align*}
For $M_{0}'$, the vectors
\begin{align*}
\begin{pmatrix}
1 \\
1
\end{pmatrix}, &&
\begin{pmatrix}
1 \\
-5
\end{pmatrix}
\end{align*}
are the eigenvectors associated to the eigenvalues $10$ and $-2$, respectively.
For $M_{0}''$, the vectors
\begin{align*}
\begin{pmatrix}
1 \\
\sqrt{5}
\end{pmatrix}, &&
\begin{pmatrix}
1 \\
-\sqrt{5}
\end{pmatrix}
\end{align*}
are the eigenvectors associated to the eigenvalues $2\sqrt{5}$ and $-2\sqrt{5}$, respectively.
For $M_{k}'$, the vectors
\begin{align*}
\begin{pmatrix}
2+\zeta_{5}^{k}+\zeta_{5}^{-k} \\
-(1+\zeta_{5}^{k})
\end{pmatrix}, &&
\begin{pmatrix}
1+\zeta_{5}^{-k} \\
-(\zeta_{5}^{2k}+\zeta_{5}^{-2k})
\end{pmatrix}
\end{align*}
are eigenvectors associated to the eigenvalues $-2$ and $2\left((\zeta_{5}^{k}+\zeta_{5}^{-k})-(\zeta_{5}^{2k}+\zeta_{5}^{-2k})\right)$, respectively.
For $k\in\Set{1,2,3,4}$, we have
\begin{align*}
\left((\zeta_{5}^{k}+\zeta_{5}^{-k})-(\zeta_{5}^{2k}+\zeta_{5}^{-2k})\right)^{2}=5.
\end{align*}
If $k=1,4$, then
\begin{align*}
(\zeta_{5}^{k}+\zeta_{5}^{-k})-(\zeta_{5}^{2k}+\zeta_{5}^{-2k})>0.
\end{align*}
If $k=2,3$, then
\begin{align*}
(\zeta_{5}^{k}+\zeta_{5}^{-k})-(\zeta_{5}^{2k}+\zeta_{5}^{-2k})<0.
\end{align*}
Hence the eigenvalues of $M_1=M_4$ are $-2$, $2\sqrt{5}$, and the eigenvalues of $M_2=M_3$ are $-2$, $-2\sqrt{5}$.
For $k\in\Set{0,1,2,3,4}$, define
\begin{align*}
\boldsymbol{z}_{k}=
\begin{pmatrix}
1 \\
\zeta_{5}^{k} \\
\zeta_{5}^{2k} \\
\zeta_{5}^{3k} \\
\zeta_{5}^{4k} \\
\end{pmatrix}.
\end{align*}
Then, by \cite[Lemma 2.1]{Y2018}, the vector
\begin{align*}
\begin{pmatrix}
\boldsymbol{z}_{0} \\
\boldsymbol{z}_{0} \\
1 \\
1
\end{pmatrix}
\end{align*}
is a eigenvector of $H_{F_\mathcal{P}}^{1}(1,1,\ldots, 1)$ associated to the eigenvalue $10$.
The vectors
\begin{align*}
\begin{pmatrix}
\boldsymbol{z}_{0} \\
\boldsymbol{z}_{0} \\
-5 \\
-5
\end{pmatrix},
\begin{pmatrix}
\left(2+\zeta_{5}+\zeta_{5}^{-1}\right) \boldsymbol{z}_{1} \\
-(1+\zeta_{5})\boldsymbol{z}_{1} \\
0 \\
0
\end{pmatrix},
\begin{pmatrix}
\left(2+\zeta_{5}^{2}+\zeta_{5}^{-2}\right) \boldsymbol{z}_{2} \\
-(1+\zeta_{5}^{2})\boldsymbol{z}_{2} \\
0 \\
0
\end{pmatrix}, \\
\begin{pmatrix}
\left(2+\zeta_{5}^{-2}+\zeta_{5}^{2}\right) \boldsymbol{z}_{3} \\
-(1+\zeta_{5}^{-2})\boldsymbol{z}_{3} \\
0 \\
0
\end{pmatrix},
\begin{pmatrix}
\left(2+\zeta_{5}^{-1}+\zeta_{5}\right) \boldsymbol{z}_{4} \\
-(1+\zeta_{5}^{-1})\boldsymbol{z}_{4} \\
0 \\
0
\end{pmatrix}
\end{align*}
are the eigenvectors of $H_{F_\mathcal{P}}^{1}(1,1,\ldots, 1)$ associated to the eigenvalue $-2$.
The vectors
\begin{align*}
\begin{pmatrix}
\boldsymbol{z}_{0} \\
-\boldsymbol{z}_{0} \\
\sqrt{5} \\
-\sqrt{5}
\end{pmatrix},
\begin{pmatrix}
\left(1+\zeta_{5}^{-1}\right) \boldsymbol{z}_{1} \\
-(\zeta_{5}^{2}+\zeta_{5}^{-2})\boldsymbol{z}_{1} \\
0 \\
0
\end{pmatrix},
\begin{pmatrix}
\left(1+\zeta_{5}\right) \boldsymbol{z}_{4} \\
-(\zeta_{5}^{-2}+\zeta_{5}^{2})\boldsymbol{z}_{4} \\
0 \\
0
\end{pmatrix}.
\end{align*}
are the eigenvectors of $H_{F_\mathcal{P}}^{1}(1,1,\ldots, 1)$ associated to the eigenvalue $2\sqrt{5}$.
The vectors
\begin{align*}
\begin{pmatrix}
\boldsymbol{z}_{0} \\
-\boldsymbol{z}_{0} \\
-\sqrt{5} \\
\sqrt{5}
\end{pmatrix},
\begin{pmatrix}
\left(1+\zeta_{5}^{-2}\right) \boldsymbol{z}_{2} \\
-(\zeta_{5}^{-1}+\zeta_{5})\boldsymbol{z}_{2} \\
0 \\
0
\end{pmatrix},
\begin{pmatrix}
\left(1+\zeta_{5}^{2}\right) \boldsymbol{z}_{3} \\
-(\zeta_{5}+\zeta_{5}^{-1})\boldsymbol{z}_{3} \\
0 \\
0
\end{pmatrix}
\end{align*}
are the eigenvectors of $H_{F_\mathcal{P}}^{1}(1,1,\ldots, 1)$ associated to the eigenvalues $-2\sqrt{5}$.

Finally, we obtain Proposition \ref{eigen n=20}.

\bibliographystyle{amsplain-url}


\section{\Large{Supporting Material}}
\setlength{\intextsep}{0.8cm}
\setlength{\textfloatsep}{0.8cm}

\paragraph{\textbf{Modulating the disorder strength.---}}To realize the periodic drive, we  sinusoidally modulate the intensity of the laser that realizes the disorder lattice. The intensity of this laser is linearly proportional to the onsite disorder and hence modulates the onsite disorder in a sinusoidal fashion~\cite{Schreiber15}.\\

\begin{figure}[b!]
	\centering
	\includegraphics[width=01.0\linewidth]{Preparation_and_detection}
	\caption{\textbf{Lattice sequence for the experiment.} In order to keep the system effectively one-dimensional we freeze the motion along the perpendicular directions by ramping the $y,z$-direction lattices deep to 40$E_{\text{r}}$. Along the $x$-direction we prepare a density-wave  pattern using the superlattice where only the even sublattice is occupied. We then initiate the dynamics by switching off the long lattice, and quickly ramping the short lattice to 8$\,E_{\text{r}}$. Simultaneously, we modulate the disorder lattice at a given frequency and amplitude. Afterwards, we freeze the system by suddenly ramping up the lattices and initiate the bandmapping procedure to extract the total number of atoms on even and odd sublattices. Note that the $x$-axis is not to scale.}
	\label{sequenceprepdet}
\end{figure}

\paragraph{\textbf{General experimental sequence.---}}We produce a gas of ultracold fermionic Potassium-40 ($^{40}$K) atoms by sympathetically cooling it with bosonic Rubidium-87 ($^{87}$Rb). By lowering the trapping potential depth, we let the gravitational force selectively remove the $^{87}$Rb atoms, leaving behind a pure fermionic gas. This Fermi gas consists of an equal mixture of the two spin components. We evaporate the gas further and obtain a final temperature of $0.15(2)\,T_F$, where $T_F$ is the Fermi temperature. The two spin components are the two lowest hyperfine states of the $^4S_{1/2}$  manifold ($\ket{F,m_F} = \ket{9/2,-9/2} \equiv \ket{\downarrow}$ and $\ket{9/2,-7/2} \equiv \ket{\uparrow}$). Interactions between the two spin components are tuned via an $s$-wave Feshbach resonance at 202.1G~\cite{Regal03}.

We load the gas in to a deep three dimensional lattice where tunneling can be neglected. The lattice is formed by lasers of wavelengths $\lambda_{\text{l}} = 1064\,$nm in the $x$-direction and  $\lambda_{\text{y,z}} = 738.20\,$nm in the perpendicular directions. In order to avoid double occupancies in the initial state, the interactions between the two species are set to be repulsive during loading with a magnitude of 140$\,$a$_0$, where a$_0$ is the Bohr radius. This restricts the double occupancies in the initial state to less than $10\%$. Next, we prepare a density wave by utilizing a superlattice setup in the $x$-direction~\cite{Trotzky08}, consisting of a short $\lambda_{\text{s}}=532.0\,$nm lattice laser in addition to the long lattice laser $\lambda_{\text{l}}$ . Subsequently, we set the interactions to the desired value using the Feshbach resonance and initiate the dynamics by lowering the $x$-lattice to $8\,E_{\text{r}}^{\text{s}}$~\cite{Schreiber15}. The system remains effectively one-dimensional during the time scales probed in the experiment as quantum tunneling is appreciable only in the $x$-direction. The perpendicular lattices remain deep at $40\,E_{\text{r}}^{\text{y,z}}$ and hence quantum tunneling is strongly suppressed. Here $E_{\text{r}}^i = h^2/(2 m \lambda_i ^2)$ is the recoil energy of the laser with wavelength $\lambda_i$ and $m$ is the atomic mass.

Simultaneous to the reduction of the $x$-lattice depth, we introduce quasi-random disorder by illuminating the atoms with a second, incommensurate lattice with $\lambda_d = 738\,\text{nm}$, which we refer to as the disorder lattice. By modulating the intensity of the disorder lattice, we synchronously modulate all onsite disorder potentials, see Eq.(1). Different phases $\phi$ of the disorder are realized by changing the detuning of the disorder laser. After variable evolution times, we suddenly ramp up the short lattice within 10$\,\mu \text{s}$ to 20$\,E_{\text{r}}^{\text{s}}$ followed by ramping the long lattice to 20$\,E_\text{r}^{\text{l}}$ to suppress the tunneling. Then we initiate our bandmapping sequence to measure the population of even and odd sublattices and perform absorption imaging on the bandmapped atomic gas after 8$\,$ms of time-of-flight~\cite{Trotzky12,Schreiber15}. A brief schematic of this lattice sequence is given in \fig{sequenceprepdet}.\\

\begin{figure*}
	\centering
	\includegraphics[width=1.0\linewidth]{PD_A_nu_v3}
	\caption{\textbf{Dynamical phase diagram as a function of frequency and amplitude.} The asymptotic imbalance is presented for disorder strength $\Delta = 4\,J$ and interaction strength $U = 4\,J$. We observe that the system remains localized except for large drive amplitudes and low frequencies. In the experimental graph, the gray dots denote the parameters where the data is measured.}
	\label{pdampnu}
\end{figure*}

\paragraph{\textbf{Atom losses and band resonances.---}} We measure no discernible atom loss with or without shaking for all measurements shown in the main text. This is because higher band resonances are off-resonant in energy and hence transitions to higher bands are suppressed. Due to the parity of the Wannier functions, the lowest band to which the atom could be excited via amplitude modulation is the third band (second excited band) along the $x$-direction which starts with a gap of $210\,J$ above the lowest band. This is much larger than the largest frequency probed in the experiment of $40\,J$. In addition, the background lifetimes of the imbalance and atom number are very long compared to the time scales probed in the main text~\cite{Bordia16}.\\

\paragraph{\textbf{Amplitude-frequency phase diagram.---}} The full phase diagram at $\Delta = 4\,J$ and $U = 4\,J$ is shown in \fig{pdampnu} as a function of amplitude and frequency of the drive for both experiment and theory. We observe that the system remains localized except for large amplitudes ($A \sim \Delta$) and low frequencies.\\

\paragraph{\textbf{SU(2) symmetry.---}}In the experiment, a weak breaking of the SU(2) symmetry between the spin states is expected due to the different polarizabilities of the two hyperfine states at finite detuning of the lasers. For the numerical studies, we assume a 2$\%$ difference in the disorder strength for the two spin species which breaks the SU(2) symmetry.\\

\paragraph{\textbf{Asymptotic imbalance value $\mathcal{I}_0$ in the undriven system.---}}The asymptotic values of the undriven system, $\mathcal{I}_0$, is also obtained from such time traces both for theory and experiment. For disorder $\Delta = 7.5\, J$ and $U = 4\,J$, the undriven asymptotic value measured in the experiment is $\mathcal{I}_0^{\text{expt}} = 0.56$ and for the theory is $\mathcal{I}_0^{\text{theory}} = 0.66$. The difference between theory and experiment is primarily attributed to the finite background lifetime of the experiment \cite{Bordia16} and to the fidelity of the initial state imbalance of $\mathcal{I}_\text{expt}(t=0) = 0.91$, compared to the numerical calculations which has unity imbalance.\\

\paragraph{\textbf{Numerical simulations.---}}We employ matrix product states~\cite{vidal_efficient_2004} to numerically investigate the time-dependent imbalance of the periodically modulated interacting Aubry-Andr\'e model. In all the presented data, we choose systems of size $L=32$. We discretize the time evolution operator using a second order Trotter decomposition with timestep $0.05\, \tau$ and take a bond dimension of 800 for the matrix product states. This guarantees that the truncated weight remains small; typically smaller than $5 \times 10^{-6}$. For the initial state, we prepare a density-wave pattern in which every other site is occupied by a fermion in a random spin state. For each time trace of the imbalance we average over $20-40$ disorder realizations. The asymptotic imbalance shown in the phase diagrams is obtained by averaging the traces between 30$\, \tau$ and 50$\, \tau$, which is based on a slightly larger time window compared to the experimentally determined value, in order to remove temporal oscillations in the imbalance.

\begin{figure}[h]
	\centering
	\includegraphics[width=1\linewidth]{LT_TT_v10}
	\caption{\textbf{Long-time dynamics of the imbalance for different driving modes.} For disorder $\Delta=5\,J$ and interactions $U=4\,J$, we calculate the time trace of the imbalance up to times $500 \, \tau$. We compare the dynamics without the drive $A=0$ \textbf{(a)}, where we observe the imbalance to saturate at a finite value, to finite amplitude drive $A=0.1\, \Delta$ for different driving modes \textbf{(b--d)}. While a field gradient modulation with drive frequency $\nu=\frac{1}{40}\times\Delta/h$ results in a power law decay in the imbalance \textbf{(b)}, a globally homogeneous modulation of the on-site energies shown for \textbf{(c)} $\nu = \frac{1}{40}\times\Delta/h$ and \textbf{(d)} $\nu = \frac{1}{20}\times\Delta/h$, indicates slow relaxation to a finite imbalance.}
	\label{gradientamplitude}
\end{figure}

We compute the level statistics of the effective Floquet Hamiltonian by approximating the sinusoidal drive with a two step function. Using exact diagonalization, we obtain all the eigenstates and eigenenergies of the stroboscopic time evolution operator for systems of $L=8$ sites at quarter filling. This allows us to compute the level statistics $r$ for the quasi-energies of the Floquet Hamiltonian. The level statistics distribution $P(r)$ shown in the main text consists of data from 2000 disorder realizations.\\

\begin{figure}
	\centering
	\includegraphics[width=1.0\linewidth]{TT_Long_v4}
	\caption{\textbf{Long-time dynamics of large amplitude drive}. Evolution of the imbalance for disorder $\Delta = 7.5\, J$ and interaction $U = 4\, J$ with amplitude $A= \Delta$, as in \fig{time_traces} but for longer times supporting the observation that the system is localized for high frequency $\nu = 2.0\,\Delta/h$ but delocalizes for low frequency $\nu = 0.4\,\Delta/h$. Part \textbf{(a)} and \textbf{(b)} show the same data but on a linear-linear and log-log plot, respectively. While \textbf{(a)} allows direct comparison with \fig{time_traces}, \textbf{(b)} demonstrates the slow power law decay of small residual imbalance at low drive frequency of $\nu = 0.4 \, \Delta/h$.}
	\label{tt_long}
\end{figure}

\paragraph{\textbf{Long-time dynamics of weakly driven system.---}} We numerically simulate the long time dynamics of the imbalance for different driving modes and particularly focus on the case of weak drive amplitude $A \ll \Delta$ and small frequencies $\nu\ll \Delta/h$, \fig{gradientamplitude}. We set the disorder to $\Delta = 5\,J$ and the interaction strength to $U = 4\,J$. We compare the relaxation of the \textbf{(a)} undriven system, where we observe saturation at a finite imbalance, to \textbf{(b--d)} weakly driven systems $A=0.1\,\Delta$ with different driving modes. For this simulation, the starting state is a product state with alternating spin states (unlike the rest of the paper which has random configuration of initial spins states). This fixed initial state has been chosen to remove the uncertainty resulting from sampling different spin configurations and hence reduces the number of samples required to obtain a converged time trace. The modulation of a field gradient, corresponding to a linearly increasing potential in space, leads to a power-law decrease of the imbalance at late times \textbf{\fc{b}}. In contrast, a globally homogeneous drive that synchronously modulates all on-site potentials, leads to a saturation of the imbalance to a finite value. This is shown for two small frequencies in subplot \textbf{(c)} for $\nu = \frac{1}{40}\times\Delta/h$, and in \textbf{(d)} for $\nu=\frac{1}{20}\times\Delta/h$. Such behavior may be explained by the fact that the system follows the drive quasi-adiabatically. While this leads to a periodic modulation of the localization length, set by the spatial extent of the locally conserved quantities~\cite{Nandkishore15}, this drive seems not to be able to create a percolating network of resonances and therefore does not lead to global particle rearrangements, which renders a persistent localization plausible. Studying such extreme long-time dynamics remains challenging for the current experimental setup due to the residual background effects, such as coupling to the neighboring tubes~\cite{Bordia16}.\\

\begin{figure}
	\centering
	\includegraphics[width=1\linewidth]{nuSpectrum_v2}
	\caption{\textbf{Frequency resolved spectrum.} The figure shows normalized imbalance for the non-interacting system with disorder $\Delta = 7.5\,J$ as a function of drive frequency $\nu$ at amplitude $A=0.2\,\Delta$.}
	\label{nuSpectrum}
\end{figure}

\begin{figure}
	\centering
	\includegraphics[width=1\linewidth]{AAHistogram_v1}
	\caption{\textbf{Nearest-neighbor energy differences for quasi-random disorder.} The energy difference of nearest-neighbor on-site energies realized by quasi-random disorder of the Aubry-Andr\'e model is peaked at around $1.5\,\Delta$ which gives rise to the small substructure in the frequency resolved spectrum.}
	\label{aaediff}
\end{figure}

\paragraph{\textbf{Long-time dynamics of the strongly driven system.---}}We compute the long-time dynamics corresponding to \fig{time_traces}, which is shown in \fig{tt_long}. These results support the observation that the system is localized at large frequency $\nu = 2.0\,\Delta/h$ while it is delocalized at small frequency $\nu = 0.4\,\Delta/h$. For strong drive and small frequency, the numerical data suggests that after a fast initial drop, the residual imbalance decays as a powerlaw toward zero with a small exponent, while it rapidly saturates for high drive frequencies.\\

\paragraph{\textbf{Frequency resolved absorption.---}}In order to determine the absorption spectrum of the system, we measure the imbalance as a function of the drive frequency for relatively small drive amplitude $A = 0.2\,\Delta$, \fig{nuSpectrum}. We consider strong disorder $\Delta = 7.5\,J$ and set the interactions close to $U=0$. The absorption spectrum features a reduced imbalance for frequencies between 0.2 and 2$\,\Delta/h$. We understand the data as follows: At low frequencies the system can adiabatically follow the drive and hence the imbalance remains high. Upon increasing the frequency the system begins to absorb from the drive. However, for even larger frequency, the system cannot follow the drive anymore and only evolves with the time-averaged Hamiltonian. Hence the system effectively becomes transparent to the drive and remains localized. We attribute the peculiar line shape of a double dip, to the very asymmetric distribution of nearest-neighbor onsite energy differences in the quasi-periodic Aubry-Andr\'e disorder potential, which features a peak at $1.5\,\Delta$, see \fig{aaediff}. This strongly peaked distribution also gives rise to the relatively sharp features visible in the simulations shown in \fig{phase_diagram} and~\figc{cut}{a}. In the experiment, the resulting structure might be softened due to the presence of the harmonic confinement and averaging over multiple 1D tubes~\cite{Schreiber15}.\\

\begin{figure}
	\centering
	\includegraphics[width=1.0\linewidth]{i_ni_freq_diff_v3}
	\caption{\textbf{Interaction effects on the ratio of the asymptotic imbalance between the driven and undriven systems.} We measure the imbalance for an interacting $U = 4\,J$ and non-interacting $U=0$ system with disorder $\Delta = 7.5\,J$ and drive amplitude $A = 0.7\,\Delta$ as a function of the drive frequency $\nu$. At low drive frequencies the imbalance of the interacting system is stronger suppressed.}
	\label{i_ni_nu}
\end{figure}

\paragraph{\textbf{Interaction effects on the imbalance.---}}We find that at low frequencies  $\nu \lesssim 2\,J/h$ and strong drive amplitude $A=0.7\,\Delta$, the ratio of the saturated imbalance of the driven and undriven systems depends on the interaction strength, even at strong disorder $\Delta = 7.5\,J$, \fig{i_ni_nu}. We attribute this observation to the shift of the critical disorder strength of the MBL transition toward larger values when increasing the interactions. We have checked that in the weak drive limit ($A \ll \Delta$), the saturated imbalance does not display a relevant interaction dependence (data not shown).\\

\end{document}