text:
\section{Introduction}

Matrix multiplication is a fundamental operation with applications in nearly any
area of science and engineering. However, after more than 50 years of work on
matrix multiplication techniques (see,
e.g.,~\cite{buergisser-book,pan-survey2018,blaeser-survey2013,landsberg2017geometry}),
the complexity of matrix multiplication is still a mystery. Even for small matrices, the problem
is not completely understood, and understanding these cases better can provide valuable hints
towards more efficient algorithms for large matrices.

The naive way for computing the product $C$ of two $2\times 2$ matrices
$A,B$ requires 8 multiplications:
\[
 \begin{pmatrix}
 a_{11} & a_{12} \\
 a_{21} & a_{22}
 \end{pmatrix}
 \begin{pmatrix}
 b_{11} & b_{12} \\
 b_{21} & b_{22}
 \end{pmatrix} =
 \begin{pmatrix}
 a_{11}b_{11} + a_{12}b_{21} &\kern.5em a_{11}b_{12} + a_{21}b_{22}\\
 a_{21}b_{11} + a_{22}b_{21} &\kern.5em a_{21}b_{12} + a_{22}b_{22}
 \end{pmatrix}
 =
 \begin{pmatrix}
 c_{11} & c_{12} \\
 c_{21} & c_{22}
 \end{pmatrix}
\]
Strassen observed 50 years ago that $C$ can also be computed with only 7
multiplications~\cite{strassen1969gaussian}.
His scheme proceeds in two steps.
In the first step he introduces auxiliary variables $M_1, \ldots, M_7$
which are defined as the product of certain linear combinations of
the entries of $A$ and~$B$. In the second step the entries of $C$ are obtained as
certain linear combinations of the~$M_i$:
\begin{alignat*}{5}
 M_1 &= (a_{11} + a_{22}) (b_{11} + b_{22})
 \qquad\qquad & c_{11} & =  M_1 + M_4 - M_5 + M_7 \\[-2pt]
 M_2 &= (a_{21} + a_{22}) (b_{11})
  & c_{12} & =  M_3 + M_5 \\[-2pt]
 M_3 &= (a_{11}) (b_{12} - b_{22})
  & c_{21} & =   M_2 + M_4 \\[-2pt]
 M_4 &= (a_{22}) (b_{21} - b_{11})
  & c_{22} & =   M_1 - M_2 + M_3 + M_6 \\[-2pt]
 M_5 &= (a_{11} + a_{12})(b_{22})\\[-2pt]
 M_6 &= (a_{21} - a_{11}) (b_{11}+ b_{12})\\[-2pt]
 M_7 &= (a_{12} - a_{22}) (b_{21} + b_{22})
\end{alignat*}
Recursive application of this scheme gave rise to the first algorithm for multiplying
arbitrary $n\times n$ matrices in subcubic complexity.
Winograd~\cite{winograd1971multiplication} showed
that Strassen's scheme is optimal in the sense
that there does not exist a similar scheme with fewer than 7 multiplications,
and de Groote~\cite{de1978varieties}
showed that Strassen's scheme is essentially unique.

Less is known about $3\times 3$ matrices.
The naive scheme requires 27 multiplications, and in 1976
Laderman~\cite{laderman1976noncommutative} found one with~23.
Similar as Strassen, he defines $M_1,\dots,M_{23}$ as
products of certain linear combinations of
the entries of $A$ and~$B$. The entries of $C=AB$ are then obtained
as linear combinations of $M_1,\dots,M_{23}$.
It is not known whether 23 is optimal (the best lower bound is 19~\cite{blaser2003complexity}).
It is known however that Laderman's scheme is \emph{not} unique.
A small number of intrinsically different schemes have been found
over the years. Of particular interest are schemes in which all
coefficients in the linear combinations are $+1$, $-1$, or~$0$.
The only four such schemes (up to equivalence) we are aware of
are due to Laderman, Smirnov~\cite{smirnov2013bilinear}, Oh et al.~\cite{oh2013inequivalence}, and Courtois
et al.~\cite{DBLP:journals/corr/abs-1108-2830}.

While Smirnov and Oh et al. found their multiplicatoin schemes
with computer-based search using non-linear numerical optimization
methods, Courtois found his multiplication scheme using a SAT solver. This is
also what we do here. We present two approaches which allowed us to generate more
than 13,000 mutually inequivalent new matrix multiplication schemes for $3\times
3$ matrices, using altogether about 35 years of CPU years. We believe that the
new schemes are of interest to the matrix multiplication community. We therefore
make them publicly available in various formats and grouped by invariants at
\begin{center}
\url{http://www.algebra.uni-linz.ac.at/research/matrix-multiplication/}.
\end{center}

\section{Encoding and Workflow}

\def\x{^{\vphantom(}}

To search for multiplication schemes of $3 \times 3$ matrices having
the above form, we define the $M_i$ as product of linear combination
of all entries of $A$ and $B$ with undetermined coefficients $\alpha_{ij}^{(\ell)}, \beta_{ij}^{(\ell)}$:
 \begin{alignat*}1
   M_1\x &=
   (\alpha_{11}^{(1)}a_{11}\x + \cdots + \alpha_{33}^{(1)}a_{33}\x)
   (\beta_{11}^{(1)}b_{11}\x + \cdots + \beta_{33}^{(1)}b_{33}\x)\\[-2pt]
   &\vdots\\[-2pt]
   M_{23}\x &=
   (\alpha_{11}^{(23)}a_{11}\x + \cdots + \alpha_{33}^{(23)}a_{33}\x)
   (\beta_{11}^{(23)}b_{11}\x + \cdots + \beta_{33}^{(23)}b_{33}\x)
 \end{alignat*}
 Similarly, we define the $c_{ij}$ as linear combinations of the
 $M_i$ with undetermined coefficients $\gamma_{i,j}^{(\ell)}$:
 \begin{alignat*}3
   c_{11}\x &= \gamma_{11}^{(1)}M_1\x + \cdots + \gamma_{11}^{(23)}M_{23}\x,
   &\quad \ldots,\quad  &
   c_{33}\x &= \gamma_{33}^{(1)}M_1\x + \cdots + \gamma_{33}^{(23)}M_{23}\x
 \end{alignat*}
 Comparing the coefficients of all terms $a_{i_1i_2}b_{j_1j_2}c_{k_1k_2}$ in the equations $c_{ij}=\sum_k a_{ik}b_{kj}$
 leads to the polynomial equations
 \[
 \sum_{\ell=1}^{23} \alpha_{i_1i_2}^{(\ell)}\beta_{j_1j_2}^{(\ell)}\gamma_{k_1k_2}^{(\ell)} =
 \delta_{i_2j_1}\x\delta_{i_1k_1}\x\delta_{j_2k_2}\x
 \]
 for $i_1,i_2,j_1,j_2,k_1,k_2\in\{1,2,3\}$.  These 729 cubic equations with 621
 variables are also known as \emph{Brent equations}~\cite{brent1970algorithms}.  The
 $\delta_{uv}$ on the right are Kronecker-deltas, i.e., $\delta_{uv}=1$ if $u=v$
 and $\delta_{uv}=0$ otherwise.  Each solution of the system of these equations
 corresponds to a matrix multiplication scheme.  The equations become slightly
 more symmetric if we flip the indices of the~$\gamma_{ij}$, and since this is
 the variant mostly used in the literature, we will also adopt it from now on.

 Another view on the Brent equations is as follows. View the
 $\alpha_{i_1i_2}^{(\ell)}$, $\beta_{j_1j_2}^{(\ell)}$,
 $\gamma_{k_1k_2}^{(\ell)}$ as variables, as before, and regard
 $a_{i_1i_2},b_{j_1j_2},c_{k_1k_2}$ as polynomial indeterminants. Then the task
 consists of instantiating the variables in such a way that
 \[
 \sum_{\ell=1}^{23}(\alpha_{11}^{(\ell)}a_{11}\x+\cdots)(\beta_{11}^{(\ell)}b_{11}\x+\cdots)(\gamma_{11}^{(\ell)}c_{11}\x+\cdots)
 =\sum_{i=1}^3\sum_{j=1}^3\sum_{k=1}^3 a_{ij}b_{jk}c_{ki}
 \]
 holds as equation of polynomials in the variables
 $a_{i_1i_2},b_{j_1j_2},c_{k_1k_2}$. Expanding the left hand side and equating
 coefficients leads to the Brent equations as stated before (but with indices of
 $\gamma$ flipped, as agreed). In other words, expanding the left hand side,
 all terms have to cancel except for the terms on the right. We found it
 convenient to say that a term $a_{i_1i_2}b_{j_1j_2}c_{k_1k_2}$ has ``type $m$''
 if $m = \delta_{i_2j_1}+\delta_{j_2k_1}+\delta_{k_2i_1}$.  With this
 terminology, all terms of types 0, 1,~2 have to cancel each other, and all
 terms of type~3 have to survive. Note that since all 27 type~3 terms
 must be produced by the 23 summands on the left, some summands must produce
 more than one type~3 term.

 For solving the Brent equations with a SAT solver, we use $\set Z_2$ as
 coefficient domain, so that multiplication translates into `and' and
 addition translates into `xor'. When, for example, the variable $\alpha_{i_1i_2}^{(\ell)}$ is true
 in a solution of the corresponding SAT instance, this indicates that
 the term $a_{i_1i_2}$ occurs in~$M_\ell$, and likewise for the $b$-variables.
 If $\gamma_{k_1k_2}^{(\ell)}$ is true, this means that $M_\ell$ appears in the
 linear combination for~$c_{k_1k_2}$. We call  $\alpha_{i_1i_2}^{(\ell)}$, $\beta_{j_1j_2}^{(\ell)}$, and $\gamma_{k_1k_2}^{(\ell)}$
 the \emph{base variables}.

 In order to bring the Brent equations into CNF, we use Tseitin transformation, i.e., we introduce definitions for
 subformulas to avoid exponential blow-up. To keep the number of fresh variables
 low, we do not introduce one new variable for every cube but only for pairs of literals,
 i.e., we encode a cube $(\alpha \land \beta \land \gamma)$ as $u \leftrightarrow (\alpha \land \beta)$ and
 $v \leftrightarrow (u \land \gamma)$. In this way, we can reuse~$u$.
 Furthermore, a sum $v_1 \oplus \dots \oplus v_m$ with $m\geq4$ is encoded by
 $w\leftrightarrow(v_1\oplus v_2\oplus v_3)$ and $v_4 \oplus \dots \oplus v_m\oplus w$,
 with the latter sum being encoded recursively. This encoding seems to require the
 smallest sum of the number of variables and the number of clauses---a commonly used optimality heuristic. The used scripts are available at
\begin{center}
\url{https://github.com/marijnheule/matrix-challenges/tree/master/src}.
\end{center}
 The generation of new schemes proceeds in several steps, with SAT solving being
 the first and main step. If the SAT solver finds a solution, we next check
 whether it is equivalent to any known or previously found solution modulo de Groote's symmetry group~\cite{de1978varieties}.
 If so, we discard it. Otherwise, we next try to simplify the new scheme by searching for an element in
 its orbit which has a smaller number of terms. The scheme can then be used to initiate
 a new search. In the fourth step, we use
 Gr\"obner bases to lift the scheme from the coefficient domain $\set Z_2$ to~$\set Z$.
 Finally, we cluster large sets of similar solutions into parameterized families.
 \begin{center}
   \begin{tikzpicture}[xscale=1.7]
     \node (b) at (1,0) {$\bullet$} (1,0) node[left,xshift=-2mm] {\vbox{\llap{\strut known}\kern-2pt\llap{\strut schemes}}};
     \node (c) at (2,0) {$\bullet$};
     \node (d) at (3,0) {$\bullet$};
     \node (e) at (4,0) {$\bullet$};
     \node (f) at (5,0) {$\bullet$};
     \node (g) at (6,0) {$\bullet$} (6,0) node[right,xshift=2mm] {\vbox{\rlap{\strut new}\kern-2pt\rlap{\strut schemes}}};
     \draw[->] (b)--(c) node[midway,above] {\footnotesize\strut solve};
     \draw[->] (c)--(d) node[midway,above] {\footnotesize\strut filter};
     \draw[->] (d)--(e) node[midway,above] {\footnotesize\strut simplify};
     \draw[->] (e)--(f) node[midway,above] {\footnotesize\strut lift};
     \draw[->] (f)--(g) node[midway,above] {\footnotesize\strut cluster};
     \draw[->] (e)--++(0,-.5)--++(-3,0)--(b);
   \end{tikzpicture}
 \end{center}
 In the present paper, we give a detailed description of the first step in this workflow.
 The subsequent steps use algebraic techniques unrelated to SAT and will be described in~\cite{mmr-jsc}.

 \section{Solving Methods}\label{sec:methods}

 The \emph{core} of a scheme is the pairing of the type~3 terms.

 Our first method focuses on finding schemes with new cores, while our second method searches for schemes that are similar to an
 existing one and generally has the same core.
 For all experiments we used the local search SAT solver {\tt yalsat}~\cite{yalsat} as this solver performed best on instances from this application.
 We also tried solving these instances using CDCL solvers, but the performance was disappointing. We observed a possible explanation:
 The runtime of CDCL solvers tends to be exponential in the average backtrack level (ABL) on unsatisfiable instances.
 For most formulas arising from other applications, ABL is small ($<50$), while on the matrix multiplication instances ABL is large ($> 100$).

 \subsection{Random Pairings of Type 3 Terms and Streamlining}

 Two of the known schemes, those of Smirnov~\cite{smirnov2013bilinear} and Courtois et al.~\cite{DBLP:journals/corr/abs-1108-2830},
 have the property that each type~3 term occurs exactly once and at most two type~3 terms occur in the same summand. We decided to use this pattern
  to search for new schemes: randomly pair four type~3 terms and assign the remaining type~3 terms to the other 19 summands.
  Only in very rare cases, random pairing could be extended to a valid scheme in reasonable time, say a few minutes.
  In the other cases it is not known whether the pairing cannot be extended to a valid scheme or whether finding such a scheme is very hard.

  Since the number of random pairings that could be extended to a valid scheme was very low, we
  tried
  adding \emph{streamlining constraints}~\cite{streamlining} to formulas. A streamlining constraint is a set of clauses that
  guides the solver to a solution, but these clauses may not (and generally are not) implied by the formula. Streamlining constraints are
  usually patterns observed in solutions of a given problem, potentially of  smaller sizes. We experimented with various
  streamlining constraints, such as enforcing that each type~0, type~1, and type~2 term occurs either zero times or twice in a scheme
  (instead of an even number of times).  The most effective streamlining constraint that we came up with was observed in the Smirnov scheme:
  for each summand that is assigned a single type~3 term, enforce that (i) one matrix has either two rows, two columns
  or a row and a column fully assigned to zero and (ii) another matrix has two rows and two columns assigned to zero,
  i.e., the matrix has a single nonzero entry. This streamlining constraint reduced the runtime from minutes to seconds. Yet
  some random pairings may only be extended to a valid scheme that does not satisfy the streamlining constraint.

\subsection{Neighborhood Search}

The second method is based on neighborhood search: we select a scheme, randomly fix some the corresponding base variables,
and search for an assignment for the remaining base variables. This simple method turned out to be remarkably effective to find
new schemes. The only parameter for this method is the number of base variables that will be fixed. The lower the number of
fixed base variables, the higher the probability to find a different scheme and the higher the costs to find an assignment for the remaining base variables.
We experimented with various values and it turned out that fixing 2/3 of the 621 base variables (414) is effective to
find many new schemes in a reasonable time.

The neighborhood search is able to find many new schemes, but in almost all cases they have the same pairing
of type~3 terms. Only in some rare cases the pairing of type~3 terms is different. Figure~\ref{fig:neighbors}
shows such an example: scheme {\tt A} has term $a_{13}b_{31}c_{11}$ in summand 22 and
term $a_{23}b_{33}c_{32}$ in summand 23, while the neighboring scheme {\tt B} has term $a_{13}b_{33}c_{31}$ in summand 22
and terms $a_{13}b_{31}c_{11}$, $a_{23}b_{33}c_{32}$, and $a_{13}b_{33}c_{31}$ in summand~23.

\begin{figure}[ht]
\centering
\begin{minipage}{0.7\textwidth}\footnotesize
\begin{eqnarray*}
{\tt 1~\,}&~&(a_{11} + a_{13} + a_{21} + a_{22} + a_{23})(b_{13})(c_{22} + c_{32})\\[-2pt]
{\tt 2~\,}&~&(a_{11} + a_{13} + a_{23})(b_{13} + b_{32})(c_{11} + c_{22} + c_{31} + c_{32})\\[-2pt]
{\tt 3~\,}&~&(a_{11} + a_{13})(b_{32})(c_{21} + c_{22} + c_{31} + c_{32})\\[-2pt]
{\tt 4~\,}&~&(a_{11} + a_{31})(b_{11} + b_{12} + b_{13})(c_{23})\\[-2pt]
{\tt 5~\,}&~&(a_{11} + a_{33})(b_{11} + b_{13} + b_{32})(c_{11} + c_{23})\\[-2pt]
{\tt 6~\,}&~&(a_{12} + a_{13} + a_{23})(b_{13} + b_{33})(c_{11} + c_{31})\\[-2pt]
{\tt 7~\,}&~&(a_{12} + a_{22} + a_{32})(b_{21} + b_{22} + b_{23})(c_{33})\\[-2pt]
{\tt 8~\,}&~&(a_{12} + a_{31} + a_{32} + a_{33})(b_{22})(c_{23} + c_{33})\\[-2pt]
{\tt 9~\,}&~&(a_{12} + a_{33})(b_{13} + b_{21} + b_{33})(c_{11} + c_{33})\\[-2pt]
{\tt 10~\,}&~&(a_{12})(b_{13} + b_{23} + b_{33})(c_{31} + c_{33})\\[-2pt]
{\tt 11~\,}&~&(a_{21} + a_{31} + a_{33})(b_{11})(c_{12} + c_{22})\\[-2pt]
{\tt 12~\,}&~&(a_{21})(b_{11} + b_{12} + b_{13})(c_{22})\\[-2pt]
{\tt 13~\,}&~&(a_{22} + a_{31} + a_{33})(b_{13} + b_{22})(c_{12} + c_{13} + c_{22} + c_{33})\\[-2pt]
{\tt 14~\,}&~&(a_{22} + a_{32} + a_{33})(b_{21})(c_{13} + c_{33})\\[-2pt]
{\tt 15~\,}&~&(a_{22})(b_{13} + b_{21} + b_{22})(c_{12} + c_{13})\\[-2pt]
{\tt 16~\,}&~&(a_{22})(b_{13} + b_{23})(c_{32} + c_{33})\\[-2pt]
{\tt 17~\,}&~&(a_{23})(b_{31})(c_{11} + c_{12} + c_{31} + c_{32})\\[-2pt]
{\tt 18~\,}&~&(a_{31} + a_{33})(b_{11} + b_{13} + b_{22})(c_{12} + c_{13} + c_{22} + c_{23})\\[-2pt]
{\tt 19~\,}&~&(a_{33})(b_{11} + b_{21} + b_{31})(c_{11} + c_{13})\\ \hline
{\tt 20{A}}&~&(a_{12})(b_{22})(c_{21} + c_{23})\\[-2pt]
{\tt 21{A}}&~&(a_{11})(b_{12} + b_{32})(c_{21} + c_{23})\\[-2pt]
{\tt 22{A}}&~&(a_{13} + a_{33})(b_{31} + b_{32} + b_{33})(c_{11})\\[-2pt]
{\tt 23{A}}&~&(a_{23})(b_{31} + b_{32} + b_{33})(c_{11} + c_{31} + c_{32})\\ \hline
{\tt 20{B}}&~&(a_{11} + a_{12})(b_{22})(c_{21} + c_{23})\\[-2pt]
{\tt 21{B}}&~&(a_{11})(b_{12} + b_{22} + b_{32})(c_{21} + c_{23})\\[-2pt]
{\tt 22{B}}&~&(a_{13} + a_{33})(b_{31} + b_{32} + b_{33})(c_{31} + c_{32})\\[-2pt]
{\tt 23{B}}&~&(a_{13} + a_{23} + a_{33})(b_{31} + b_{32} + b_{33})(c_{11} + c_{31} + c_{32})
\end{eqnarray*}
\end{minipage}
\caption{Two neighboring schemes
  with 19 identical summands and 4 different ones
}
\label{fig:neighbors}
\end{figure}

 \section{Evaluation and Analysis}\label{sec:evaluation}

The methods presented in Section~\ref{sec:methods} enabled us to find several hundreds of solutions individually, but they were particularly
effective when combined.
The first method allows finding schemes that can be quite different compared to the known schemes. However, finding
a scheme using that method may require a few CPU hours as most pairings of type~3 terms cannot be extended to a valid scheme that
satisfies the streamlining constraints. The second method can find schemes that are very similar to known ones with a second. The
neighborhood of known solutions turned out to be limited to a few hundred of new schemes.

In contrast, some of the schemes found using the first method have a large neighborhood. We approximated the size of the neighborhood
of a scheme using the following experiment: Start with a given scheme $S$ and find a neighboring scheme by randomly fixing 2/3 of the base variables. Once
a neighboring scheme $S'$ is found, find a neighboring scheme of~$S'$, etc. We ran this experiment on a machine with 48 cores of the Lonestar 5 cluster of
Texas Advanced Computing Center. We started each experiment using 48 threads with each thread assigned a different seed.
Figure~\ref{fig:results} shows the number of different schemes (after sorting) found in 1000 seconds when
starting with one of the four known schemes and a scheme that was computed from the streamlining method.
Some of these different schemes are new, while others are equivalent to each other or known ones. We only assure here that they are not identical after sorting the summands.

Observe that the number of different schemes found in 1000 seconds depends a lot on the starting scheme. No different neighboring scheme was found for Laderman's scheme, only 9 different schemes were found for the scheme of Courtois et al., 94 different schemes were found for the scheme of Oh et al., 561 new schemes were found for Smirnov's scheme, and 3359 schemes were found using a randomly selected new scheme obtained with the streamlining method.

\begin{figure}[t]
\centering
\input{plot.tex}
\caption{The number of different schemes (vertical axis in logscale) found within a period of time (horizontal axis in seconds) during a random walk in the neighborhood of a given scheme.}
\label{fig:results}
\end{figure}

In view of the large number of solutions we found, it is also interesting to compare them with each other. For example,
if we define the \emph{support} of a solution as the number of base variables set to~$1$, we observe that the support
seems to follow a normal distribution with mean around 160, see Fig.~\ref{fig:support} for a histogram. We can also see
that the Laderman scheme differs in many ways from all the other solutions. It is, for example, the only scheme whose
core consists of four quadruples of type~3 terms. In 89\% of the solutions, the core consists of four pairs of type~3 terms,
about 10\% of the solution have three pairs and one quadrupel, and less than 1\% of the schemes have cores of the form 2-2-2-2-3
or 2-2-2-3-4.

\begin{figure}[t]
\centering
\input{support-histo}
\caption{Number of non-equivalent schemes found, arranged by support.}
\label{fig:support}
\end{figure}

 \section{Challenges}\label{sec:challenges}

The many thousands of new schemes
 that we found may still be just the tip of the iceberg. However, we also observed that the state-of-the-art SAT solving techniques are unable to answer
 several other questions. This section provides four challenges for SAT solvers with increasing difficulty. For each challenge we constructed one or
 more formulas that are available at
\begin{center}
{\tt \url{https://github.com/marijnheule/matrix-challenges}}.
\end{center}
The challenges are hard, but they may be
 doable in the coming years.

\paragraph{Challenge 1: Local search without streamlining.}

Our first method combines randomly pairing the type~3 terms with streamlining constraints. The latter was required to
limit the search. We expect that local search solvers can be optimized to efficiently solve the formulas without the streamlining
constraints. This may result in schemes that are significantly different compared to ones we found. We prepared ten
satisfiable formulas with hardcoded pairings of type~3 terms. Five of these formulas can be solved using {\tt yalsat} in a few
minutes. All of these formulas appear hard for CDCL solvers (and many local search solvers).

\paragraph{Challenge 2: Prove unsatisfiability of subproblems.}

We observed that complete SAT solvers performed weakly on our
matrix multiplication instances. It seems therefore unlikely that one could prove any optimality results for
the product of two $3 \times 3$ matrices using SAT solvers in the near future. A more realistic challenge concerns proving
unsatisfiability of some subproblems. We prepared ten formulas with 23 multiplications and hardcoded pairings of type~3 terms.
We expect that these formulas are unsatisfiable.

\paragraph{Challenge 3: Avoiding a type~3 term in a summand.}

All known schemes have the following property: each summand has at least one type~3 term.
We do not know whether there exists a scheme with 23 multiplications such that one of the
summands contains no type~3 term. The challenge problem blocks the existence of a type~3 term
in the last summand and does not have any additional (streamlining) constraints.

\paragraph{Challenge 4: Existence of a scheme with 22 multiplications.}

The main challenge concerns finding a scheme with only 22 multiplications. The hardness
of this challenge strongly depends on whether there exists such a scheme. The repository
contains a plain formula for a scheme with 22 multiplications.

\paragraph*{\bf Acknowledgments.}
The authors acknowledge the Texas Advanced Computing Center at The University of Texas at Austin for providing HPC
resources that have contributed to the research results reported within this paper.

\bibliographystyle{plain}


text:
\section{introduction}

In the literature, there are many papers on the research of homological properties of connected cochain DG algebras. For example, Gorenstein properties of DG algebras are studied in \cite{FHT1,FIJ, FJ1,FJ2,FM,Gam,Jor,Mao,MW2,Sch};
  He-Wu (cf. \cite{HW}) introduce and study Koszul  connected cochain DG algebras; and recently,  the first  author and J.-W. He (cf. \cite{HM}) give a criterion for a connected cochain DG algebra to be $0$-Calabi-Yau, and prove that a locally finite connected cochain DG algebra is $0$-Calabi-Yau if and only if it is defined by a potential. In spite of these, it is still difficult to judge whether a given DG algebra has some good homological properties such as formality, homologically smoothness, Gorensteinness  and
   Calabi-Yau property.

   Generally, the homological properties of a DG algebra are determined by the joint effects of its underlying graded algebra structure and differential structure. However, it is feasible,  at least in some special
cases, to judge some homological properties of a DG algebra from its underlying graded algebra. For example, it is shown in \cite{Mao} that
 a connected cochain DG algebra $\mathcal{B}$ is Gorenstein if its underlying graded algebra $\mathcal{B}^{\#}$ is an Artin-Schelter regular algebra of dimension $2$. Especially, if $\mathcal{B}^{\#}$
is generated by degree $1$ elements $x,y$ and subject to the
relation $xy+yx=0$, then $\mathcal{B}$ is a Koszul Calabi-Yau DG algebra (see \cite{MH}).  Recently, DG down-up algebras are introduced and studied in \cite{MHLX}. It is proved that all Non-trivial Noetherian DG down-up algebras are Calabi-Yau DG algebras.

This paper deals with DG polynomial algebras, which are connected cochain DG algebras
whose underlying graded algebras are polynomial algebras generated by degree $1$ elements.
We describe all possible differential structures on such DG polynomial algebras by the following theorem (see Theorem \ref{diffstructure}).
\\
\begin{bfseries}
Theorem \ A.
\end{bfseries}
Let $(\mathcal{A},\partial_{\mathcal{A}})$ be a connected cochain DG algebra such that $\mathcal{A}^{\#}$ is a polynomial graded algebra $\k[x_1,x_2,\cdots,x_n]$ with $|x_i|=1$, for any $i\in \{1,2,\cdots,n\}$.
Then there exist some $t_1,t_2,\cdots,t_n\in \k$ such that $\partial_{\mathcal{A}}$ is defined  by
$$\partial_{\mathcal{A}}(x_i)=\sum\limits_{j=1}^nt_jx_ix_j=\sum\limits_{j=1}^{i-1}t_jx_jx_i+t_ix_i^2+\sum\limits_{j=i+1}^nt_jx_ix_j,$$ for any $i\in \{1,2,\cdots,n\}$.
 Conversely, for any point $(t_1,t_2,\cdots, t_n)$ in the affine $n$-space $ \Bbb{A}_{\k}^n$, we can define a differential $\partial$ on $\k[x_1,x_2,\cdots,x_n]$ by
$$\partial(x_i)=\sum\limits_{j=1}^{i-1}t_jx_jx_i+t_ix_i^2+\sum\limits_{j=i+1}^nt_jx_ix_j,   \forall i\in \{1,2,\cdots,n\},$$
such that $(\k[x_1,x_2,\cdots,x_n],\partial)$ is a DG polynomial algebra.

By Theorem A, we can define $\mathcal{A}(t_1,t_2,\cdots, t_n)$ as a cochain DG algebra such that
$\mathcal{A}(t_1,t_2,\cdots, t_n)^{\#}=\k[x_1,x_2,\cdots,x_n]$ and its differential $\partial_{\mathcal{A}}$ is defined by $$\partial_{\mathcal{A}}(x_i)=\sum\limits_{j=1}^{i-1}t_jx_jx_i+t_ix_i^2+\sum\limits_{j=i+1}^nt_jx_ix_j, \forall i\in \{1,2,\cdots,n\}.$$
Then the set
$$\Omega(x_1,x_2,\cdots,x_n)=\{\mathcal{A}(t_1,t_2,\cdots,t_n)|t_i\in
\k, i=1,2,\cdots, n\}\cong \Bbb{A}_{\k}^n.$$

To consider the homological properties of $\mathcal{A}(t_1,t_2,\cdots, t_n)$, it is necessary to study the isomorphism problem of DG polynomial algebras.  We have the following theorem (see Theorem \ref{isom}).
\\
\begin{bfseries}
Theorem \ B.
\end{bfseries}
Let $\mathcal{A}(t_1,t_2,\cdots,t_n)$ and
$\mathcal{A}(t_1',t_2',\cdots,t_n')$ be two points in the
space $\Omega(x_1,x_2,\cdots,x_n)$. Then the DG algebras
$$\mathcal{A}(t_1,t_2,\cdots,t_n)\cong
\mathcal{A}(t_1',t_2',\cdots,t_n')$$ if and only if there
is a matrix $M\in \mathrm{GL}_n(\k)$ such that
$$(t_1',t_2',\cdots,t_n')=(t_1,t_2,\cdots,t_n)M.$$

By Theorem B,  we have only two isomorphism classes in $\Omega(x_1,x_2,\cdots,x_n)$ represented by $\mathcal{A}(0,0,\cdots, 0)$ and $\mathcal{A}(1,0,\cdots,0)$ (see Corollary \ref{isomor}), and we obtain the automorphism group (see Corollary \ref{auto})
$$\mathrm{Aut}_{dg}(\mathcal{A}(t_1,t_2,\cdots,t_n))\cong \{M\in \mathrm{GL}_n(\k)| (t_1,t_2,\cdots,t_n)=(t_1,t_2,\cdots, t_n)M   \}.$$
Generally, it is difficult to determine whether  a given DG algebra has some nice homological properties.
 Comparatively speaking, it is much easier to compute its cohomology graded algebra. For this, we compute the cohomology graded algebra of $\mathcal{A}(1,0,\cdots, 0)$, which is (see Proposition \ref{cohomology})
$$\k[\lceil x_2^2\rceil,\lceil x_2x_3\rceil, \cdots,\lceil x_2x_n\rceil, \lceil x_3^2\rceil, \cdots, \lceil x_3x_n\rceil, \cdots,  \lceil x_{n-1}^2\rceil, \lceil x_{n-1}x_n\rceil, \lceil x_n^2\rceil ].$$
It implies that any DG polynomial algebra $\mathcal{A}(t_1,t_2,\cdots,t_n)$ is a formal, homologically smooth and Gorenstein DG algebra (see Theorem \ref{asreg}).

It is natural for one to ask whether DG polynomial algebras are Calabi-Yau. In \cite{MH}, it is proved that a connected cochain DG algebra $\mathcal{A}$ is a Kozul Calabi-Yau DG algebra if $H(\mathcal{A})$  belongs to one of the following cases:
\begin{align*}
& (a) H(\mathcal{A})\cong \k;  \quad \quad (b) H(\mathcal{A})= \k[\lceil z\rceil], z\in \mathrm{ker}(\partial_{\mathcal{A}}^1); \\
& (c) H(\mathcal{A})= \frac{\k\langle \lceil z_1\rceil, \lceil z_2\rceil\rangle}{(\lceil z_1\rceil\lceil z_2\rceil +\lceil z_2\rceil \lceil z_1\rceil)}, z_1,z_2\in \mathrm{ker}(\partial_{\mathcal{A}}^1).
\end{align*}
Recently, it is proved in \cite[Proposition 6.5]{MHLX} that a connected cochain DG algebra $\mathcal{A}$ is Calabi-Yau if $H(\mathcal{A})=\k[\lceil z_1\rceil, \lceil z_2\rceil]$ where $z_1\in \mathrm{ker}(\partial_{\mathcal{A}}^1)$ and $z_2\in \mathrm{ker}(\partial_{\mathcal{A}}^2)$. And a connected cochain DG algebra $\mathcal{A}$ is not Calabi-Yau if $H(\mathcal{A})=\k[\lceil z_1\rceil, \lceil z_2\rceil]$ where $z_1\in \mathrm{ker}(\partial_{\mathcal{A}}^1)$ and $z_2\in \mathrm{ker}(\partial_{\mathcal{A}}^1)$ (see \cite[Theorem B]{MH}). These motivate us to consider more general cases.

We have the following two theorems (see Theorem \ref{cyprop} and Theorem \ref{degreeone}).
 \\
\begin{bfseries}
Theorem \ C.
\end{bfseries}
 Let $\mathcal{A}$ be a connected cochain DG algebra such that $$H(\mathcal{A})=\k[\lceil y_1\rceil, \cdots, \lceil y_m\rceil ],$$ for some central,  cocycle and degree $2$ elements $y_1,\cdots, y_m$  in $\mathcal{A}$. Then $\mathcal{A}$ is a $-m$-Calabi-Yau DG algebra.
 \\
 \begin{bfseries}
Theorem \ D.
\end{bfseries}
Let $\mathcal{A}$ be a connected cochain DG algebra such that $$H(\mathcal{A})=\k[\lceil y_1\rceil, \cdots, \lceil y_m\rceil ],$$ for some central, cocycle and degree $1$ elements $y_1,\cdots, y_m$  in $\mathcal{A}$. Then $\mathcal{A}$ is a Koszul, homologically smooth and Gorenstein DG algebra. Moreover, $\mathcal{A}$ is Calabi-Yau if and only if $n$ is an odd integer.

With the help of Theorem C and Theorem D, we get the following conclusion (see Corollary \ref{non-zero} and Corollary \ref{zerodiff}): $\mathcal{A}(t_1,t_2,\cdots, t_n)$ is a Calabi-Yau DG algebra if $(t_1,t_2,\cdots, t_n)\neq (0,0,\cdots, 0)$ and $\mathcal{A}(0,0,\cdots, 0)$ is a Calabi-Yau DG algebra if and only if $n$ is an odd integer.

----------------------------------------------------------------------
\section{Notations and conventions}
 We assume that the reader is familiar with basic definitions concerning DG homological
algebra.  If this is not the case, we refer to \cite{AFH, FHT2, MW1,MW2,FJ2} for more details on them. We begin by fixing some notations and terminology.
There are some overlaps here in \cite{MHLX}.

Throughout this paper, $\k$ is an algebraically closed field of characteristic $0$.
For any $\k$-vector space $V$, we write $V^*=\Hom_{\k}(V,\k)$. Let $\{e_i|i\in I\}$ be a basis of a finite dimensional $\k$-vector space $V$.  We denote the dual basis of $V$ by $\{e_i^*|i\in I\}$, i.e., $\{e_i^*|i\in I\}$ is a basis of $V^*$ such that $e_i^*(e_j)=\delta_{i,j}$. For any graded vector space $W$ and $j\in\Bbb{Z}$,  the $j$-th suspension $\Sigma^j W$ of $W$ is a graded vector space defined by $(\Sigma^j W)^i=W^{i+j}$.

A cochain DG algebra is
a graded
$\k$-algebra $\mathcal{A}$ together with a differential $\partial_{\mathcal{A}}: \mathcal{A}\to \mathcal{A}$  of
degree $1$ such that
\begin{align*}
\partial_{\mathcal{A}}(ab) = (\partial_{\mathcal{A}} a)b + (-1)^{|a|}a(\partial_{\mathcal{A}} b)
\end{align*}
for all graded elements $a, b\in \mathcal{A}$.
For any DG algebra $\mathcal{A}$,  we denote $\mathcal{A}\!^{op}$ as its opposite DG
algebra, whose multiplication is defined as
 $a \cdot b = (-1)^{|a|\cdot|b|}ba$ for all
graded elements $a$ and $b$ in $\mathcal{A}$.  A cochain DG algebra $\mathcal{A}$ is called
non-trivial if $\partial_{\mathcal{A}}\neq 0$, and $\mathcal{A}$ is said to be connected if its underlying graded algebra $\mathcal{A}^{\#}$ is a connected  graded algebra. Given a cochain DG algebra $\mathcal{A}$, we denote by $\mathcal{A}^i$ its $i$-th homogeneous component.  The differential $\partial_{\mathcal{A}}$ is a sequence of linear maps $\partial_{\mathcal{A}}^i: \mathcal{A}^i\to \mathcal{A}^{i+1}$ such that $\partial_{\mathcal{A}}^{i+1}\circ \partial_{\mathcal{A}}^i=0$, for all $i\in \Bbb{Z}$. The cohomology graded algebra of $\mathcal{A}$ is the graded algebra $$H(\mathcal{A})=\bigoplus_{i\in \Bbb{Z}}\frac{\mathrm{ker}(\partial_{\mathcal{A}}^i)}{\mathrm{im}(\partial_{\mathcal{A}}^{i-1})}.$$
 For any cocycle element $z\in \mathrm{ker}(\partial_{\mathcal{A}}^i)$, we write $\lceil z \rceil$ as the cohomology class in $H(\mathcal{A})$ represented by $z$. One sees that $H(\mathcal{A})$ is a connected graded algebra if $\mathcal{A}$ is a connected cochain DG algebra.
  For any connected cochain DG algebra $\mathcal{A}$, we denote by $\frak{m}_{\mathcal{A}}$ its maximal DG
ideal $$ \cdots\to 0\to \mathcal{A}^1\stackrel{\partial^1_{\mathcal{A}}}{\to}
\mathcal{A}^2\stackrel{\partial^2_{\mathcal{A}}}{\to} \cdots \stackrel{\partial^{n-1}_{\mathcal{A}}}{\to}
\mathcal{A}^n\stackrel{\partial^n_{\mathcal{A}}}{\to}
 \cdots .$$
 Clearly,  $\k$ has a structure of DG $\mathcal{A}$-module via the augmentation map $$\varepsilon: \mathcal{A}\to \mathcal{A}/\frak{m}_{\mathcal{A}}=\k.$$
It is easy to check that the enveloping DG algebra $\mathcal{A}^e = \mathcal{A}\otimes \mathcal{A}\!^{op}$ of $\mathcal{A}$
is also a connected cochain DG algebra with $H(\mathcal{A}^e)\cong H(\mathcal{A})^e$, and $$\frak{m}_{\mathcal{A}^e}=\frak{m}_{\mathcal{A}}\otimes \mathcal{A}^{op} + \mathcal{A}\otimes
\frak{m}_{\mathcal{A}^{op}}.$$

A morphism $f:\mathcal{A}\to \mathcal{A}'$ of DG algebras is a chain map of complexes which respects multiplication and unit; $f$ is said to be a DG algebra isomorphism (resp. quasi-isomorphism) if $f$ (resp. $H(f)$) is an isomorphism.  A DG algebra isomorphism $f$ is called a DG automorphism when $\mathcal{A}'=\mathcal{A}$. The set of all DG algebra automorphisms of $\mathcal{A}$ is a group, denoted by $\mathrm{Aut}_{dg}(\mathcal{A})$.

Let $\mathcal{A}$ be a connected cochain DG algebra. The derived category of left DG modules over $\mathcal{A}$ (DG $\mathcal{A}$-modules for short) is denoted by $\mathrm{D}(\mathcal{A})$.  A DG $\mathcal{A}$-module  $M$ is compact if the functor $\Hom_{\mathrm{D}(A)}(M,-)$ preserves
all coproducts in $\mathrm{D}(\mathcal{A})$.
 By \cite[Proposition 3.3]{MW1},
a DG $\mathcal{A}$-module  is compact if and only if it admits a minimal semi-free resolution with a finite semi-basis. The full subcategory of $\mathrm{D}(\mathcal{A})$ consisting of compact DG $\mathcal{A}$-modules is denoted by $\mathrm{D}^c(\mathcal{A})$.

In the rest of this section, we review some important homological properties for DG algebras.
 We have the following definitions.
\begin{defn}\label{basicdef}
{\rm Let $\mathcal{A}$ be a connected cochain DG algebra.
\begin{enumerate}
\item  If $\dim_{\k}H(R\Hom_{\mathcal{A}}(\k,\mathcal{A}))=1$, then $\mathcal{A}$ is called Gorenstein (cf. \cite{FHT1});
\item  If $\mathcal{A}$ can be connected with the trivial DG algebra $(H(\mathcal{A}),0)$ by a zig-zag $\leftarrow \rightarrow\leftarrow \cdots\rightarrow $ of quasi-isomorphisms, then $\mathcal{A}$ is called formal (cf.\cite{Kal,Lunt});
\item  If ${}_{\mathcal{A}}\k$, or equivalently ${}_{\mathcal{A}^e}\mathcal{A}$, has a minimal semi-free resolution with a semi-basis concentrated in degree $0$, then $\mathcal{A}$ is called Koszul (cf. \cite{HW});
\item If ${}_{\mathcal{A}}\k$, or equivalently the DG $\mathcal{A}^e$-module $\mathcal{A}$ is compact, then $\mathcal{A}$ is called homologically smooth (cf. \cite[Corollary 2.7]{MW3});
\item If $\mathcal{A}$ is homologically smooth and $$R\Hom_{\mathcal{A}^e}(\mathcal{A}, \mathcal{A}^e)\cong
\Sigma^{-n}\mathcal{A}$$ in  the derived category $\mathrm{D}((\mathcal{A}^e)^{op})$ of right DG $\mathcal{A}^e$-modules, then $\mathcal{A}$ is called an $n$-Calabi-Yau DG algebra  (cf. \cite{Gin,VdB}).
\end{enumerate}}
 \end{defn}
 The motivation of this paper is to study whether DG polynomial algebras have these homological properties mentioned in Definition \ref{basicdef}.

\section{differential structures on polynomial algebra}
In this section, we study the differential structures of DG polynomial algebras. We have the following theorem.
\begin{thm}\label{diffstructure}
Let $(\mathcal{A},\partial_{\mathcal{A}})$ be a connected cochain DG algebra such that $\mathcal{A}^{\#}$ is a polynomial graded algebra $\k[x_1,x_2,\cdots,x_n]$ with $|x_i|=1$, for any $i\in \{1,2,\cdots,n\}$.
Then there exist some $t_1,t_2,\cdots,t_n\in \k$ such that $\partial_{\mathcal{A}}$ is defined  by
$$\partial_{\mathcal{A}}(x_i)=\sum\limits_{j=1}^nt_jx_ix_j=\sum\limits_{j=1}^{i-1}t_jx_jx_i+t_ix_i^2+\sum\limits_{j=i+1}^nt_jx_ix_j, \forall i\in \{1,2,\cdots,n\}. $$  Conversely, for any point $(t_1,t_2,\cdots, t_n)\in \Bbb{A}_{\k}^n$, we can define a differential $\partial$ on $\k[x_1,x_2,\cdots,x_n]$ by
$$\partial(x_i)=\sum\limits_{j=1}^{i-1}t_jx_jx_i+t_ix_i^2+\sum\limits_{j=i+1}^nt_jx_ix_j,   \forall i\in \{1,2,\cdots,n\},$$
such that $(\k[x_1,x_2,\cdots,x_n],\partial)$ is a DG polynomial algebra.
\end{thm}
\begin{proof}
Since the differential $\partial_{\mathcal{A}}$ of $\mathcal{A}$ is a $\k$-linear map of degree $1$, we may let
$$
\partial_{\mathcal{A}}(x_i) = \sum\limits_{j=1}^n\sum\limits_{k=j}^nc^i_{j,k}x_jx_k
$$
where $c^i_{j,k}\in \k$, for any $i, j\in \{1,2,\cdots,n\}$ and $k\in \{j+1,\cdots, n\}$
 By
definition, $(\mathcal{A},\partial_{\mathcal{A}})$ is a cochain DG
algebra if and only if $\partial_{\mathcal{A}}$ satisfies the
Leibniz rule and
\begin{align}\label{eqs}
\begin{cases}
\partial_{\mathcal{A}}\circ \partial_{\mathcal{A}}(x_i) = 0, \forall i\in\{1,2,\cdots,n\}   \\
\partial_{\mathcal{A}}(x_jx_k-x_kx_j) = 0, \forall 1\le j<k\le n.
\end{cases}
\end{align}
Since \begin{align*} \partial_{\mathcal{A}}(x_jx_k-x_kx_j)&=\partial_{\mathcal{A}}(x_j)x_k-x_j\partial_{\mathcal{A}}(x_k)-\partial_{\mathcal{A}}(x_k)x_j+x_k\partial_{\mathcal{A}}(x_j)\\
&=2\partial_{\mathcal{A}}(x_j)x_k-2x_j\partial_{\mathcal{A}}(x_k)\\
&=2[\sum\limits_{p=1}^n\sum\limits_{q=p}^nc^j_{p,q}x_px_q]x_k-2x_j[\sum\limits_{p=1}^n\sum\limits_{q=p}^nc^k_{p,q}x_px_q]\\
&=2[\sum\limits_{p=1}^n\sum\limits_{q=p}^nc^j_{p,q}x_px_q]x_k-2x_j[\sum\limits_{q=1}^n\sum\limits_{p=1}^qc^k_{p,q}x_px_q],
\end{align*}
\begin{align*}
&\quad\quad\quad [\sum\limits_{p=1}^n\sum\limits_{q=p}^nc^j_{p,q}x_px_q]x_k \\
&=\sum\limits_{p=1}^{j-1}\sum\limits_{q=p}^{j-1}c_{p,q}^jx_px_qx_k +\sum\limits_{p=1}^{j-1}c_{p,j}^jx_px_jx_k +\sum\limits_{p=1}^{j-1}\sum\limits_{q=j+1}^nc_{p,q}^jx_px_qx_k+c_{j,j}^jx_j^2x_k\\
&\quad \quad +\sum\limits_{q=j+1}^nc_{j,q}^jx_jx_qx_k +\sum\limits_{p=j+1}^n\sum\limits_{q=p}^nc_{p,q}^jx_px_qx_k\\
&=R_1+R_2
\end{align*}
and
\begin{align*}
&\quad\quad\quad x_j[\sum\limits_{q=1}^n\sum\limits_{p=1}^qc^k_{p,q}x_px_q] \\
&=\sum\limits_{q=1}^{k-1}\sum\limits_{p=1}^qc_{p,q}^kx_jx_px_q + \sum\limits_{p=1}^{k-1}c_{p,k}^kx_jx_px_k +c_{k,k}^kx_jx_k^2+\sum\limits_{q=k+1}^n\sum\limits_{p=1}^{k-1}c_{p,q}^kx_jx_px_q\\
&\quad\quad +\sum\limits_{q=k+1}^nc_{k,q}^kx_jx_kx_q+\sum\limits_{q=k+1}^n\sum\limits_{p=k+1}^qc_{p,q}^kx_jx_px_q\\
&=S_1+S_2,
\end{align*}
where
\begin{align*}
\begin{cases}
R_1= \sum\limits_{p=1}^{j-1}\sum\limits_{q=p}^{j-1}c_{p,q}^jx_px_qx_k+\sum\limits_{p=1}^{j-1}\sum\limits_{q=j+1}^nc_{p,q}^jx_px_qx_k+\sum\limits_{p=j+1}^n\sum\limits_{q=p}^nc_{p,q}^jx_px_qx_k \\
R_2= \sum\limits_{p=1}^{j-1}c_{p,j}^jx_px_jx_k + c_{j,j}^jx_j^2x_k +\sum\limits_{q=j+1}^nc_{j,q}^jx_jx_qx_k  \\
S_1= \sum\limits_{q=1}^{k-1}\sum\limits_{p=1}^qc_{p,q}^kx_jx_px_q + \sum\limits_{q=k+1}^n\sum\limits_{p=1}^{k-1}c_{p,q}^kx_jx_px_q + \sum\limits_{q=k+1}^n\sum\limits_{p=k+1}^qc_{p,q}^kx_jx_px_q \\
S_2=\sum\limits_{p=1}^{k-1}c_{p,k}^kx_jx_px_k +c_{k,k}^kx_jx_k^2+\sum\limits_{q=k+1}^nc_{k,q}^kx_jx_kx_q,
\end{cases}
\end{align*}
we obtain that $\partial_{\mathcal{A}}(x_jx_k-x_kx_j) = 0$ if and only if $R_1+R_2-S_1-S_2=0$, which is equivalent to
\begin{align*}
\begin{cases}
R_1=0\\
S_1=0\\
R_2-S_2=0.
\end{cases}
\end{align*}
Since
$$R_1=[\sum\limits_{p=1}^{j-1}\sum\limits_{q=p}^{j-1}c_{p,q}^jx_px_q+\sum\limits_{p=1}^{j-1}\sum\limits_{q=j+1}^nc_{p,q}^jx_px_q+\sum\limits_{p=j+1}^n\sum\limits_{q=p}^nc_{p,q}^jx_px_q]x_k
$$
and $$S_1= x_j[\sum\limits_{q=1}^{k-1}\sum\limits_{p=1}^qc_{p,q}^kx_px_q + \sum\limits_{q=k+1}^n\sum\limits_{p=1}^{k-1}c_{p,q}^kx_px_q + \sum\limits_{q=k+1}^n\sum\limits_{p=k+1}^qc_{p,q}^kx_px_q],$$
we get that $R_1=S_1=0$ if and only if
\begin{align*}
\begin{cases}
 c_{p,q}^j=0, \forall p\in \{1,\cdots, j-1\}, q\in \{p,\cdots,j-1\}\\
c_{p,q}^j=0, \forall p\in \{1,\cdots,j-1\}, q\in \{j+1,\cdots,n\} \\
c_{p,q}^j=0, \forall p\in \{j+1,\cdots, n\}, q\in\{p,\cdots,n\}
\end{cases}
\end{align*}
and
\begin{align*}
\begin{cases}
c_{p,q}^k=0, \forall q\in \{1,\cdots, k-1\}, p\in \{1,\cdots, q\}\\
c_{p,q}^k=0, \forall q\in \{k+1,\cdots, n\}, p\in \{1,\cdots, k-1\}\\
c_{p,q}^k=0, \forall q\in \{k+1,\cdots, n\}, p\in \{k+1,\cdots, q\},
\end{cases}
\end{align*}
which are equivalent to
\begin{align*}
\begin{cases}
c_{p,q}^j=0, \text{if}\quad p\neq j \quad \text{and}\quad q\neq j\\
c_{p,q}^k=0, \text{if}\quad p\neq k \quad \text{and}\quad q\neq k.
\end{cases}
\end{align*}
Since \begin{align*}
R_2-S_2&=\sum\limits_{p=1}^{j-1}c_{p,j}^jx_px_jx_k + c_{j,j}^jx_j^2x_k +\sum\limits_{q=j+1}^nc_{j,q}^jx_jx_qx_k \\
 & \quad\quad- [\sum\limits_{p=1}^{k-1}c_{p,k}^kx_jx_px_k +c_{k,k}^kx_jx_k^2+\sum\limits_{q=k+1}^nc_{k,q}^kx_jx_kx_q]\\
&=\sum\limits_{p=1}^{j-1}(c_{p,j}^j-c_{p,k}^k)x_px_jx_k +c_{j,j}^jx_j^2x_k + \sum\limits_{q=k+1}^n(c_{j,q}^j-c_{k,q}^k)x_jx_kx_q  \\
& \quad\quad+\sum\limits_{q=j+1}^{k-1}c_{j,q}^jx_jx_qx_k + c_{j,k}^jx_jx_k^2- c_{j,k}^kx_j^2x_k -\sum\limits_{p=j+1}^{k-1}c_{p,k}^kx_jx_px_k-c_{k,k}^kx_jx_k^2 \\
&=\sum\limits_{p=1}^{j-1}(c_{p,j}^j-c_{p,k}^k)x_px_jx_k +(c_{j,j}^j-c_{j,k}^k)x_j^2x_k+\sum\limits_{q=k+1}^n(c_{j,q}^j-c_{k,q}^k)x_jx_kx_q\\
& \quad\quad+ \sum\limits_{r=j+1}^{k-1}(c_{j,r}^j- c_{r,k}^k )x_jx_rx_k +(c_{j,k}^j -c_{k,k}^k )x_jx_k^2,
\end{align*}
we conclude that $R_2-S_2=0$ if and only if
$$
\begin{cases}
c_{p,j}^j=c_{p,k}^k, \forall p\in \{1,2,\cdots, j-1\}\\
c_{j,j}^j=c_{j,k}^k \\
c_{j,q}^j=c_{k,q}^k, \forall q\in \{k+1,k+2,\cdots, n\} \\
c_{j,r}^j=c_{r,k}^k, \forall r\in \{ j+1,j+2,\cdots, k-1\}\\
c_{j,k}^j=c_{k,k}^k,
\end{cases}
$$
which is equivalent to
\begin{align*}
\begin{cases}
c_{p,j}^j=c_{p,k}^k, \forall p\in \{1,2,\cdots, j\}\\
c_{j,r}^j=c_{r,k}^k, \forall r\in \{ j+1,j+2,\cdots, k-1\}\\
c_{j,q}^j=c_{k,q}^k, \forall q\in \{k,k+2,\cdots, n\}.
\end{cases}
\end{align*}
Therefore, for any $1\le j<k\le n$, $\partial_{\mathcal{A}}(x_jx_k-x_kx_j) = 0$ if and only if
\begin{align}\label{fincond}
\begin{cases}
c_{p,q}^j=0, \text{if}\quad p\neq j \quad \text{and}\quad q\neq j\\
c_{p,q}^k=0, \text{if}\quad p\neq k \quad \text{and}\quad q\neq k\\
c_{p,j}^j=c_{p,k}^k, \forall p\in \{1,2,\cdots, j\}\\
c_{j,r}^j=c_{r,k}^k, \forall r\in \{ j+1,j+2,\cdots, k-1\}\\
c_{j,q}^j=c_{k,q}^k, \forall q\in \{k,k+2,\cdots, n\}.
\end{cases}
\end{align}\label{diff}
Let $t_i=c_{1,i}^1, i=1,2,\cdots, n$. Then (\ref{fincond}) implies \begin{align}
\partial_{\mathcal{A}}(x_i)=\sum\limits_{j=1}^{i-1}t_jx_jx_i+t_ix_i^2+\sum\limits_{j=i+1}^nt_jx_ix_j, \forall i\in \{1,2,\cdots,n\}.
\end{align}
For any $(t_1,t_2,\cdots, t_n)\in \Bbb{A}_{\k}^n$, we claim that $\partial_{\mathcal{A}}\circ\partial_{\mathcal{A}}(x_i)=0$, if (\ref{diff}) holds,
$\forall i\in \{1,2,\cdots, n\}$. Indeed, if (\ref{diff}) holds, then
\begin{align*}
&\quad\quad\quad \partial_{\mathcal{A}}\circ\partial_{\mathcal{A}}(x_i)=\partial_{\mathcal{A}}[\sum\limits_{j=1}^{i-1}t_jx_jx_i+t_ix_i^2+\sum\limits_{j=i+1}^nt_jx_ix_j]\\
&=\sum\limits_{j=1}^{i-1}t_j\partial_{\mathcal{A}}(x_j)x_i-\sum\limits_{j=1}^{i-1}t_jx_j\partial_{\mathcal{A}}(x_i)+\sum\limits_{j=i+1}^nt_j\partial_{\mathcal{A}}(x_i)x_j-\sum\limits_{j=i+1}^nt_jx_i\partial_{\mathcal{A}}(x_j)\\
&=\sum\limits_{j=1}^{i-1}t_j[(\sum\limits_{l=1}^{j-1}t_lx_lx_j+t_jx_j^2+\sum\limits_{l=j+1}^nt_lx_jx_l)x_i-x_j(\sum\limits_{l=1}^{i-1}t_lx_lx_i+t_ix_i^2+ \sum\limits_{l=i+1}^nt_lx_ix_l)]\\
& + \sum\limits_{j=i+1}^nt_j[(\sum\limits_{l=1}^{i-1}t_lx_lx_i+t_ix_i^2+ \sum\limits_{l=i+1}^nt_lx_ix_l)x_j-x_i(\sum\limits_{l=1}^{j-1}t_lx_lx_j+t_jx_j^2+\sum\limits_{l=j+1}^nt_lx_jx_l)]\\
&=\sum\limits_{j=1}^{i-1}t_jx_i[\sum\limits_{l=1}^{j-1}t_lx_lx_j+t_jx_j^2+\sum\limits_{l=j+1}^nt_lx_jx_l-x_j(\sum\limits_{l=1}^{i-1}t_lx_l+t_ix_i+ \sum\limits_{l=i+1}^nt_lx_l)]\\
& + \sum\limits_{j=i+1}^nt_jx_i[(\sum\limits_{l=1}^{i-1}t_lx_l+t_ix_i+ \sum\limits_{l=i+1}^nt_lx_l)x_j-(\sum\limits_{l=1}^{j-1}t_lx_lx_j+t_jx_j^2+\sum\limits_{l=j+1}^nt_lx_jx_l)].
\end{align*}
Since
\begin{align*}
&\sum\limits_{j=1}^{i-1}t_j[\sum\limits_{l=1}^{j-1}t_lx_lx_j+t_jx_j^2+\sum\limits_{l=j+1}^nt_lx_jx_l-x_j(\sum\limits_{l=1}^{i-1}t_lx_l+t_ix_i+ \sum\limits_{l=i+1}^nt_lx_l)]\\
& + \sum\limits_{j=i+1}^nt_j[(\sum\limits_{l=1}^{i-1}t_lx_l+t_ix_i+ \sum\limits_{l=i+1}^nt_lx_l)x_j-(\sum\limits_{l=1}^{j-1}t_lx_lx_j+t_jx_j^2+\sum\limits_{l=j+1}^nt_lx_jx_l)]\\
&=\sum\limits_{j=1}^{i-1}t_j[-\sum\limits_{l=j}^{i-1}t_lx_lx_j+t_jx_j^2-t_ix_ix_j + \sum\limits_{l=j+1}^it_lx_jx_l]\\
&\quad +\sum\limits_{j=i+1}^nt_j[ -\sum\limits_{l=i}^{j-1}t_lx_lx_j +t_ix_ix_j+\sum\limits_{l=i+1}^jt_lx_lx_j-t_jx_j^2                                                                           ]\\
&=\sum\limits_{j=1}^{i-1}t_j[-\sum\limits_{l=j+1}^{i-1}t_lx_lx_j +\sum\limits_{l=j+1}^{i-1}t_lx_jx_l]+\sum\limits_{j=i+1}^nt_j[ -\sum\limits_{l=i+1}^{j-1}t_lx_lx_j+\sum\limits_{l=i+1}^{j-1}t_lx_lx_j] \\
&=0,
\end{align*}
we have $\partial_{\mathcal{A}}^2(x_i)=0$,  $\forall i\in \{1,2,\cdots,n\}$.

Therefore, $(\mathcal{A},\partial_{\mathcal{A}})$ is a cochain DG algebra if there exist some $t_1,t_2,\cdots,t_n\in \k$ such that
$$\partial_{\mathcal{A}}(x_i)=\sum\limits_{j=1}^{i-1}t_jx_jx_i+t_ix_i^2+\sum\limits_{j=i+1}^nt_jx_ix_j, \forall i\in \{1,2,\cdots,n\}.$$
Conversely, $\forall (t_1,t_2,\cdots, t_n)\in \Bbb{A}_{\k}^n$, we can define a differential $\partial$ on $\k[x_1,x_2,\cdots,x_n]$ by
$$\partial(x_i)=\sum\limits_{j=1}^{i-1}t_jx_jx_i+t_ix_i^2+\sum\limits_{j=i+1}^nt_jx_ix_j,   \forall i\in \{1,2,\cdots,n\},$$
such that $(\k[x_1,x_2,\cdots,x_n],\partial)$ is a DG polynomial algebra, since one can check as above that $$
\begin{cases}
\partial \circ \partial (x_i) = 0, \forall i\in\{1,2,\cdots,n\}   \\
\partial (x_jx_k-x_kx_j) = 0, \forall 1\le j<k\le n,
\end{cases}
$$
if $\partial$ satisfies the
Leibniz rule.
\end{proof}
By Theorem \ref{diffstructure}, the following definition is reasonable.
\begin{defn}{\rm
 Define $\mathcal{A}(t_1,t_2,\cdots, t_n)$ as a cochain DG algebra such that
$$\mathcal{A}(t_1,t_2,\cdots, t_n)^{\#}=\k[x_1,x_2,\cdots,x_n],\quad |x_i|=1,$$ and its differential $\partial_{\mathcal{A}}$ is defined by $$\partial_{\mathcal{A}}(x_i)=\sum\limits_{j=1}^{i-1}t_jx_jx_i+t_ix_i^2+\sum\limits_{j=i+1}^nt_jx_ix_j, \forall i\in \{1,2,\cdots,n\}.$$
Define
$\Omega(x_1,x_2,\cdots,x_n)=\{\mathcal{A}(t_1,t_2,\cdots,t_n)|t_i\in
\k, i=1,2,\cdots, n\}$.
Clearly, $$\Omega(x_1,x_2,\cdots,x_n)\cong \Bbb{A}_{\k}^n.$$}
\end{defn}

\section{isomorphism classes of  dg  polynomial algebras}
In this section, we consider the isomorphism problem for DG polynomial algebras in $\Omega(x_1,x_2,\cdots,x_n)$. We have the following theorem.
\begin{thm}\label{isom}
Let $\mathcal{A}(t_1,t_2,\cdots,t_n)$ and
$\mathcal{A}(t_1',t_2',\cdots,t_n')$ be two points in the
space $\Omega(x_1,x_2,\cdots,x_n)$. Then the DG algebras
$$\mathcal{A}(t_1,t_2,\cdots,t_n)\cong
\mathcal{A}(t_1',t_2',\cdots,t_n')$$ if and only if there
is a matrix $M\in \mathrm{GL}_n(\k)$ such that
$$(t_1',t_2',\cdots,t_n')=(t_1,t_2,\cdots,t_n)M.$$
\end{thm}
\begin{proof}
We write
$\mathcal{A}=\mathcal{A}(t_1,t_2,\cdots,t_n)$ and
$\mathcal{A}'= \mathcal{A}(t_1',t_2',\cdots, t_n')$ for
simplicity. In order to distinguish, we assume that $\mathcal{A}'^{\#}=\k[x_1',x_2',\cdots,x_n']$ with  $|x_i'|=1$, for any $i\in \{1,2,\cdots,n\}$.
If the DG
algebras $\mathcal{A}\cong \mathcal{A}'$, then there exists an
isomorphism $f: \mathcal{A}\to \mathcal{A}'$ of DG algebras. Since
$f^1: \mathcal{A}^1 \to \mathcal{A}'^1$ is a $\k$-linear
isomorphism, we may let $$\left(\begin{array}{c}
                           f(x_1) \\
                           f(x_2) \\
                           \vdots \\
                           f(x_n)
                         \end{array}
                       \right)= M\left(
                         \begin{array}{c}
                           x_1' \\
                           x_2' \\
                           \vdots \\
                           x_n'
                         \end{array}
                       \right)$$
for some  $M=(a_{ij})_{n\times n} \in \mathrm{GL}_n(\k)$.
Since $f$ is a chain map, we have $f\circ \partial_{\mathcal{A}}=\partial_{\mathcal{A}'}\circ f$.
For any $i\in \{1,2,\cdots,n\}$, we have
\begin{align*}\label{Eq1}\tag{Eq1}
&\quad\quad\quad \partial_{\mathcal{A}'}\circ f(x_i)=\partial_{\mathcal{A}'}(\sum\limits_{j=1}^na_{ij}x_j')\\
&=\sum\limits_{j=1}^na_{ij}(\sum\limits_{l=1}^{j-1}t_l'x_l'x_j'+t_j'x_j'^2+\sum\limits_{l=j+1}^nt_l'x_j'x_l')\\
&=\sum\limits_{j=1}^na_{ij}x_j'(\sum\limits_{l=1}^{j-1}t_l'x_l'+t_j'x_j'+\sum\limits_{l=j+1}^nt_l'x_l')\\
&=(\sum\limits_{j=1}^na_{ij}x_j')(\sum\limits_{l=1}^nt_l'x_l')\\
&=(\sum\limits_{l=1}^na_{il}x_l')(\sum\limits_{s=1}^nt_s'x_s')
\end{align*}
and
\begin{align*}\label{Eq2}\tag{Eq2}
&\quad \quad\quad\quad f\circ \partial_{\mathcal{A}}(x_i)=f(\sum\limits_{j=1}^{i-1}t_jx_jx_i+t_ix_i^2+\sum\limits_{j=i+1}^nt_jx_ix_j)\\
&=\sum\limits_{j=1}^{i-1}t_j(\sum\limits_{l=1}^na_{jl}x_l')(\sum\limits_{l=1}^na_{il}x_l')+t_i(\sum\limits_{l=1}^na_{il}x_l')^2+\sum\limits_{j=i+1}^nt_j(\sum\limits_{l=1}^na_{il}x_l')(\sum\limits_{l=1}^na_{jl}x_l')\\
&=\sum\limits_{j=1}^{i-1}t_j(\sum\limits_{s=1}^na_{js}x_s')(\sum\limits_{l=1}^na_{il}x_l')+t_i(\sum\limits_{l=1}^na_{il}x_l')^2+\sum\limits_{j=i+1}^nt_j(\sum\limits_{l=1}^na_{il}x_l')(\sum\limits_{s=1}^na_{js}x_s')\\
&=(\sum\limits_{l=1}^na_{il}x_l')[\sum\limits_{j=1}^{i-1}t_j(\sum\limits_{s=1}^na_{js}x_s')+t_i(\sum\limits_{l=1}^na_{il}x_l')+\sum\limits_{j=i+1}^nt_j(\sum\limits_{s=1}^na_{js}x_s')]\\
&=(\sum\limits_{l=1}^na_{il}x_l')[\sum\limits_{j=1}^{i-1}t_j(\sum\limits_{s=1}^na_{js}x_s')+t_i(\sum\limits_{s=1}^na_{is}x_s')+\sum\limits_{j=i+1}^nt_j(\sum\limits_{s=1}^na_{js}x_s')]\\
&=(\sum\limits_{l=1}^na_{il}x_l')[\sum\limits_{j=1}^nt_j(\sum\limits_{s=1}^na_{js}x_s')]\\
&=(\sum\limits_{l=1}^na_{il}x_l')[\sum\limits_{s=1}^n(\sum\limits_{j=1}^na_{js}t_j)x_s'].
\end{align*}
Since $(a_{ij})_{n\times n} \in \mathrm{GL}_n(\k)$, we have $(a_{i1},a_{i2},\cdots,a_{in})\neq (0,0,\cdots, 0)$. So $\sum\limits_{l=1}^na_{il}x_l'\neq 0$ and $ \partial_{\mathcal{A}'}\circ f(x_i)=f\circ \partial_{\mathcal{A}}(x_i)$ implies that $t_s'=\sum\limits_{j=1}^na_{js}t_j$, for any $s\in \{1,2,\cdots,n\}$. Then we get
 $$(t_1',t_2',\cdots,t_n')=(t_1,t_2,\cdots,t_n)M.$$

 Conversely, if there exists a matrix $M=(a_{ij})_{n\times n}\in \mathrm{GL}_n(\k)$ such that $$(t_1',t_2',\cdots,t_n')=(t_1,t_2,\cdots,t_n)M,$$ then $t_s'=\sum\limits_{j=1}^na_{js}t_j$, for any $s\in \{1,2,\cdots,n\}$. Hence
 \begin{equation}\label{chainmap}(\sum\limits_{l=1}^na_{il}x_l')(\sum\limits_{s=1}^nt_s'x_s')=(\sum\limits_{l=1}^na_{il}x_l')[\sum\limits_{s=1}^n(\sum\limits_{j=1}^na_{js}t_j)x_s'], \forall i\in \{1,2,\cdots,n\}.
 \end{equation}
 Define a linear map $f: \mathcal{A}^1 \to \mathcal{A}'^1$
                        by
 $$\left(\begin{array}{c}
                           f(x_1) \\
                           f(x_2) \\
                           \vdots \\
                           f(x_n)
                         \end{array}
                       \right)= M\left(
                         \begin{array}{c}
                           x_1' \\
                           x_2' \\
                           \vdots \\
                           x_n'
                         \end{array}
                       \right).$$
 Obviously, $f$ is invertible since $M\in \mathrm{GL}_n(\k)$.  We have $$f(x_i)f(x_j)=f(x_j)f(x_i), \quad \forall 1\le i< j\le n,$$ since $\mathcal{A}'$ is a commutative algebra. Hence $f: \mathcal{A}^1\to \mathcal{A}'^1$ can be extended to a morphism of graded algebras between $\k[x_1,x_2,\cdots, x_n]$ and $\k[x_1',x_2',\cdots, x_n']$. We still denote it by $f$. For any $i\in \{1,2,\cdots,n\}$, we still have (\ref{Eq1}) and (\ref{Eq2}). Then (\ref{chainmap}) indicates that $f\circ \partial_{\mathcal{A}}(x_i)=\partial_{\mathcal{A}'}\circ f(x_i)$, which implies that $f$ is a chain map. Hence, $f:A\to A'$ is an isomorphism of DG algebras.
\end{proof}

\begin{cor}\label{isomor}
In space $\Omega(x_1,x_2,\cdots, x_n)$, there are only two isomorphism
 classes $\mathcal{A}(0,0,\cdots,0)$ and $\mathcal{A}(1,0,\cdots, 0)$.
\end{cor}
\begin{proof}
By Theorem \ref{isom}, $\mathcal{A}(t_1,t_2,\cdots,t_n)\cong \mathcal{A}(t_1',t_2',\cdots,t_n')$ if and only if there is $M\in \mathrm{GL}_n(\k)$ such that $(t_1',t_2',\cdots,t_n')=(t_1,t_2,\cdots,t_n)M$. For any $(t_1,t_2,\cdots,t_n)\neq (0,0,\cdots,0)$, there exists $i\in \{1,2,\cdots, n\}$ such that $t_i\neq 0$.
We have $$(t_1,t_2,\cdots, t_n)=(1,0,\cdots,0)\left(
                         \begin{array}{cccccccc}
                           t_1 & t_2& \cdots &t_{i-1} &t_i&t_{i+1} &\cdots & t_n \\
                           1 & 0  &\cdots  &0&0&0 &\cdots & 0 \\
                           0& 1&\cdots &0&0&0&\cdots & 0 \\
                           \vdots &\vdots &\ddots &\vdots &\vdots &\vdots &\ddots &\vdots \\
                           0&0&\cdots &1&0&0&\cdots & 0\\
                           0&0&\cdots &0&0&1&\cdots &0 \\
                           \vdots &\vdots &\ddots &\vdots &\vdots &\vdots &\ddots &\vdots \\
                           0&0&\cdots &0&0&0&\cdots &1 \\
                         \end{array}
                       \right).$$
Since $$\left|
                       \begin{array}{cccccccc}
                           t_1 & t_2& \cdots &t_{i-1} &t_i&t_{i+1} &\cdots & t_n \\
                           1 & 0  &\cdots  &0&0&0 &\cdots & 0 \\
                           0& 1&\cdots &0&0&0&\cdots & 0 \\
                           \vdots &\vdots &\ddots &\vdots &\vdots &\vdots &\ddots &\vdots \\
                           0&0&\cdots &1&0&0&\cdots & 0\\
                           0&0&\cdots &0&0&1&\cdots &0 \\
                           \vdots &\vdots &\ddots &\vdots &\vdots &\vdots &\ddots &\vdots \\
                           0&0&\cdots &0&0&0&\cdots &1 \\
                         \end{array}
                        \right| = (-1)^{i+1}t_i\neq 0,$$
we have $\mathcal{A}(t_1,t_2,\cdots,t_n)\cong \mathcal{A}(1,0,\cdots,0)$.
Clearly, $\mathcal{A}(1,0,\cdots,0)\neq \mathcal{A}(0,0,\cdots,0)$
since $(1,0,\cdots,0)\neq (0,0,\cdots,0)M$ for any $M\in \mathrm{GL}_n(\k)$. Therefore, there are only two isomorphism classes $\mathcal{A}(0,0,\cdots,0)$ and $\mathcal{A}(1,0,\cdots,0)$ in $\Omega(x_1,x_2,\cdots,x_n)$.
\end{proof}

\begin{cor}\label{auto} For any $\mathcal{A}(t_1,t_2,\cdots,t_n)\in \Omega(x_1,x_2,\cdots,x_n)$, we have
$$\mathrm{Aut}_{dg}(\mathcal{A}(t_1,t_2,\cdots,t_n))\cong \{M\in \mathrm{GL}_n(\k)| (t_1,t_2,\cdots,t_n)=(t_1,t_2,\cdots, t_n)M   \}.$$
\end{cor}

\begin{rem}
We want to consider the homological properties of DG polynomial algebras. This can be reduced to studying that of $\mathcal{A}(0,0,\cdots,0)$ and $\mathcal{A}(1,0,\cdots, 0)$, by Corollary \ref{isomor}. Corollary \ref{auto} characterizes the automorphism group of any DG polynomial algebra.
\end{rem}

\section{cohomology of polynomial dg algebras}
Generally, the cohomology algebra of a DG algebra usually contains much useful information on its properties (cf.\cite{AT,DGI}). In this section, we will compute the cohomology algebra of non-trivial DG polynomial algebra. By Corollary \ref{isomor}, we only need to compute $H(\mathcal{A}(1,0,\cdots,0))$. We have the following proposition.
\begin{prop}\label{cohomology}
 The cohomology algebra of $\mathcal{A}(1,0,\cdots,0)$ is the polynomial algebra  $$\k[\lceil x_2^2\rceil,\lceil x_2x_3\rceil, \cdots,\lceil x_2x_n\rceil, \lceil x_3^2\rceil, \cdots, \lceil x_3x_n\rceil, \cdots,  \lceil x_{n-1}^2\rceil, \lceil x_{n-1}x_n\rceil, \lceil x_n^2\rceil ].$$
\end{prop}
\begin{proof}
For simplicity, let $\mathcal{A}=\mathcal{A}(1,0,\cdots,0)$. We have $\partial_{\mathcal{A}}(x_i)=x_1x_i$, for any $i\in \{1,2,\cdots, n\}$. Hence, $$\mathrm{im}(\partial_{\mathcal{A}}^1)=\bigoplus_{i=1}^n \k x_1x_i.$$
For any $i,j\in \{1,2,\cdots,n\}$, we have
$$\partial_{\mathcal{A}}(x_ix_j)=\partial_{\mathcal{A}}(x_i)x_j-x_i\partial_{\mathcal{A}}(x_j) =x_1x_ix_j-x_ix_1x_j
                              =0.$$
Then $\mathrm{ker}(\partial_{\mathcal{A}}^2)=\mathcal{A}^2$ and hence $\mathrm{ker} (\partial_{\mathcal{A}}^{2k})=\mathcal{A}^{2k}$ by the Leibniz rule, for any $k\ge 2$.
Since $\mathcal{A}^2=\bigoplus_{i=1}^n\bigoplus_{j=i}^n \k x_ix_j$, we have $$H^2(\mathcal{A})=\bigoplus_{i=2}^n\bigoplus_{j=i}^n \k x_ix_j.$$
Since $\partial_{\mathcal{A}}(x_i)=x_1x_i$ and $\mathrm{ker}(\partial_{\mathcal{A}}^{2k-2})=\mathcal{A}^{2k-2}$, it is easy to check that
$$\mathrm{im}(\partial_{\mathcal{A}}^{2k-1})=\bigoplus_{\omega_1=1}^{2k}\bigoplus_{\stackrel{\sum\limits_{j=2}^n\omega_j=2k-\omega_1}{x_j\ge 0, j=2,\cdots,n}}\k x_1^{\omega_1}x_2^{\omega_2}\cdots x_n^{\omega_n}.$$ Since $$\mathcal{A}^{2k}=\bigoplus_{\stackrel{\sum\limits_{j=1}^n\omega_j=2k}{x_j\ge 0, j=1,\cdots,n}}\k x_1^{\omega_1}x_2^{\omega_2}\cdots x_n^{\omega_n},$$ we have $$H^{2k}(\mathcal{A})=\bigoplus_{\stackrel{\sum\limits_{j=2}^n\omega_j=2k}{x_j\ge 0, j=2,\cdots,n}}\k x_2^{\omega_2}\cdots x_n^{\omega_n}.$$
For any $k\ge 2$, any cocycle element in $\mathcal{A}^{2k+1}$ can be written as
$\sum\limits_{i=1}^nx_if_i$ for some $f_i\in \mathcal{A}^{2k}, i=1,2,\cdots, n$. We have $\partial_{\mathcal{A}}(\sum\limits_{i=1}^nx_if_i)=\sum\limits_{i=1}^nx_1x_if_i=x_1\sum\limits_{i=1}^nx_if_i=0$.
So $\sum\limits_{i=1}^nx_if_i=0$. Hence, $\mathrm{ker}(\partial_{\mathcal{A}}^{2k+1})=0$ and then $H^{2k+1}(\mathcal{A})=0$. Therefore,
$$H(\mathcal{A})=\k[\lceil x_2^2\rceil,\lceil x_2x_3\rceil, \cdots,\lceil x_2x_n\rceil, \lceil x_3^2\rceil, \cdots, \lceil x_3x_n\rceil, \cdots,  \lceil x_{n-1}^2\rceil, \lceil x_{n-1}x_n\rceil, \lceil x_n^2\rceil ].$$

\end{proof}

By Proposition \ref{cohomology}, one can deduce the following interesting results for DG polynomial algebras.

\begin{thm}\label{asreg}
Any DG polynomial algebra $\mathcal{A}(t_1,t_2,\cdots, t_n)$ is a formal, homologically smooth and Gorenstein DG algebra.
\end{thm}
\begin{proof}
For briefness, let $\mathcal{A}=\mathcal{A}(t_1,t_2,\cdots, t_n)$.

 If $(t_1,t_2,\cdots, t_n)=(0,0,\cdots, 0)$, then $\partial_{A}=0$ and $H(\mathcal{A})=\mathcal{A}^{\#}=k[x_1,x_2,\cdots, x_n]$.  Hence $H(\mathcal{A})$ is a Noetherian, Gorenstein graded algebra with $\mathrm{gl.dim}H(\mathcal{A})=n$. Therefore, $\mathcal{A}$ is Gorenstein and $\k\in \mathrm{D}^c(\mathcal{A})$ by \cite[Proposition 1]{Gam} and \cite[Corollary 3.6]{MW2}, respectively.  Hence $\mathcal{A}$ is homologically smooth by \cite[Corollary 2.7]{MW3}.

If $(t_1,t_2,\cdots, t_n)\neq (0,0,\cdots, 0)$, then $\mathcal{A}\cong \mathcal{A}(1,0,\cdots, 0)$ by Corollary \ref{isomor}. We have $$H(\mathcal{A})\cong\k[\lceil x_2^2\rceil,\lceil x_2x_3\rceil, \cdots,\lceil x_2x_n\rceil, \lceil x_3^2\rceil, \cdots, \lceil x_3x_n\rceil, \cdots,  \lceil x_{n-1}^2\rceil, \lceil x_{n-1}x_n\rceil, \lceil x_n^2\rceil ]$$
by Theorem \ref{cohomology}. So $H(\mathcal{A})$ is a Noetherian, Gorenstein graded algebra with $$\mathrm{gl.dim}H(\mathcal{A})=\frac{n(n-1)}{2}<\infty.$$ By \cite[Proposition 1]{Gam}, $\mathcal{A}$ is Gorenstein.  And $\mathcal{A}$ is homologically smooth by \cite[Corollary 3.6]{MW2} and \cite[Corollary 2.7]{MW3}. We can define a morphism of DG algebras $
\iota: (H(\mathcal{A}),0) \to (\mathcal{A},\partial_{\mathcal{A}})$ by
          $\iota(\lceil x_ix_j\rceil) = x_ix_j$, for all $2\le i\le j\le n$. One sees easily that $\iota$
is a quasi-isomorphism.  So $\mathcal{A}$ is formal.
\end{proof}

\section{Calabi-Yau properties of polynomial dg algebras}
By Theorem \ref{asreg}, we know that any DG polynomial algebra $\mathcal{A}(t_1,t_2,\cdots, t_n)$ is a formal, homologically smooth and Gorenstein DG algebra. It is natural for one to ask whether $\mathcal{A}(t_1,t_2,\cdots, t_n)$ is Calabi-Yau. We completely solve this problem in this section.
\begin{rem}
For any connected cochain DG algebra $\mathcal{A}$, one sees that $H(\mathcal{A}^e)=H(\mathcal{A})^e$ is a connected graded algebra.
We should emphasize that the multiplication of $H(\mathcal{A})^e$ is defined by $$(\lceil c\rceil\otimes \lceil d\rceil)\cdot (\lceil e\rceil\otimes \lceil f\rceil) =   (-1)^{|d|\cdot |e|} \lceil c\rceil \lceil e\rceil\otimes \lceil d\rceil\diamond \lceil f\rceil=(-1)^{|d|\cdot |e|+|d|\cdot |f|} \lceil ce\rceil\otimes \lceil fd\rceil,$$ for any cocycle elements $c, e\in \mathcal{A}$ and $d, f\in \mathcal{A}^{op}$. In the proof of the following two theorems, we need to construct minimal free resolutions of ${}_{H(\mathcal{A})^e}H(\mathcal{A})$ and $H(\mathcal{A})_{H(\mathcal{A})^e}$. We remind the readers to remember our emphasis above.
\end{rem}
\begin{thm}\label{cyprop}
Let $\mathcal{A}$ be a connected cochain DG algebra such that $$H(\mathcal{A})=\k[\lceil y_1\rceil, \cdots, \lceil y_m\rceil ],$$ for some central, cocycle and degree $2$ elements $y_1,\cdots, y_m$  in $\mathcal{A}$. Then $\mathcal{A}$ is a $(-m)$-Calabi-Yau DG algebra.
\end{thm}
\begin{proof}
The graded left $H(\mathcal{A})^e$-module $H(\mathcal{A})$ admits a minimal free resolution:
\begin{align*}
0\leftarrow H(\mathcal{A})\stackrel{\mu}{\leftarrow}H(\mathcal{A})^e\stackrel{d_1}{\leftarrow}F_1\stackrel{d_2}{\leftarrow}F_2\stackrel{d_3}{\leftarrow}\cdots\stackrel{d_{m-1}}{\leftarrow} F_{m-1}\stackrel{d_m}{\leftarrow} F_m  \stackrel{0}{\leftarrow} 0,
\end{align*}
where $\mu: H(\mathcal{A})^e\to H(\mathcal{A})$ is defined by $\mu(\lceil a\rceil \otimes \lceil b\rceil)=\lceil ab\rceil$, each $F_k$ is a free $H(\mathcal{A})^e$-module $$\bigoplus_{e_{i_1\cdots i_k}\in E_k} H(\mathcal{A})^ee_{i_1\cdots i_k}$$ of rank $C_m^k$ with basis $$E_k=\{e_{i_1\cdots i_k}|1\le i_1<i_2<\cdots <i_k\le m\},  \quad |e_{i_1\cdots i_k}|=2k,$$ and $d_k$ is defined by
\begin{align*}
d_k(e_{i_1\cdots i_k})=\sum\limits_{j=1}^k(-1)^{j-1} (\lceil y_{i_j}\rceil\otimes 1-1\otimes \lceil y_{i_j}\rceil)e_{i_1\cdots \hat{i_j}\cdots i_k},
\end{align*}
for any $k=1,2,\cdots, m$.
Applying the constructing procedure of Eilenberg-Moore resolution, we can construct a minimal semi-free resolution $F$ of the DG $\mathcal{A}^e$-module $\mathcal{A}$. Since $y_1,y_2,\cdots, y_m$ are central elements in $\mathcal{A}$, it is not difficult to check that
$$F^{\#}=\mathcal{A}^{e\#}\oplus [\bigoplus_{k=1}^m \bigoplus_{e_{i_1\cdots i_k}\in E_k}\mathcal{A}^{e\#}\Sigma^ke_{i_1\cdots i_k}]$$ and
$\partial_{F}$ is defined by
\begin{align*}
\partial_{F}(\Sigma^ke_{i_1\cdots i_k})=\sum\limits_{j=1}^k(-1)^{j-1} (y_{i_j}\otimes 1-1\otimes y_{i_j})\Sigma^{k-1}e_{i_1\cdots \hat{i_j}\cdots i_k},
\end{align*}
for any $e_{i_1\cdots i_k}\in E_k$ and $k=1,2,\cdots, m$. Clearly, $\mathcal{A}$ is homologically smooth since $F$ has a finite semi-basis.

Similarly, the right graded $H(\mathcal{A})^e$-module $ H(\mathcal{A})$ has a minimal free resolution
\begin{align}\label{right}
0\leftarrow  H(\mathcal{A})\stackrel{\tau}{\leftarrow} H(\mathcal{A})^e\stackrel{d_1}{\leftarrow}G_1\stackrel{d_2}{\leftarrow}G_2\stackrel{d_3}{\leftarrow}\cdots\stackrel{d_{m-1}}{\leftarrow} G_{m-1}\stackrel{d_m}{\leftarrow} G_m  \stackrel{0}{\leftarrow} 0,
\end{align}
where $\tau: H(\mathcal{A})^e\to H(\mathcal{A})$ is defined by $\tau(\lceil a\rceil \otimes \lceil b\rceil)=(-1)^{|a|\cdot|b|}\lceil ba\rceil$, each $G_k$ is a free right $H(\mathcal{A})^e$-module $$\bigoplus_{\lambda_{i_1\cdots i_k}\in \Lambda_k}\lambda_{i_1\cdots i_k} H(\mathcal{A})^e$$ of rank $C_m^k$ with basis $$\Lambda_k=\{\lambda_{i_1\cdots i_k}|1\le i_1<i_2<\cdots <i_k\le m\}, \quad |\lambda_{i_1\cdots i_k}|=2k,$$ and the differentials are defined by
\begin{align*}
&d_1(\lambda_{i_1})= (1\otimes \lceil y_{i_1}\rceil-\lceil y_{i_1}\rceil\otimes 1),\quad   \forall \lambda_{i_1}\in \Lambda_1                \\
&d_k(\lambda_{i_1\cdots i_k})=\sum\limits_{j=1}^k(-1)^{k-j}\lambda_{i_1\cdots \hat{i_j}\cdots i_k} (1\otimes \lceil y_{i_j}\rceil-\lceil y_{i_j}\rceil\otimes 1), \quad \forall \lambda_{i_1\cdots i_k} \in \Lambda_k,
\end{align*}
$k=2,\cdots, m.$ From the free resolution (\ref{right}), we can construct the Eilenberg-Moore resolution $G$ of $\mathcal{A}_{\mathcal{A}^e}$. We have
$$G^{\#}=\mathcal{A}^{e\#}\oplus [\bigoplus_{k=1}^m \bigoplus_{\lambda_{i_1\cdots i_k}\in \Lambda_k}\Sigma^k\lambda_{i_1\cdots i_k}\mathcal{A}^{e\#}]$$ and
$\partial_{G}$ is defined by
\begin{align*}
&\partial_{G}(1)=0, \quad\partial_{G}(\Sigma \lambda_{i_1})=(1\otimes y_{i_1}-y_{i_1}\otimes 1), \quad \forall \lambda_{i_1}\in \Lambda_1\\
&\partial_{G}(\Sigma^k\lambda_{i_1\cdots i_k})=\sum\limits_{j=1}^k(-1)^{k-j} \Sigma^{k-1}\lambda_{i_1\cdots \hat{i_j}\cdots i_k}(1\otimes y_{i_j}-y_{i_j}\otimes 1),
\end{align*}
for any $\lambda_{i_1\cdots i_k}\in \Lambda_k$ and $k=2,\cdots, m$. Hence $$\Sigma^m G^{\#}= \Sigma^m1\mathcal{A}^{e\#}\oplus [\bigoplus_{k=1}^m \bigoplus_{\lambda_{i_1\cdots i_k}\in \Lambda_k}\Sigma^{m+k}\lambda_{i_1\cdots i_k}\mathcal{A}^{e\#}]$$ with
\begin{align*}
&\partial_{\Sigma^mG}(\Sigma^m1)=0, \quad\partial_{\Sigma^mG}(\Sigma^{m+1} \lambda_{i_1})=(-1)^m\Sigma^m1(1\otimes y_{i_1}-y_{i_1}\otimes 1), \quad \forall \lambda_{i_1}\in \Lambda_1\\
&\partial_{\Sigma^m G}(\Sigma^{m+k}\lambda_{i_1\cdots i_k})=\sum\limits_{j=1}^k(-1)^{m+k-j} \Sigma^{m+k-1}\lambda_{i_1\cdots \hat{i_j}\cdots i_k}(1\otimes y_{i_j}-y_{i_j}\otimes 1),
\end{align*}

The DG right $\mathcal{A}^e$-module $\Hom_{\mathcal{A}^e}(F,\mathcal{A}^e)$ is a minimal semi-free DG module whose underlying graded module is
\begin{align*}
\{\k1^*\oplus [\bigoplus_{k=1}^m \bigoplus_{e_{i_1\cdots i_k}\in E_k}\k(\Sigma^ke_{i_1\cdots i_k})^*]\}\otimes (\mathcal{A}^e)^{\#}.
\end{align*}
The differential $\partial_{\Hom}$ of $\Hom_{\mathcal{A}^e}(F,\mathcal{A}^e)$ is defined by $$\partial_{\Hom}(f)=\partial_{\mathcal{A}^e} \circ f -(-1)^{|f|}f\circ \partial_F,$$
for any graded element $f\in \Hom_{\mathcal{A}^e}(F,\mathcal{A}^e)$. So we have  $\partial_{\Hom}[(\Sigma^me_{1\cdots m})^*]=0$,
\begin{small}
\begin{align*}
&\partial_{\Hom}[(\Sigma^ke_{i_1\cdots i_k})^*]=\sum\limits_{e_{i_1\cdots i_jli_{j+1}\cdots i_k}\in E_{k+1}}(-1)^{j-k}(\Sigma^{k+1}e_{i_1\cdots i_jli_{j+1}\cdots i_k})^*(1\otimes y_l-y_l\otimes 1),
\end{align*}
\end{small}
for any $k\in \{1,2,\cdots, m-1\}$, and
\begin{align*}
&\partial_{\Hom}[1^*]=\sum\limits_{i=1}^m(\Sigma e_i)^*(1\otimes y_i-y_i\otimes 1).
\end{align*}
Define an $(\mathcal{A}^e)^{op}$-linear map
$ \theta: \Hom_{\mathcal{A}^e}(F,\mathcal{A}^e)\to \Sigma^m G$
 by $$ \theta: \begin{cases}
(\Sigma^me_{1\cdots m})^* \,\, \mapsto (-1)^{\frac{m(m+1)}{2}} \Sigma^m 1,   \\
(\Sigma^k e_{i_1\cdots i_k})^* \,\, \mapsto (-1)^{\sum\limits_{j=1}^k i_j}\Sigma^{2m-k}\lambda_{\overline{i_1\cdots i_{k}}}, \quad \forall k\in \{1,\cdots, m-1\},\\
1^*  \mapsto \,\, \Sigma^{2m} \lambda_{1\cdots m},
\end{cases}
$$
where $\overline{i_1\cdots i_k}$ is the $m-k$ integers arranged from small to large obtained by deleting $i_1,i_2,\cdots, i_k$ from $\{1,2,\cdots, m\}$. We claim that $\theta$ is a chain map. Indeed, we have $$\theta\circ \partial_{\Hom} [(\Sigma^me_{1\cdots m})^*] =0=(-1)^{\frac{m(m+1)}{2}}\partial_{\Sigma^m G}[\Sigma^m 1]= \partial_{\Sigma^m G}\circ \theta [(\Sigma^me_{1\cdots m})^*]$$ and $\theta\circ \partial_{\Hom}[1^* ]=\partial_{\Sigma^m G}\circ \theta[1^*]$ since  \begin{align*}
\quad\quad \theta\circ \partial_{\Hom}[1^* ]&=\theta[\sum\limits_{i=1}^m(\Sigma e_i)^*(1\otimes y_i-y_i\otimes 1)]\\
&=\sum\limits_{i=1}^m(-1)^{i}\Sigma^{2m-1}\lambda_{1\cdots(i-1)(i+1)\cdots m}(1\otimes y_i-y_i\otimes 1)
\end{align*}
and
\begin{align*}
\partial_{\Sigma^m G}\circ \theta[1^* ]& =\partial_{\Sigma^m G}[\Sigma^{2m} \lambda_{1\cdots m}]\\
&=\sum\limits_{i=1}^{m}(-1)^{m+m-i}\Sigma^{2m-1}\lambda_{1\cdots (i-1)(i+1)\cdots m}(1\otimes y_i-y_i\otimes 1)\\
&=\sum\limits_{i=1}^{m}(-1)^{i}\Sigma^{2m-1}\lambda_{1\cdots (i-1)(i+1)\cdots  m}(1\otimes y_i-y_i\otimes 1).
\end{align*} Furthermore, for any $k\in \{1,2,\cdots m-1\}$, we have
$$\partial_{\Sigma^m G}\circ \theta [(\Sigma^ke_{i_1\cdots i_k})^*]=\theta\circ \partial_{\Hom} [(\Sigma^ke_{i_1\cdots i_k})^*]$$ since
\begin{align*}
&\quad\quad \theta\circ \partial_{\Hom} [(\Sigma^ke_{i_1\cdots i_k})^*]\\
&=\theta [\sum\limits_{e_{i_1\cdots i_jli_{j+1}\cdots i_k}\in E_{k+1}}(-1)^{j-k}(\Sigma^{k+1}e_{i_1\cdots i_jli_{j+1}\cdots i_k})^*(1\otimes y_l-y_l\otimes 1) ]\\
&=\sum\limits_{e_{i_1\cdots i_jli_{j+1}\cdots i_k}\in E_{k+1}}(-1)^{j-k+\sum\limits_{j=1}^k i_j+l}\Sigma^{2m-k-1}\lambda_{\overline{i_1\cdots i_jli_{j+1}\cdots i_k}}(1\otimes y_l-y_l\otimes 1)
\end{align*}
and
\begin{align*}
&\quad\quad \partial_{\Sigma^m G}\circ \theta [(\Sigma^ke_{i_1\cdots i_k})^*]\\
&=\partial_{\Sigma^m G}[(-1)^{\sum\limits_{j=1}^k i_j}\Sigma^{2m-k}\lambda_{\overline{i_1\cdots i_{k}}}]\\
&=\sum\limits_{e_{i_1\cdots i_jli_{j+1}\cdots i_k}\in E_{k+1}}(-1)^{\sum\limits_{j=1}^k i_j +2m-k-l+j}\Sigma^{2m-k-1}\lambda_{\overline{i_1\cdots i_jli_{j+1}\cdots i_k}}(1\otimes y_l-y_l\otimes 1).
\end{align*}
So $\theta$ is an isomorphism of DG $\mathcal{A}^{op}$-modules and
$\Hom_{\mathcal{A}^e}(F,\mathcal{A}^e)\cong \Sigma^m\mathcal{A}$ in $\mathrm{D}((\mathcal{A}^e)^{op})$.
Therefore, $\mathcal{A}$ is a $(-m)$-Calabi-Yau DG algebra.
\end{proof}

\begin{cor}\label{non-zero}
The DG polynomial algebra $\mathcal{A}(t_1,t_2,\cdots, t_n)$ is a $\frac{-n(n-1)}{2}$-Calabi-Yau DG algebra if $(t_1,t_2,\cdots, t_n)\neq (0,0,\cdots, 0)$.
\end{cor}

\begin{proof}
By Corollary \ref{isomor}, we have $\mathcal{A}(t_1,t_2,\cdots, t_n)\cong \mathcal{A}(1,0,\cdots, 0)$. So we only need to show $\mathcal{A}(1,0,\cdots, 0)$ is a $\frac{-n(n-1)}{2}$-Calabi-Yau DG algebra. By Theorem \ref{cohomology}, the cohomology graded algebra of $\mathcal{A}(1,0,\cdots, 0)$ is
$$\k[\lceil x_2^2\rceil,\lceil x_2x_3\rceil, \cdots,\lceil x_2x_n\rceil, \lceil x_3^2\rceil, \cdots, \lceil x_3x_n\rceil, \cdots,  \lceil x_{n-1}^2\rceil, \lceil x_{n-1}x_n\rceil, \lceil x_n^2\rceil ].$$ Theorem \ref{cyprop} indicates that $\mathcal{A}(1,0,\cdots, 0)$ is a $\frac{-n(n-1)}{2}$-Calabi-Yau DG algebra.
\end{proof}

\begin{thm}\label{degreeone}
Let $\mathcal{A}$ be a connected cochain DG algebra such that $$H(\mathcal{A})=\k[\lceil y_1\rceil, \cdots, \lceil y_m\rceil ],$$ for some central, cocycle and degree $1$ elements $y_1,\cdots, y_m$  in $\mathcal{A}$. Then $\mathcal{A}$ is a Koszul, homologically smooth and Gorenstein DG algebra. Moreover, $\mathcal{A}$ is $0$-Calabi-Yau if and only if $m$ is an odd integer.
\end{thm}
\begin{proof}
 By the assumption $H(\mathcal{A})=\k[\lceil y_1\rceil, \cdots, \lceil y_m\rceil ]$ is a Koszul, Gorenstein and Noetherian graded algebra with $\mathrm{gl.dim}H(A)=m<\infty$. Hence $\mathcal{A}$ is Koszul, Gorenstein and homologically smooth by \cite[Proposition2.3]{HW}, \cite[Proposition 1]{Gam} and \cite[Corollary 3.7]{MW2}, respectively.

 Now, lets consider the Calabi-Yau properties of $\mathcal{A}$.
The graded left $H(\mathcal{A})^e$-module $H(\mathcal{A})$ admits a minimal free resolution:
\begin{align*}
0\leftarrow H(\mathcal{A})\stackrel{\mu}{\leftarrow}H(\mathcal{A})^e\stackrel{d_1}{\leftarrow}F_1\stackrel{d_2}{\leftarrow}F_2\stackrel{d_3}{\leftarrow}\cdots\stackrel{d_{m-1}}{\leftarrow} F_{m-1}\stackrel{d_m}{\leftarrow} F_m  \stackrel{0}{\leftarrow} 0,
\end{align*}
where $\mu: H(\mathcal{A})^e\to H(\mathcal{A})$ is defined by $\mu(\lceil a\rceil \otimes \lceil b\rceil)=\lceil ab\rceil$, each $F_k$ is a free $H(\mathcal{A})^e$-module $$\bigoplus_{e_{i_1\cdots i_k}\in E_k} H(\mathcal{A})^ee_{i_1\cdots i_k}$$ of rank $C_m^k$ with basis $$E_k=\{e_{i_1\cdots i_k}|1\le i_1<i_2<\cdots <i_k\le m\}, \quad |e_{i_1\cdots i_k}|=k,$$ and $d_k$ is defined by
\begin{align*}
d_k(e_{i_1\cdots i_k})=\sum\limits_{j=1}^k(-1)^{j-1} (\lceil y_{i_j}\rceil\otimes 1+(-1)^k1\otimes \lceil y_{i_j}\rceil)e_{i_1\cdots \hat{i_j}\cdots i_k},
\end{align*}
for any $k=1,2,\cdots, m$.
Applying the constructing procedure of Eilenberg-Moore resolution, we can construct a minimal semi-free resolution $F$ of the DG $\mathcal{A}^e$-module $\mathcal{A}$. Since $y_1,\cdots, y_m$ are central elements in $\mathcal{A}$, it is not difficult to check that
$$F^{\#}=\mathcal{A}^{e\#}\oplus [\bigoplus_{k=1}^m \bigoplus_{e_{i_1\cdots i_k}\in E_k}\mathcal{A}^{e\#}\Sigma^ke_{i_1\cdots i_k}]$$ and
$\partial_{F}$ is defined by
\begin{align*}
\partial_{F}(\Sigma^ke_{i_1\cdots i_k})=\sum\limits_{j=1}^k(-1)^{j-1} (y_{i_j}\otimes 1+(-1)^k1\otimes y_{i_j})\Sigma^{k-1}e_{i_1\cdots \hat{i_j}\cdots i_k},
\end{align*}
for any $e_{i_1\cdots i_k}\in E_k$ and $k=1,2,\cdots, m$.

The DG right $\mathcal{A}^e$-module $\Hom_{\mathcal{A}^e}(F,\mathcal{A}^e)$ is a minimal semi-free DG module whose underlying graded module is
\begin{align*}
\{\k1^*\oplus [\bigoplus_{k=1}^m \bigoplus_{e_{i_1\cdots i_k}\in E_k}\k(\Sigma^ke_{i_1\cdots i_k})^*]\}\otimes (\mathcal{A}^e)^{\#}.
\end{align*}
The differential $\partial_{\Hom}$ of $\Hom_{\mathcal{A}^e}(F,\mathcal{A}^e)$ is defined by $$\partial_{\Hom}(f)=\partial_{\mathcal{A}^e} \circ f -(-1)^{|f|}f\circ \partial_F,$$
for any graded element $f\in \Hom_{\mathcal{A}^e}(F,\mathcal{A}^e)$. So we have  $\partial_{\Hom}[(\Sigma^me_{1\cdots m})^*]=0$,

\begin{align*}
&\quad\quad\quad \partial_{\Hom}[(\Sigma^ke_{i_1\cdots i_k})^*]\\
&=\sum\limits_{e_{i_1\cdots i_jli_{j+1}\cdots i_k}\in E_{k+1}}(-1)^{j+1}(\Sigma^{k+1}e_{i_1\cdots i_jli_{j+1}\cdots i_k})^*(y_l\otimes 1-(-1)^k 1\otimes y_l),
\end{align*}
for any $k\in \{1,2,\cdots, m-1\}$, and
\begin{align*}
&\partial_{\Hom}[1^*]=\sum\limits_{i=1}^m(\Sigma e_i)^*(1\otimes y_i-y_i\otimes 1).
\end{align*}
If $m=2t$ is an even integer, we claim that $\mathcal{A}$ is not a Calabi-Yau DG algebra. It suffices to
show that $\Hom_{\mathcal{A}^e}(F,\mathcal{A}^e)\not\cong  \mathcal{A}$ in $\mathrm{D}((\mathcal{A}^e)^{op})$.
  We prove this with a proof by contradiction. If $\Hom_{\mathcal{A}^e}(F,\mathcal{A}^e)\cong  \mathcal{A}$ in $\mathrm{D}((\mathcal{A}^e)^{op})$, then there exist a DG right $\mathcal{A}^e$-module $P$ and two quasi-isomorphisms of right DG $\mathcal{A}^e$-modules $\phi: P\to  \mathcal{A}$ and  $\psi: P\to \Hom_{\mathcal{A}^e}(F,\mathcal{A}^e)$. The DG right $\mathcal{A}^e$-module $P$ admits a minimal semi-free resolution $\varepsilon:G\to P$ since $H(P)\cong H(\mathcal{A})$ is bounded below.
Then $$\psi\circ \varepsilon: G\to \Hom_{\mathcal{A}^e}(F,\mathcal{A}^e)$$ is a quasi-isomorphism of right DG $\mathcal{A}^e$-modules.  Since both $G$ and $\Hom_{\mathcal{A}^e}(F,\mathcal{A}^e)$ are minimal, $\psi\circ \varepsilon$ is an isomorphism by \cite[\S 12, Corollary 1.3]{AFH}. Then the composition $$\Hom_{\mathcal{A}^e}(F,\mathcal{A}^e)\stackrel{(\psi\circ \varepsilon)^{-1}}{\longrightarrow} G\stackrel{\varepsilon}{\longrightarrow} P \stackrel{\phi}{\longrightarrow} \mathcal{A}$$
is a quasi-isomorphism of DG right $\mathcal{A}^e$-modules. Set $g=\phi\circ \varepsilon\circ (\psi\circ \varepsilon)^{-1}$. Since $H(g)$ is an isomorphism and $\lceil(\Sigma^me_{1\cdots m})^*\rceil \neq 0$, we have $g[(\Sigma^me_{1\cdots m})^*]= a_0\neq 0$, for some $a_0\in \mathcal{A}^0=k$. For any $j\in \{1,2,\cdots, m\}$, we have $|(\Sigma^{m-1} e_{1\cdots j-1 j+1\cdots m})^*|=0$.
So there exists $a\in \mathcal{A}^0$ such that $g[(\Sigma^{m-1} e_{1\cdots j-1 j+1\cdots m})^*]= a$. We have
\begin{align*}
& \partial_{\mathcal{A}}\circ   g[(\Sigma^{m-1} e_{1\cdots (j-1) (j+1)\cdots m})^*]= \partial_{\mathcal{A}}(a)=0,\\
& \quad\quad g\circ \partial_{\Hom}[(\Sigma^{m-1} e_{1\cdots (j-1)(j+1)\cdots m})^*]\\
&= g[(-1)^j(\Sigma^me_{1\cdots m})^*(y_j\otimes 1-(-1)^{m-1}1\otimes y_j)]\\
&= (-1)^ja_0 \cdot (y_j\otimes 1+1\otimes y_j)=(-1)^j2a_0y_j
\end{align*}
Therefore, $\partial_{\mathcal{A}}\circ g \neq g\circ \partial_{\Hom}$ since $\mathrm{char}\, \k=0$ and $a_0\neq 0$ in $\mathcal{A}^0$. This implies that $g$ is not a chain map. Then we get a contradiction since $g$ is a morphism of DG right $\mathcal{A}^e$-modules.

Now, it remains to consider the case that $m=2t-1$ is an odd integer.
 The right graded $H(\mathcal{A})^e$-module $ H(\mathcal{A})$ has a minimal free resolution
\begin{align}\label{right}
0\leftarrow  H(\mathcal{A})\stackrel{\tau}{\leftarrow} H(\mathcal{A})^e\stackrel{d_1}{\leftarrow}G_1\stackrel{d_2}{\leftarrow}G_2\stackrel{d_3}{\leftarrow}\cdots\stackrel{d_{m-1}}{\leftarrow} G_{m-1}\stackrel{d_m}{\leftarrow} G_m  \stackrel{0}{\leftarrow} 0,
\end{align}
where $\tau: H(\mathcal{A})^e\to H(\mathcal{A})$ is defined by $\tau(\lceil a\rceil \otimes \lceil b\rceil)=(-1)^{|a|\cdot|b|}\lceil ba\rceil$, each $G_k$ is a free right $H(\mathcal{A})^e$-module $$\bigoplus_{\lambda_{i_1\cdots i_k}\in \Lambda_k}\lambda_{i_1\cdots i_k} H(\mathcal{A})^e$$ of rank $C_m^k$ with basis $$\Lambda_k=\{\lambda_{i_1\cdots i_k}|1\le i_1<i_2<\cdots <i_k\le m\}, \quad |\lambda_{i_1\cdots i_k}|=k,$$ and the differentials are defined by
\begin{align*}
&d_1(\lambda_{i_1})= (1\otimes \lceil y_{i_1}\rceil-\lceil y_{i_1}\rceil\otimes 1),\quad   \forall \lambda_{i_1}\in \Lambda_1                \\
&d_k(\lambda_{i_1\cdots i_k})=\sum\limits_{j=1}^k(-1)^{k-j}\lambda_{i_1\cdots \hat{i_j}\cdots i_k} (1\otimes \lceil y_{i_j}\rceil+(-1)^k\lceil y_{i_j}\rceil\otimes 1), \quad \forall \lambda_{i_1\cdots i_k} \in \Lambda_k,
\end{align*}
$k=2,\cdots, m.$ From the free resolution (\ref{right}), we can construct the Eilenberg-Moore resolution $G$ of $\mathcal{A}_{\mathcal{A}^e}$. We have
$$G^{\#}=\mathcal{A}^{e\#}\oplus [\bigoplus_{k=1}^m \bigoplus_{\lambda_{i_1\cdots i_k}\in \Lambda_k}\Sigma^k\lambda_{i_1\cdots i_k}\mathcal{A}^{e\#}]$$ and
$\partial_{G}$ is defined by
\begin{align*}
&\partial_{G}(1)=0, \quad\partial_{G}(\Sigma \lambda_{i_1})=(1\otimes y_{i_1}-y_{i_1}\otimes 1), \quad \forall \lambda_{i_1}\in \Lambda_1\\
&\partial_{G}(\Sigma^k\lambda_{i_1\cdots i_k})=\sum\limits_{j=1}^k(-1)^{k-j} \Sigma^{k-1}\lambda_{i_1\cdots \hat{i_j}\cdots i_k}(1\otimes y_{i_j}+(-1)^ky_{i_j}\otimes 1),
\end{align*}
for any $\lambda_{i_1\cdots i_k}\in \Lambda_k$ and $k=2,\cdots, m$.

Define an $(\mathcal{A}^e)^{op}$-linear map
$ \theta: \Hom_{\mathcal{A}^e}(F,\mathcal{A}^e)\to  G$
 by $$ \theta: \begin{cases}
(\Sigma^me_{1\cdots m})^* \,\, \mapsto  (-1)^{\frac{m(m+1)}{2}+m}1,   \\
(\Sigma^k e_{i_1\cdots i_k})^* \,\, \mapsto (-1)^{\sum\limits_{j=1}^ki_j+k}\Sigma^{m-k}\lambda_{\overline{i_1\cdots i_{k}}}, \quad \forall k\in \{1,\cdots, m-1\},\\
1^*  \mapsto \,\, \Sigma^{m} \lambda_{1\cdots m},
\end{cases}
$$
where $\overline{i_1\cdots i_k}$ is the $m-k$ integers arranged from small to large obtained by deleting $i_1,i_2,\cdots, i_k$ from $\{1,2,\cdots, m\}$. We claim that $\theta$ is a chain map. Indeed, we have $$\theta\circ \partial_{\Hom} [(\Sigma^me_{1\cdots m})^*] =0=\partial_{G}[(-1)^{\frac{m(m+1)}{2}+m}1]= \partial_{G}\circ \theta [(\Sigma^me_{1\cdots m})^*]$$ and $\theta\circ \partial_{\Hom}[1^* ]=\partial_{\Sigma^m G}\circ \theta[1^*]$ since  \begin{align*}
\quad\quad \theta\circ \partial_{\Hom}[1^* ]&=\theta[\sum\limits_{i=1}^m(\Sigma e_i)^*(1\otimes y_i-y_i\otimes 1)]\\
&=\sum\limits_{i=1}^m(-1)^{i+1}\Sigma^{m-1}\lambda_{1\cdots(i-1)(i+1)\cdots m}(1\otimes y_i-y_i\otimes 1)
\end{align*}
and
\begin{align*}
\partial_{G}\circ \theta[1^*]& =\partial_{G}[\Sigma^{m} \lambda_{1\cdots m}]\\
&=\sum\limits_{i=1}^{m}(-1)^{m-i}\Sigma^{m-1}\lambda_{1\cdots (i-1)(i+1)\cdots  m}(1\otimes y_i+(-1)^my_i\otimes 1)\\
&=\sum\limits_{i=1}^{m}(-1)^{2t-1-i}\Sigma^{m-1}\lambda_{1\cdots (i-1)(i+1)\cdots m}(1\otimes y_i-y_i\otimes 1)\\
&=\sum\limits_{i=1}^{m}(-1)^{1+i}\Sigma^{m-1}\lambda_{1\cdots (i-1)(i+1)\cdots m}(1\otimes y_i-y_i\otimes 1).
\end{align*}

 Furthermore, for any $k\in \{1,2,\cdots m-1\}$, we have
$$\partial_{G}\circ \theta [(\Sigma^ke_{i_1\cdots i_k})^*]=\theta\circ \partial_{\Hom} [(\Sigma^ke_{i_1\cdots i_k})^*]$$ since
\begin{align*}
&\quad\quad \theta\circ \partial_{\Hom} [(\Sigma^ke_{i_1\cdots i_k})^*]\\
&=\theta [\sum\limits_{e_{i_1\cdots i_jli_{j+1}\cdots i_k}\in E_{k+1}}(-1)^{j+1}(\Sigma^{k+1}e_{i_1\cdots i_jli_{j+1}\cdots i_k})^*(y_l\otimes 1-(-1)^k 1\otimes y_l) ]\\
&=\sum\limits_{e_{i_1\cdots i_jli_{j+1}\cdots i_k}\in E_{k+1}}(-1)^{j+\sum\limits_{s=1}^ki_s+l+k}\Sigma^{m-k-1}\lambda_{\overline{i_1\cdots i_jli_{j+1}\cdots i_k}}(y_l\otimes 1-(-1)^k 1\otimes y_l)
\end{align*}
and
\begin{align*}
&\quad\quad \partial_{G}\circ \theta [(\Sigma^ke_{i_1\cdots i_k})^*]\\
&=\partial_{G}[(-1)^{\sum\limits_{s=1}^k i_s +k}\Sigma^{m-k}\lambda_{\overline{i_1\cdots i_{k}}}]\\
&=\sum\limits_{e_{i_1\cdots i_jli_{j+1}\cdots i_k}\in E_{k+1}}(-1)^{\sum\limits_{s=1}^k i_s +m-l+j }\Sigma^{m-k-1}\lambda_{\overline{i_1\cdots i_jli_{j+1}\cdots i_k}}(1\otimes y_l-(-1)^{-k}y_l\otimes 1)\\
&=\sum\limits_{e_{i_1\cdots i_jli_{j+1}\cdots i_k}\in E_{k+1}}(-1)^{\sum\limits_{s=1}^k i_s-l+j-k }\Sigma^{m-k-1}\lambda_{\overline{i_1\cdots i_jli_{j+1}\cdots i_k}}(y_l\otimes 1-(-1)^k1\otimes y_l).
\end{align*}
So $\theta$ is an isomorphism of DG $\mathcal{A}^{op}$-modules and
$\Hom_{\mathcal{A}^e}(F,\mathcal{A}^e)\cong \mathcal{A}$ in $\mathrm{D}((\mathcal{A}^e)^{op})$.
Therefore, $\mathcal{A}$ is a $0$-Calabi-Yau DG algebra.
\end{proof}

\begin{cor}\label{zerodiff}
The trivial DG polynomial algebra $\mathcal{A}(0,0,\cdots, 0)$ is a  Koszul, homologically smooth and Gorenstein DG algebra.  Moreover, it is $0$-Calabi-Yau if and only if $n$ is an odd integer.
\end{cor}

\begin{rem}{\rm
In Theorem \ref{cyprop} and Theorem \ref{degreeone}, we need that the cocycle elements $y_1,y_2,\cdots, y_m$ are central in $\mathcal{A}$. This simplifies the construction of Eilenberg-Moore resolution. Without this subtle condition, we are unable to get the corresponding result.
 Note that Theorem \ref{degreeone} is not a generalization of \cite[Proposition 2.4, Theorem 2.7]{MH} for this reason.}
\end{rem}

\section*{Acknowledgments}
 The first author is
supported by NSFC  (Grant No.11001056),
the
China Postdoctoral
Science Foundation  (Grant Nos.
20090450066 and 201003244),  the
Key Disciplines of Shanghai Municipality (Grant No.S30104) and the Innovation Program of
Shanghai Municipal Education Commission (Grant No.12YZ031).
\def\refname{References}

redpajama_set_name: RedPajamaArXiv