# On the Gorensteinness of broken circuit complexes and Orlik–Terao ideals

Dinh Van Le  Universität Osnabrück, Institut für Mathematik, 49069 Osnabrück,
Germany<EMAIL_ADDRESS>
###### Abstract.

It is proved that the broken circuit complex of an ordered matroid is
Gorenstein if and only if it is a complete intersection. Several
characterizations for a matroid that admits such an order are then given, with
particular interest in the $h$-vector of broken circuit complexes of the
matroid. As an application, we prove that the Orlik–Terao algebra of a
hyperplane arrangement is Gorenstein if and only if it is a complete
intersection. Interestingly, our result shows that the complete intersection
property (and hence the Gorensteinness as well) of the Orlik–Terao algebra can
be determined from the last two nonzero entries of its $h$-vector.

##  1\. Introduction and main results

The broken circuit complex was introduced for a graph by Wilf [32] based on
the idea of Whitney [31]. This notion then was extended to matroids by
Brylawski [4] and has been studied by various authors; see, e.g., [1, 2, 6].
The broken circuit complex is important because of both of its combinatorial
and algebraic aspects. On the one hand, the entries of its $f$-vector coincide
with the coefficients of the Poincaré polynomial of the matroid [1]. On the
other hand, the broken circuit complex defines two algebras which are
deformations of two important algebras arising in the theory of hyperplane
arrangements: the Orlik–Solomon algebra and the Orlik–Terao algebra; see [1,
18].

A well-known property of the broken circuit complex is that it is shellable
[19]. It follows, in particular, that the Stanley–Reisner ring of the broken
circuit complex and the Orlik–Terao algebra are Cohen–Macaulay. Natural
questions then arise: when are these algebras complete intersections? or
Gorenstein? Characterizations for the complete intersection property of these
algebras were obtained in [13]; see also [10] for related results. A partial
answer to the Gorensteinness of the broken circuit complex is also given in
[13], in which it is shown that Gorenstein broken circuit complexes of
codimension 3 are complete intersections. However, a complete answer to this
question seems, especially for the Orlik–Terao algebra, much more complicated.

00footnotetext:  Mathematics Subject Classification (2010): 05B35, 05E40,
05E45, 13F55, 13H10, 52C35. Key words and phrases: broken circuit complex,
complete intersection, Gorenstein, hyperplane arrangement, matroid,
Orlik–Terao algebra.

The aim of this paper is to investigate the Gorensteinness of the
Stanley–Reisner ring of the broken circuit complex and the Orlik–Terao
algebra. Quite surprisingly, we are able to show, among other things, that
these algebras are Gorenstein exactly when they are complete intersections,
thus giving a satisfactory complete answer to the ‘Gorenstein’ question
mentioned above. More precisely, for the broken circuit complex, we obtain the
following:

######  Theorem 1.1.

Let $\Delta:=BC(M,<)$ be the broken circuit complex of an ordered matroid
$(M,<)$. Let $K$ be an arbitrary field. Then the following conditions are
equivalent:

  1. (i)

$\Delta$ is Gorenstein over $K$;

  2. (ii)

$\Delta$ is locally Gorenstein over $K$;

  3. (iii)

for any face $F$ with $\dim\operatorname{link}_{\Delta}F=1$,
$\operatorname{link}_{\Delta}F$ is Gorenstein over $K$;

  4. (iv)

for any face $F$ with $\dim\operatorname{link}_{\Delta}F=1$,
$\operatorname{link}_{\Delta}F$ is a complete intersection complex;

  5. (v)

$\Delta$ is a locally complete intersection complex;

  6. (vi)

$\Delta$ is a complete intersection complex.

The Gorensteinness of the Orlik–Terao algebra is equivalent to the symmetry of
its $h$-vector [23, Theorem 4.4], and thus equivalent to the symmetry of the
$h$-vector of a broken circuit complex associated to that Orlik–Terao algebra.
For this latter property, we have the following characterizations:

######  Theorem 1.2.

Let $M$ be a simple matroid. Let $(h_{0},\ldots,h_{s})$ be the $h$-vector of
broken circuit complexes of $M$, where $h_{s}\neq 0$. Then the following
conditions are equivalent:

  1. (i)

There exists an ordering $<$ of the ground set of $M$ such that the broken
circuit complex $BC(M,<)$ is a complete intersection;

  2. (ii)

the $h$-vector $(h_{0},\ldots,h_{s})$ satisfies the Dehn–Sommerville
equations: $h_{i}=h_{s-i}$ for $i=0,\ldots,s$;

  3. (iii)

$h_{0}=h_{s}$ and $h_{1}=h_{s-1}$;

  4. (iv)

each connected component of $M$ is either a coloop or isomorphic to an
iterated parallel connection of uniform matroids of the form $U_{m,m+1}$
($m\geq 2$).

From this, one easily gets:

######  Theorem 1.3.

Let $\mathcal{A}$ be an essential central hyperplane arrangement. Let
$\mathbf{C}(\mathcal{A})$ be the Orlik–Terao algebra of $\mathcal{A}$. Then
the following conditions are equivalent:

  1. (i)

$\mathbf{C}(\mathcal{A})$ is a complete intersection;

  2. (ii)

$\mathbf{C}(\mathcal{A})$ is Gorenstein;

  3. (iii)

$M(\mathcal{A})$ satisfies one of the equivalent conditions in Theorem 1.2,
where $M(\mathcal{A})$ is the underlying matroid of $\mathcal{A}$.

Note that for an essential central hyperplane arrangement, the first two
entries of the $h$-vector of its Orlik–Terao algebra are fixed: $h_{0}=1$ and
$h_{1}=n-r$, where $n$ is the number of hyperplanes and $r$ is the rank of the
arrangement. Therefore, Theorem 1.3 implies an interesting consequence: the
Gorensteinness (and hence the complete intersection property) of the
Orlik–Terao algebra depends only on the last two nonzero entries of its
$h$-vector.

The paper is divided into 4 sections. In Section 2 we recall some notions and
basis facts about the Orlik–Terao algebra, the broken circuit complex, and
series-parallel networks. Section 3 contains the proof of Theorem 1.1 and
related results. Section 4 is devoted to the proofs of Theorem 1.2 and Theorem
1.3.

Acknowledgments. The author would like to thank Professor Tim Römer for
raising his interest in the topic of the paper and for valuable discussions.
He also thanks the referees for useful suggestions.

##  2\. Preliminaries

In this section we collect several notions and properties concerning
simplicial complexes, the Orlik–Terao algebra, matroids, the broken circuit
complex, and series-parallel networks which will be used later. The reader is
referred to [15, 17, 25] for unexplained terminology.

###  2.1. Simplicial complexes

Let $\Delta$ be a simplicial complex with vertices $[n]=\\{1,\ldots,n\\}$. Let
$K$ be a field and $S=K[x_{1},\ldots,x_{n}]$ a polynomial ring over $K$. The
_Stanley–Reisner ideal_ $I_{\Delta}$ of $\Delta$ (over $K$) is the ideal in
$S$ generated by all monomial $x_{i_{1}}\cdots x_{i_{p}}$ such that
$\\{i_{1},\ldots,i_{p}\\}\not\in\Delta.$ The $K$-algebra
$K[\Delta]=S/I_{\Delta}$ is then called the _Stanley–Reisner ring_ of
$\Delta$. For a face $F\in\Delta$ and a subset $W\subseteq[n]$ we define

$\displaystyle\operatorname{star}_{\Delta}F$
$\displaystyle=\\{G\in\Delta:F\cup G\in\Delta\\},$
$\displaystyle\operatorname{link}_{\Delta}F$
$\displaystyle=\\{G\in\Delta:F\cup G\in\Delta,F\cap G=\emptyset\\},$
$\displaystyle\Delta_{W}$ $\displaystyle=\\{G\in\Delta:G\subseteq W\\}.$

Moreover, we put $\operatorname{core}\Delta=\Delta_{\operatorname{core}[n]}$,
where
$\operatorname{core}[n]=\\{i\in[n]:\operatorname{star}_{\Delta}\\{i\\}\neq\Delta\\}$.

Let us say that $\Delta$ is _Cohen–Macaulay_ (resp. _Gorenstein_ , a _complete
intersection_) over $K$ if so is the ring $K[\Delta]$. We then define $\Delta$
to be _locally Gorenstein_ over $K$ (resp. a _locally complete intersection_)
if $\operatorname{link}_{\Delta}\\{i\\}$ is Gorenstein over $K$ (resp. a
complete intersection) for $i=1,\ldots,n$.

Assume that $\dim\Delta=r-1$. Then the _$f$ -vector_ of $\Delta$ is the
$(r+1)$-tuple $(f_{0},\ldots,f_{r})$, where $f_{i}$ is the number of faces of
$\Delta$ of cardinality $i$. The $f$-vector is the most natural way to encode
the combinatorics of a simplicial complex. However, in many cases it is more
convenient to work with a related vector, the _$h$ -vector_
$(h_{0},\ldots,h_{r})$, defined as follows

$h_{i}=\sum_{j=0}^{i}(-1)^{i-j}\binom{r-j}{i-j}f_{j},\quad i=0,\ldots,r.$

Equivalently, if we let $f(t)=\sum_{i=0}^{r}f_{i}t^{r-i}$ be the
$f$-polynomial and $h(t)=\sum_{i=0}^{r}h_{i}t^{r-i}$ the $h$-polynomial of
$\Delta$, then $f(t)=h(t+1)$. In this paper we usually consider the $h$-vector
with zero entries at the end removed.

###  2.2. The Orlik–Terao algebra

Let $\mathcal{A}=\\{H_{1},\ldots,H_{n}\\}$ be an essential central hyperplane
arrangement in a vector space $V$ over a field $K$. The Orlik–Terao algebra of
$\mathcal{A}$ introduced in [16] is a commutative analog of the Orlik–Solomon
algebra. Recently, it has gotten more attention because it encodes subtle
information which is missed in the Orlik–Solomon algebra; see [10, 13, 20, 21,
22, 30] for more details.

Let $\alpha_{i}\in K[V]$ be linear forms such that $\ker\alpha_{i}=H_{i}$ for
$i=1,\ldots,n$. Then the _Orlik–Terao algebra_ $\mathbf{C}(\mathcal{A})$ of
$\mathcal{A}$ is the subalgebra of $K(V)$ generated by reciprocals of the
linear forms $\alpha_{i}$, that is,
$\mathbf{C}(\mathcal{A})=K[1/\alpha_{1},\ldots,1/\alpha_{n}]$. Let
$S={K}[x_{1},\ldots,x_{n}]$ be the polynomial ring in $n$ variables. The
kernel of the surjection

$S\rightarrow\mathbf{C}(\mathcal{A}),\ x_{i}\mapsto 1/\alpha_{i}\ \text{ for
}\ i=1,\ldots,n$

is called the _Orlik–Terao ideal_ of $\mathcal{A}$, denoted by
$I(\mathcal{A})$. Thus, $\mathbf{C}(\mathcal{A})=S/I(\mathcal{A})$.

It is known that $\mathbf{C}(\mathcal{A})$ is a standard graded Cohen–Macaulay
domain; see [22, Proposition 2.1] and [18, Theorem 4]. Therefore, it follows
from a result due to Stanley [23, Theorem 4.4] that the Gorensteinness of
$\mathbf{C}(\mathcal{A})$ depends only on its Hilbert series, or equivalently
on its $h$-vector. This allows one to use combinatorial tools to study the
Gorensteinness of $\mathbf{C}(\mathcal{A})$: the $h$-vector of
$\mathbf{C}(\mathcal{A})$ can be computed via a combinatorial object - the
broken circuit complex of the underlying matroid of $\mathcal{A}$. Recall that
the _underlying matroid_ $M(\mathcal{A})$ of $\mathcal{A}$ is the matroid on
the ground set $\mathcal{A}$ whose independent sets are the independent
subsets of $\mathcal{A}$. It was proved by Proudfoot and Speyer [18, Theorem
4] that the Stanley–Reisner ideal of the broken circuit complex of
$M(\mathcal{A})$ with respect to an arbitrary ordering of $\mathcal{A}$ (see
below) is an initial ideal of $I(\mathcal{A})$. Therefore,
$\mathbf{C}(\mathcal{A})$ and that complex share the same $h$-vector.

######  Remark 2.1.

The above relation between the Orlik–Terao ideal $I(\mathcal{A})$ and the
Stanley–Reisner ideal of the broken circuit complex of $M(\mathcal{A})$ was
exploited in [13, Proposition 2.4] to give a short proof for a formula
essentially due to Terao [30, Theorem 1.2] which relates the Hilbert series of
$\mathbf{C}(\mathcal{A})$ to the Poincaré polynomial of $\mathcal{A}$: Let
$(f_{0},\ldots,f_{r})$ be the $f$-vector of the broken circuit complex of
$M(\mathcal{A})$. We call $\pi(\mathcal{A},t)=\sum_{p=0}^{r}f_{i}t^{i}$ the
_Poincaré polynomial_ of $\mathcal{A}$. Then one has the following formula for
the Hilbert series of $\mathbf{C}(\mathcal{A})$:

$H_{\mathbf{C}(\mathcal{A})}(t)=\pi\Big{(}\mathcal{A},\frac{t}{1-t}\Big{)}.$

###  2.3. Matroids

A matroid $M$ on the ground set $E$ is a collection $\mathcal{I}$ of subsets
of $E$, called _independent sets_ , satisfying the following conditions:

  1. (i)

$\emptyset\in\mathcal{I};$

  2. (ii)

If $I\in\mathcal{I}$ and $I^{\prime}\subseteq I$, then
$I^{\prime}\in\mathcal{I}$;

  3. (iii)

If $I,I^{\prime}\in\mathcal{I}$ and $|I^{\prime}|<|I|$, then there is an
element $e\in I-I^{\prime}$ such that $I^{\prime}\cup\\{e\\}\in\mathcal{I}.$

Maximal independent sets of $M$ are called _bases_. They have the same
cardinality which is called the _rank_ of $M$. _Dependent sets_ are subsets of
$E$ that are not in $\mathcal{I}$. Minimal dependent sets are called
_circuits_. Denote by $\mathcal{C}(M)$ the set of all circuits of $M$.
Clearly, $\mathcal{C}(M)$ determines $M$: $\mathcal{I}$ consists of subsets of
$E$ that do not contain any member of $\mathcal{C}(M)$.

Let $e\in M$ (by abuse of notation, we also write $e\in M$ if $e\in E$). Then
$e$ is a _loop_ if $\\{e\\}$ is a circuit of $M$. We call $e$ a _coloop_ if it
is contained in every basis of $M$. Two elements $e,f\in M$ are _parallel_ if
they form a circuit. A _parallel class_ of $M$ is a maximal subset of $E$
whose any two distinct members are parallel and whose no member is a loop. A
parallel class is _non-trivial_ if it contains at least two elements. The
matroid $M$ is _simple_ if it has no loops and no non-trivial parallel
classes. In general, one may associate to an arbitrary matroid $M$ a simple
matroid $\overline{M}$, called the _simplification_ of $M$, by identifying all
members of each parallel class and deleting all the loops from $M$.

The _deletion_ of $M$ at an element $e$, denoted $M-e$, is the matroid on the
ground set $E-e$ whose independent sets are those members of $\mathcal{I}$
which do not contain $e$. The _contraction_ of $M$ at $e$, denoted $M/e$, is
the matroid whose ground set is also $E-e$ and whose circuits consist of the
minimal non-empty members of $\\{C-e:C\in\mathcal{C}(M)\\}$.

A typical example of a matroid is the underlying matroid of a central
hyperplane arrangement introduced above. There are two other common examples
which will be concerned in this paper.

######  Example 2.2.

(i) Let $m\leq n$ be non-negative integers and let $E$ be an $n$-element set.
The _uniform matroid_ $U_{m,n}$ on $E$ is the matroid whose independent sets
are the subsets of $E$ of cardinality at most $m$. This matroid has rank $m$
and its circuits are the $(m+1)$-element subsets of $E$. Thus in particular,
$U_{m,n}$ is simple if and only if $m\geq 2$. Moreover, when $n=m+1$, the
matroid $U_{m,m+1}$ has a unique circuit which is often denoted by $C_{m+1}$.

(ii) Let $G$ be a graph whose edge set is $E.$ Let $\mathcal{C}$ be the set of
edge sets of cycles of $G$. Then $\mathcal{C}$ forms the set of circuits of a
matroid $M(G)$ on $E$. We call $M(G)$ the _cycle matroid_ (or _polygon
matroid_) of $G$. This matroid is simple if and only if $G$ is a simple graph.

Let $M_{1}$ and $M_{2}$ be two matroids on disjoint ground sets $E_{1}$ and
$E_{2}$. We define their _direct sum_ $M_{1}\oplus M_{2}$ to be the matroid on
the ground set $E_{1}\cup E_{2}$ whose independent sets are the unions of an
independent set of $M_{1}$ and an independent set of $M_{2}$. A matroid is
called _separable_ if it is the direct sum of two smaller matroids. A matroid
is _connected_ if it is not separable. Every matroid has a unique
decomposition as a direct sum $M=M_{1}\oplus\cdots\oplus M_{k}$ of connected
matroids; we call $M_{1},\ldots,M_{k}$ _connected components_ of $M$.

It is apparent from the definition that the collection of independent sets of
a matroid $M$ forms a simplicial complex. We call it the _matroid complex_ (or
_independence complex_) of $M$. When $M$ has rank $r$, this complex is a pure
complex of dimension $r-1$.

###  2.4. The broken circuit complex

Let $M$ be a matroid on the ground set $E$. Given a linear order $<$ on $E$, a
_broken circuit_ is a circuit with its least element removed. For each circuit
$C$ of $M$, the broken circuit corresponding to $C$ will be denoted by
$bc(C)$. The _broken circuit complex_ of $M$ with respect to $<$ is the family
of all subsets of $E$ that contain no broken circuit, denoted by $BC(M,<)$ (or
$BC(M)$ when the order $<$ is specified). It is a subcomplex of the same
dimension as the matroid complex of $M$. Note that the broken circuit complex
of $M$ is isomorphic to that of its simplification [1, Proposition 7.4.1], so
one always may assume that the matroid $M$ is simple when working with its
broken circuit complex.

Fix a linear order on $E$ and let $e_{0}$ be the smallest element of $E$. Then
$BC(M)$ is the cone over the _reduced broken circuit complex_
$\overline{BC}(M)$ with apex $e_{0}$. Here, $\overline{BC}(M)$ is the family
of all subsets of $E-e_{0}$ that do not contain any broken circuit. It is
well-known that $BC(M)$ and $\overline{BC}(M)$ are shellable complexes; see
[19] and also [1, 7.4]. Hence, when $M$ has rank $r$, these complexes are
Cohen–Macaulay of dimension $r-1$ and $r-2$, respectively. Clearly, $BC(M)$ is
a Gorenstein (resp. complete intersection) complex if and only if so is
$\overline{BC}(M)$.

Observe that the broken circuit complex $BC(M,<)$ depends on the given order
$<$ in the sense that another order may yield a non-isomorphic broken circuit
complex. However, its $h$-vector is independent of the choice of order; see
[1, 7.4]. We summarize some properties of the $h$-vector of the broken circuit
complex in the next proposition.

######  Proposition 2.3.

Assume that $M$ is a simple matroid of rank $r$ on an $n$-element ground set.
Let $(h_{0},h_{1},\ldots,h_{r})$ and
$h_{M}(t)=h_{0}t^{r}+h_{1}t^{r-1}+\cdots+h_{r}$ be respectively the $h$-vector
and the $h$-polynomial of a broken circuit complex of $M$. Then the following
statements hold:

  1. (i)

$h_{0}=1,\ h_{1}=n-r,\ h_{r-1}=\beta(M)$, and $h_{r}=0$.

  2. (ii)

If $M=M_{1}\oplus M_{2}$, then $h_{M}(t)=h_{M_{1}}(t)h_{M_{2}}(t)$.

  3. (iii)

$M$ has $k$ connected components if and only if $k$ is the smallest number
such that $h_{r-k}\neq 0$. In particular, $\beta(M)>0$ if and only if $M$ is
connected.

  4. (iv)

If $e\in M$ is not a coloop, then $h_{M}(t)=h_{M-e}(t)+h_{\overline{M/e}}(t)$,
where $\overline{M/e}$ is the simplification of $M/e$. In particular, if $M$
is connected then either $M-e$ or $\overline{M/e}$ is connected. Moreover, if
$\beta(M)=1$ then exactly one of $M-e$ and $\overline{M/e}$ is connected.

  5. (v)

Assume that $M$ is representable. Let $s$ be the largest index such that
$h_{s}\neq 0$. Then $\sum_{j=0}^{i}h_{j}\leq\sum_{j=0}^{i}h_{s-j}$ for all
$i=0,\ldots,s$.

In the above proposition, $\beta(M)$ is the _beta invariant_ of $M$, defined
by Crapo [9]. The reader is referred to [33] for more information on this
invariant. The case $\beta(M)=1$ will be discussed in the next subsection.

###### Proof of Proposition 2.3.

Note that $h_{M}(t)=T_{M}(t,0)$, where $T_{M}(t,u)$ is the Tutte polynomial of
$M$ [1, p. 240]. So (i)-(iv) can be deduced from the corresponding properties
of the Tutte polynomial which are contained in [7, 6.2]. To prove (v), let
$\mathcal{A}$ be a central hyperplane arrangement such that
$M=M(\mathcal{A})$. Then the $h$-vector of the broken circuit complex of $M$
coincides with the $h$-vector of the Orlik–Terao algebra
$\mathbf{C}(\mathcal{A})$ of $\mathcal{A}$. Since $\mathbf{C}(\mathcal{A})$ is
a standard graded Cohen–Macaulay domain, the result follows from [24, Theorem
2.1]. ∎

######  Remark 2.4.

Regarding Proposition 2.3(v), a major open question in matroid theory asks
whether the stronger inequalities $h_{i}\leq h_{s-i},$ $i=0,\ldots,\lfloor
s/2\rfloor$, hold for arbitrary matroids; see [27]. As far as we know, the
question has not been answered even for the case $h_{s}=1$. In this context,
it is natural to ask whether Proposition 2.3(v) holds without the assumption
on the representability of the matroid. We personally expect an affirmative
answer. But it would be much more fascinating if the answer was negative.
Because in this case the answer to the above open question would also be
negative and Proposition 2.3(v) would give a non-trivial necessary condition
on the Tutte polynomial for the representability of matroids.

###  2.5. Series-parallel networks

The notion of series-parallel networks has its origin in electrical network
theory. We recall here several properties of series and parallel connections.
A full treatment of this topic may be found in [3] or [17].

Let $M_{1},M_{2}$ be matroids on the ground sets $E_{1},E_{2}$ with $E_{1}\cap
E_{2}=\\{e\\}$, where $e$ is neither a loop nor a coloop in $M_{1}$ or
$M_{2}$. Denote by $\mathcal{C}(M)$ the set of circuits of a matroid $M$. Then
the _series connection_ $S(M_{1},M_{2})$ and the _parallel connection_
$P(M_{1},M_{2})$ of $M_{1},M_{2}$ relative to $e$ are the matroids on the
ground set $E_{1}\cup E_{2}$ whose sets of circuits are respectively:

$\displaystyle\mathcal{C}(S(M_{1},M_{2}))$
$\displaystyle=\mathcal{C}(M_{1}-e)\cup\mathcal{C}(M_{2}-e)\cup\\{C_{1}\cup
C_{2}:e\in C_{i}\in\mathcal{C}(M_{i})\ \text{ for }\ i=1,2\\},$
$\displaystyle\mathcal{C}(P(M_{1},M_{2}))$
$\displaystyle=\mathcal{C}(M_{1})\cup\mathcal{C}(M_{2})\cup\\{C_{1}\cup
C_{2}-e:e\in C_{i}\in\mathcal{C}(M_{i})\ \text{ for }\ i=1,2\\}.$

The following properties of series and parallel connections are given in [3,
Proposition 4.6, Corollary 4.11, Theorem 6.16(v)] and [5, Proposition 2.3].

######  Proposition 2.5.

Let $M,M^{\prime}$ be matroids on the ground sets $E,E^{\prime}$ with $E\cap
E^{\prime}=\\{e\\}$, where $e$ is neither a loop nor a coloop in $M$ or
$M^{\prime}$. Let $S(M,M^{\prime})$ and $P(M,M^{\prime})$ be the series and
parallel connections of $M,M^{\prime}$ relative to $e$, respectively. Then the
following statements hold:

  1. (i)

$P(M,M^{\prime})$ is connected (resp. simple) if and only if both $M$ and
$M^{\prime}$ are connected (resp. simple).

  2. (ii)

$S(M,M^{\prime})$ is simple if and only if $M$ and $M^{\prime}$ are loopless,
each has no $2$-element circuit which do not contain $e$, and each has at most
one $2$-element circuit which contains $e$. In particular, if this is the
case, then $S(M,M^{\prime})$ has at most one $3$-element circuit which
contains $e$.

  3. (iii)

$h_{P(M,M^{\prime})}(t)=t^{-1}h_{M}(t)h_{M^{\prime}}(t)$, where $h_{M}(t)$ is
the $h$-polynomial of a broken circuit complex of $M$.

  4. (iv)

Assume $M$ is connected and $f\in M$. Then $M-f$ is separable if and only if
$M$ is a series connection relative to $f$; $\overline{M/f}$ is separable if
and only if $M$ is a parallel connection relative to $f$, where
$\overline{M/f}$ is the simplification of $M/f$.

  5. (v)

If $M$ is connected, then $M$ can be decomposed as an iterated parallel
connection of parallel irreducible connected matroids.

A matroid $M$ is a _series-parallel network_ if it can be obtained from the
$2$-element circuit $C_{2}$ by a sequence of operations each of which is
either a series or a parallel connection. Some characterizations of series-
parallel networks are provided in the next proposition. For the proof, see [3,
Theorem 7.6] and [17, Theorem 5.4.10, Corollary 11.2.15].

######  Proposition 2.6.

Let $M$ be a connected matroid on a ground set of two or more elements. Then
the following statements are equivalent:

  1. (i)

$M$ is a series-parallel network;

  2. (ii)

$\beta(M)=1$;

  3. (iii)

$M$ is the cycle matroid of a block $G$ having no subgraph that is a
subdivision of the complete graph $K_{4}$.

Here by a _block_ we mean a connected graph (in the usual sense of graph
theory) whose cycle matroid is connected. The graph $G$ in the last statement
of the above proposition is also called a (graphical) series-parallel network.
Note that such a graph which is simple always contains a vertex of degree 2,
by a result due to Dirac [11]; see also [17, Lemma 5.4.1].

##  3\. The Gorensteinness of the broken circuit complex

In this section we prove Theorem 1.1 and discuss some related results. We will
need a characterization of Gorenstein complexes due to Hochster [12,
Proposition 5.5, Theorem 6.7] (see also [25, Theorem 5.1]):

######  Lemma 3.1.

Let $K$ be a field. Let $\Delta$ be a simplicial complex and
$\Gamma:=\operatorname{core}\Delta$. Denote by $\widetilde{\chi}(\Gamma)$ the
reduced Euler characteristic of $\Gamma$. Then $\Delta$ is Gorenstein over $K$
if and only if one of the following conditions holds:

  1. (i)

$\Delta=\emptyset$;

  2. (ii)

$\Delta$ consists of one or two vertices;

  3. (iii)

$\Delta$ is Cohen–Macaulay over $K$ of dimension $\geq 1$,
$\widetilde{\chi}(\Gamma)=(-1)^{\dim\Gamma}$, and for any face $F$ with
$\dim\operatorname{link}_{\Delta}F=1$, $\operatorname{link}_{\Delta}F$ is
either an $n$-gon ($n\geq 3$), or a path with at most 3 vertices.

We will also need the following characterization of (locally) complete
intersection complexes due to Terai and Yoshida [29, Corollary 1.10,
Proposition 1.11]:

######  Lemma 3.2.

Let $K$ be a field. Let $\Delta$ be a simplicial complex of dimension $\geq
1$. Assume that $K[\Delta]$ satisfies Serre’s condition $(S_{2})$.

  1. (a)

If $\dim\Delta=1$, then the following conditions are equivalent:

    1. (i)

$\Delta$ is a locally complete intersection complex;

    2. (ii)

$\Delta$ is a locally Gorenstein complex;

    3. (iii)

$\Delta$ is either an $n$-gon ($n\geq 3$), or an $m$-vertex path ($m\geq 2$).

  2. (b)

If $\dim\Delta\geq 2$, then the following conditions are equivalent:

    1. (i)

$\Delta$ is a complete intersection complex;

    2. (ii)

$\Delta$ is a locally complete intersection complex;

    3. (iii)

for any face $F$ with $\dim\operatorname{link}_{\Delta}F=1$,
$\operatorname{link}_{\Delta}F$ is a complete intersection complex.

Recall that an arbitrary Noetherian ring $R$ is said to _satisfy Serre’s
condition $(S_{2})$_ if $\operatorname{depth}R_{P}\geq\min\\{\dim R_{P},2\\}$
for every prime ideal $P$ of $R$. So for example, all Cohen–Macaulay rings
satisfy $(S_{2})$. For an ideal $I$ in a polynomial ring $S$, we also say that
$I$ satisfies $(S_{2})$ when so does the quotient ring $S/I$.

From the above results one might see that the main part in the proof of
Theorem 1.1 is the implication (iii)$\Rightarrow$(iv). For this, some more
preparations are needed.

######  Lemma 3.3.

Let $\Delta$ and $\Sigma$ be respectively the broken circuit complex and the
matroid complex of an ordered matroid $(M,<)$. Let $F$ be a face of $\Delta$.
Denote by $V_{1}$ and $V_{2}$ the vertex sets of
$\operatorname{link}_{\Delta}F$ and $\operatorname{link}_{\Sigma}F$,
respectively. Assume $V_{2}-V_{1}\neq\emptyset.$ Then there exists $u\in
V_{1}$ such that $u<v$ for all $v\in V_{2}-V_{1}$.

###### Proof.

As $\Delta\subseteq\Sigma$, it is obvious that $V_{1}\subseteq V_{2}$. Let
$v=\min(V_{2}-V_{1})$. Since $v\not\in V_{1}$, $F\cup\\{v\\}\not\in\Delta$.
Thus there exists a circuit $C$ of $M$ such that $bc(C)\subseteq
F\cup\\{v\\}$. We have $v\in bc(C)$, since otherwise $bc(C)\subseteq F$,
contradicting the fact that $F\in\Delta$. Hence $u:=\min C<v.$ We will show
that $u\in V_{1}.$ Indeed, since $v\in V_{2}$, $F\cup\\{v\\}\in\Sigma$, i.e.,
$F\cup\\{v\\}$ is an independent set of $M$. Therefore, $C$ is the unique
circuit contained in $F\cup\\{v,u\\}$; see [17, Proposition 1.1.6]. It follows
that $F\cup\\{u\\}$ is an independent set of $M$, or in other words, $u\in
V_{2}.$ Note that $u\not\in V_{2}-V_{1}$ (since $u<v$), we obtain $u\in
V_{1}$, as desired. ∎

######  Lemma 3.4.

Let $\Delta$ be the broken circuit complex of an ordered matroid $(M,<)$.
Assume $\dim\Delta\geq 1$. If $F$ is a face of $\Delta$ with
$\operatorname{link}_{\Delta}F$ an $n$-gon, then $n\leq 4$.

###### Proof.

Suppose on the contrary that there exists a face $F\in\Delta$ such that
$\operatorname{link}_{\Delta}F$ is an $n$-gon $v_{1}v_{2}\ldots v_{n}$ with
$n\geq 5$ ($\\{v_{1},v_{2}\\},\\{v_{2},v_{3}\\},\ldots,\\{v_{n},v_{1}\\}$ are
the edges of $v_{1}v_{2}\ldots v_{n}$). Assume
$v_{1}=\min\\{v_{i}:i=1,\ldots,n\\}$. Since $B_{1}=F\cup\\{v_{1},v_{2}\\}$ and
$B_{2}=F\cup\\{v_{3},v_{4}\\}$ are bases of $M$, either
$B_{3}=F\cup\\{v_{1},v_{3}\\}$ or $B_{4}=F\cup\\{v_{1},v_{4}\\}$ is a basis of
$M$; see, e.g., [17, Corollary 1.2.5]. Note that neither $B_{3}$ nor $B_{4}$
belongs to $\Delta$ as $n\geq 5$. So in the following argument we may assume
without loss of generality that $B_{3}$ is a basis of $M$ and
$B_{3}\not\in\Delta$. Denote by $\Sigma$ the matroid complex of $M$. Then
$v_{1}\in\operatorname{link}_{\Sigma}(F\cup\\{v_{3}\\})-\operatorname{link}_{\Delta}(F\cup\\{v_{3}\\})$.
By Lemma 3.3, there exists
$u\in\operatorname{link}_{\Delta}(F\cup\\{v_{3}\\})$ with $u<v_{1}$. In
particular, $u\in\operatorname{link}_{\Delta}F$ and $u<v_{1}$. But this is
impossible because it contradicts our assumption on $v_{1}$. ∎

######  Lemma 3.5.

Let $\Delta$ be the broken circuit complex of an ordered matroid $(M,<)$. If
$\Delta$ is an $m$-vertex path, then $m\leq 3$.

###### Proof.

Recall that $\Delta$ is a cone over the reduced broken circuit complex
$\overline{BC}(M)$ which consists of vertices since $\dim\Delta=1$. It follows
that if $\Delta$ is a path then $\overline{BC}(M)$ has at most 2 vertices, and
hence, $\Delta$ has at most 3 vertices. ∎

We are now ready to prove Theorem 1.1.

###### Proof of Theorem 1.1.

The case $\Delta=\emptyset$ is trivial. The case $\dim\Delta=0$ is also
trivial, because in this case $\Delta$ consists of only one vertex (since
$\Delta$ is a cone over the empty complex $\overline{BC}(M)$).

If $\dim\Delta=1$, then $F=\emptyset$ is the only face of $\Delta$ with
$\dim\operatorname{link}_{\Delta}F=1$, and
$\operatorname{link}_{\Delta}\emptyset=\Delta$. So we have
(i)$\Leftrightarrow$(iii) and (iv)$\Leftrightarrow$(vi). Moreover, the
equivalence (ii)$\Leftrightarrow$(v) follows from Lemma 3.2, and the
implication (i)$\Rightarrow$(ii) is well-known; see [12, Proposition 5.6].
Thus it remains to show (ii)$\Rightarrow$(vi). By Lemma 3.2, $\Delta$ is
either an $n$-gon ($n\geq 3$) or an $m$-vertex path ($m\geq 2$). Now by Lemma
3.4 and Lemma 3.5, $n\leq 4$ and $m\leq 3$. In any case, it is easy to see
that $\Delta$ is a complete intersection.

Now assume that $\dim\Delta\geq 2$. Then we have
(iv)$\Leftrightarrow$(v)$\Leftrightarrow$(vi) from Lemma 3.2 and
(vi)$\Rightarrow$(i)$\Rightarrow$(ii)$\Rightarrow$(iii) as it is well-known;
see again [12, Proposition 5.6]. Finally, to prove (iii)$\Rightarrow$(iv), let
$F$ be a face of $\Delta$ with $\dim\operatorname{link}_{\Delta}F=1$. Then by
Lemma 3.1, $\operatorname{link}_{\Delta}F$ is either an $n$-gon or a path with
at most 3 vertices. Using Lemma 3.4, one gets that $n\leq 4$ when
$\operatorname{link}_{\Delta}F$ is an $n$-gon. So we come to the same
situation as in the case $\dim\Delta=1$ above, in which
$\operatorname{link}_{\Delta}F$ is easily seen to be a complete intersection.
∎

A classical theorem due to Cowsik and Nori [8] implies that a simplicial
complex $\Delta$ is a complete intersection if and only if all powers of its
Stanley–Reisner ideal $I_{\Delta}^{m}$ are Cohen–Macaulay. In [28], Terai and
Trung give a refinement for this result: they show that $\Delta$ is a complete
intersection if and only if $I_{\Delta}^{m}$ is Cohen–Macaulay if and only if
$I_{\Delta}^{m}$ satisfies Serre’s condition $(S_{2})$ for some $m\geq 3$.
They also point out that there are simplicial complexes $\Delta$, e.g., the
$5$-gon, for which $I_{\Delta}^{2}$ is Cohen–Macaulay but $I_{\Delta}^{m}$ is
not Cohen–Macaulay for every $m\geq 3$. As one might have seen from above,
this is not the case for the broken circuit complex.

######  Proposition 3.6.

Let $\Delta$ be the broken circuit complex of an ordered matroid $(M,<)$. Let
$K$ be an arbitrary field. Denote by $I_{\Delta}$ the Stanley–Reisner ideal of
$\Delta$ over $K$. Then the following conditions are equivalent:

  1. (i)

$\Delta$ is a complete intersection complex;

  2. (ii)

$I_{\Delta}^{m}$ is Cohen–Macaulay for every $m\geq 1$;

  3. (iii)

$I_{\Delta}^{m}$ is Cohen–Macaulay for some $m\geq 2$;

  4. (iv)

$I_{\Delta}^{m}$ satisfies $(S_{2})$ for some $m\geq 2$.

###### Proof.

As in the proof of Theorem 1.1, the cases $\Delta=\emptyset$ or $\dim\Delta=0$
are trivial. Let $\dim\Delta\geq 1$. By [28, Theorem 4.3], it suffices to show
that $\Delta$ is a complete intersection when $I_{\Delta}^{2}$ satisfies
$(S_{2})$. If $\dim\Delta=1$, then $(S_{2})$ means that $I_{\Delta}^{2}$ is
Cohen–Macaulay. Hence by [14, Corollary 3.4], either $\Delta$ has at most 3
vertices, or $\Delta$ is a 4-gon or 5-gon. But by Lemma 3.4, $\Delta$ cannot
be a 5-gon. For the remaining possibilities of $\Delta$, one checks easily
that $\Delta$ is a complete intersection. Now assume $\dim\Delta\geq 2$. Let
$F$ be a face of $\Delta$ with $\dim\operatorname{link}_{\Delta}F=1$. Since
$I_{\Delta}^{2}$ satisfies $(S_{2})$, $I_{\operatorname{link}_{\Delta}F}^{2}$
also satisfies $(S_{2})$; see [28, Corollary 4.2]. Note that
$\operatorname{link}_{\Delta}F$ cannot be a 5-gon by Lemma 3.4. So one may use
the same argument as in the case $\dim\Delta=1$ to get that
$\operatorname{link}_{\Delta}F$ is a complete intersection. Now by Lemma 3.2,
$\Delta$ is a complete intersection. ∎

The equivalence of conditions (i) and (ii) in the following consequence was
proved in [26, Theorem 4.4.10].

######  Corollary 3.7.

Let $\Sigma$ be the matroid complex of a matroid $M$. Let $K$ be an arbitrary
field. Denote by $I_{\Sigma}$ the Stanley–Reisner ideal of $\Sigma$ over $K$.
Then the following conditions are equivalent:

  1. (i)

$\Sigma$ is Gorenstein over $K$;

  2. (ii)

$\Sigma$ is a complete intersection complex;

  3. (iii)

$I_{\Sigma}^{m}$ is Cohen–Macaulay for every $m\geq 1$;

  4. (iv)

$I_{\Sigma}^{m}$ is Cohen–Macaulay for some $m\geq 2$;

  5. (v)

$I_{\Sigma}^{m}$ satisfies $(S_{2})$ for some $m\geq 2$.

Moreover, if $\dim\Sigma\geq 1$, then each of the above conditions is
equivalent to any one of the following:

  1. (vi)

$\Sigma$ is locally Gorenstein over $K$;

  2. (vii)

for any face $F$ with $\dim\operatorname{link}_{\Sigma}F=1$,
$\operatorname{link}_{\Sigma}F$ is Gorenstein over $K$;

  3. (viii)

for any face $F$ with $\dim\operatorname{link}_{\Sigma}F=1$,
$\operatorname{link}_{\Sigma}F$ is a complete intersection complex;

  4. (ix)

$\Sigma$ is a locally complete intersection complex.

###### Proof.

Let $\tilde{M}$ be the free dual extension of $M$. Then $\Sigma$ is isomorphic
to the reduced broken circuit complex of $\tilde{M}$; see [4, Theorem 4.2]. So
we may consider the broken circuit complex $\tilde{\Delta}$ of $\tilde{M}$ as
a cone over $\Sigma$. It follows that $\Sigma$ is Gorenstein (resp. a complete
intersection) if and only if $\tilde{\Delta}$ is. Moreover, if
$I_{\Sigma}^{m}$ satisfies $(S_{2})$ then so does $I_{\tilde{\Delta}}^{m}$.
Therefore, from Theorem 1.1 and Proposition 3.6 we get the equivalence of
conditions (i)-(v).

Observe that
$\operatorname{link}_{\Sigma}F=\operatorname{link}_{\tilde{\Delta}}(F\cup
e_{0})$ for all $F\in\Sigma$, where $e_{0}$ is the apex of $\tilde{\Delta}$
over $\Sigma$. So if $\dim\Sigma\geq 2$, then the implication
(vii)$\Rightarrow$(viii) can be proved as in the proof of Theorem 1.1. From
this the equivalence of conditions (i) and (ii) with (vi)-(ix) follows easily.
Now assume that $\dim\Sigma=1$. The only thing we need to prove here is the
implication (vi)$\Rightarrow$(i). Similar to the proof of Theorem 1.1, this
will be done once we have shown that $m\leq 3$ when $\Sigma$ is an $m$-vertex
path. Suppose $\Sigma$ is a path containing 4 consecutive vertices
$v_{1},v_{2},v_{3},v_{4}$ in that order. Then $\\{v_{1},v_{2}\\}$ and
$\\{v_{3},v_{4}\\}$ are bases of $M$. It follows that either
$\\{v_{1},v_{3}\\}$ or $\\{v_{1},v_{4}\\}$ is a basis of $M$; see [17,
Corollary 1.2.5]. Thus $\\{v_{1},v_{3}\\}\in\Sigma$ or
$\\{v_{1},v_{4}\\}\in\Sigma$, and hence, $\Sigma$ cannot be a path. This
contradiction concludes the proof. ∎

######  Remark 3.8.

In Corollary 3.7, without the assumption on the dimension of the matroid
complex, conditions (i)-(v) may not be equivalent to conditions (vi)-(ix). For
instance, let $M$ be the uniform matroid $U_{1,n}$ with $n\geq 3$. Then
$\Sigma$ is a 0-dimensional complex consists of $n$ vertices. So $\Sigma$ is
locally Gorenstein (and a locally complete intersection), but not Gorenstein
by Lemma 3.1(ii). Of course in this case, conditions (vii), (viii) in
Corollary 3.7 hold vacuously.

##  4\. The Gorensteinness of the Orlik–Terao algebra

We present the proofs of Theorem 1.2 and Theorem 1.3 in this section. For
this, the following simple lemma will play an important role.

######  Lemma 4.1.

Let $M_{1},M_{2},M_{3}$ be simple matroids such that $M_{1}$ is either a
direct sum or a parallel connection of $M_{2}$ and $M_{3}$. Let
$(h_{i,0},\ldots,h_{i,s_{i}})$ denote the $h$-vector of a broken circuit
complex of $M_{i}$, where $h_{i,s_{i}}\neq 0$. Then the following conditions
are equivalent:

  1. (i)

$h_{1,0}=h_{1,s_{1}}$ and $h_{1,1}=h_{1,s_{1}-1}$;

  2. (ii)

$h_{i,0}=h_{i,s_{i}}$ and $h_{i,1}=h_{i,s_{i}-1}$ for $i=2,3$.

###### Proof.

Let $r_{i}=\operatorname{rank}(M_{i})$ and denote by
$h_{M_{i}}(t)=\sum_{j=0}^{s_{i}}h_{i,j}t^{r_{i}-j}$ the $h$-polynomial of the
broken circuit complex of $M_{i}$. Then by Proposition 2.3(ii) and Proposition
2.5(iii), we have

$h_{M_{1}}(t)=\begin{cases}h_{M_{2}}(t)h_{M_{3}}(t)&\text{if
}M_{1}=M_{2}\oplus M_{3},\\\ t^{-1}h_{M_{2}}(t)h_{M_{3}}(t)&\text{if
}M_{1}=P(M_{2},M_{3}).\end{cases}$

It follows that in both cases,

$\displaystyle h_{1,0}$ $\displaystyle=h_{2,0}h_{3,0},$ $\displaystyle
h_{1,1}$ $\displaystyle=h_{2,0}h_{3,1}+h_{2,1}h_{3,0},$ $\displaystyle
h_{1,s_{1}}$ $\displaystyle=h_{2,s_{2}}h_{3,s_{3}},$ $\displaystyle
h_{1,s_{1}-1}$
$\displaystyle=h_{2,s_{2}}h_{3,s_{3}-1}+h_{2,s_{2}-1}h_{3,s_{3}}.$

Thus, the implication (ii)$\Rightarrow$(i) is obvious. For the converse, we
first have $h_{i,s_{i}}=h_{i,0}=1$ for $i=2,3$ because
$h_{1,s_{1}}=h_{1,0}=1$. Let $N$ be a connected component of $M_{2}$. Then
$h_{N}(t)$, the $h$-polynomial of the broken circuit complex of $N$, divides
$h_{M_{2}}(t)$ by Proposition 2.3(ii). It follows that $\beta(N)$, which is
the last nonzero coefficient of $h_{N}(t)$ by Proposition 2.3(i) and (iii),
must divide $h_{2,s_{2}}=1$. Hence, $\beta(N)=1$. From Proposition 2.6, this
means that $N$ is either a coloop or a series-parallel network. Thus every
connected component of $M_{2}$ is a graphic matroid, which implies that
$M_{2}$ itself is also graphic. The same conclusion holds for $M_{3}$. In
particular, $M_{2}$ and $M_{3}$ are representable; see [17, Proposition
5.1.2]. Now applying Proposition 2.3(v) we get $h_{i,1}\leq h_{i,s_{i}-1}$ for
$i=2,3$. The equalities are now forced by the relation
$h_{1,1}=h_{1,s_{1}-1}$. ∎

###### Proof of Theorem 1.2.

(i)$\Rightarrow$(ii)$\Rightarrow$(iii) is trivial.

(iii)$\Rightarrow$(iv): By Proposition 2.5(i), (v) and Lemma 4.1 we may assume
that $M$ is a simple, connected, and parallel irreducible. Let
$r=\operatorname{rank}M$. If $r=1$, then $M$ is a coloop. Suppose $r\geq 2$.
We prove by induction on $r$ that $M\cong U_{r,r+1}$.

If $r=2$, then the $h$-polynomial of $M$ is $h_{M}(t)=t^{2}+t$. This can be
the case only when $M\cong U_{2,3}$; see [5, Theorem 3.1(1b)]. Assume now that
$r\geq 3$. Since $M$ is connected, $h_{r-1}\neq 0$ by Proposition 2.3(iii). So
by assumption, $\beta(M)=h_{r-1}=h_{0}=1$. It follows from Proposition 2.6
that $M=M(G)$, where $G$ is a simple, $2$-connected graphical series-parallel
network. Note that $G$ has a vertex $v$ of degree 2. Let $e_{1},e_{2}$ be the
two incident edges of $v$. By Proposition 2.3(iv),

(1) $h_{M}(t)=h_{M-e_{1}}(t)+h_{\overline{M/e_{1}}}(t),$

where $\overline{M/e_{1}}$ is the simplification of $M/e_{1}.$ Since
$\operatorname{rank}(M-e_{1})=r$ and
$\operatorname{rank}(\overline{M/e_{1}})=r-1$ [4, Proposition 2.4], we may
write

$\displaystyle h_{M-e_{1}}(t)$
$\displaystyle=h^{\prime}_{0}t^{r}+h^{\prime}_{1}t^{r-1}+\cdots+h^{\prime}_{r-1}t,$
$\displaystyle h_{\overline{M/e_{1}}}(t)$
$\displaystyle=h^{\prime\prime}_{0}t^{r-1}+h^{\prime\prime}_{1}t^{r-2}+\cdots+h^{\prime\prime}_{r-2}t.$

We will show that the matroid $\overline{M/e_{1}}$ satisfies the induction
hypothesis. First of all, $\overline{M/e_{1}}$ is simple by definition. The
remaining conditions will be shown through the following claims.

######  Claim 1.

$\overline{M/e_{1}}$ is connected.

This follows from Proposition 2.5(iv) and the assumption that $M$ is parallel
irreducible.

######  Claim 2.

$h^{\prime\prime}_{r-2}=h^{\prime\prime}_{0}=1$.

Since $\overline{M/e_{1}}$ is connected, $h^{\prime\prime}_{r-2}\neq 0$. On
the other hand, $h^{\prime\prime}_{r-2}\leq h_{r-1}=1$ by (1). Hence,
$h^{\prime\prime}_{r-2}=1$.

######  Claim 3.

$h^{\prime\prime}_{1}=h^{\prime\prime}_{r-3}.$

We have $h^{\prime}_{r-1}=h_{r-1}-h^{\prime\prime}_{r-2}=0$, thus $M-e_{1}$ is
separable by Proposition 2.3(iii). It then follows from Proposition 2.5(iv)
that $M$ is a series connection relative to $e_{1}$. Hence $M$ has at most one
3-element circuit containing $e_{1}$ by Proposition 2.5(ii). This implies that
$|\overline{M/e_{1}}|\geq|M/e_{1}|-1=|M|-2$ since the contraction $M/e_{1}$
contains at most one 2-element circuit. Now by Proposition 2.3(i),

$h^{\prime\prime}_{1}=|\overline{M/e_{1}}|-\operatorname{rank}(\overline{M/e_{1}})\geq(|M|-2)-(r-1)=h_{1}-1.$

On the other hand, from Proposition 2.3(v) we have
$h^{\prime\prime}_{0}+h^{\prime\prime}_{1}\leq
h^{\prime\prime}_{r-2}+h^{\prime\prime}_{r-3}$, which implies
$h^{\prime\prime}_{1}\leq h^{\prime\prime}_{r-3}$ by Claim 2. Moreover, it
follows from (1) that $h^{\prime\prime}_{r-3}\leq
h^{\prime\prime}_{r-3}+h^{\prime}_{r-2}=h_{r-2}$. Thus we have shown

$h_{1}-1\leq h^{\prime\prime}_{1}\leq h^{\prime\prime}_{r-3}\leq
h_{r-2}=h_{1}.$

Therefore, the equality $h^{\prime\prime}_{1}=h^{\prime\prime}_{r-3}$ will
follow if we prove that $h^{\prime\prime}_{1}\neq h_{1}-1$. Indeed, if
$h^{\prime\prime}_{1}=h_{1}-1$ then $M/e_{1}$ contains one 2-element circuit.
This means that $M$ has one 3-element circuit $C$ which contains $e_{1}$. Note
that every circuit of $M$ containing $e_{1}$ must also contain $e_{2}$ since
the vertex $v$ has degree 2. So $C=\\{e_{1},e_{2},e\\}$ with $e\in M$. As
$r\geq 3$, $M$ properly contains $C$. Now one may easily check that $M$ is the
parallel connection of $M-\\{e_{1},e_{2}\\}$ and $C$ relative to $e$. But this
contradicts the assumption that $M$ is parallel irreducible.

######  Claim 4.

$\overline{M/e_{1}}$ is parallel irreducible.

It follows from the proof of Claim 3 that $\overline{M/e_{1}}=M/e_{1}$. Thus
by Proposition 2.5(iv), we need to prove that $\overline{(M/e_{1})/e}$ is
connected for all $e\in M/e_{1}$. It then suffices to show that $M/e_{1}-e$ is
separable for all $e\in M/e_{1}$, by Proposition 2.3(iv) and Claim 1. Suppose
there exists $e\in M/e_{1}$ with $M/e_{1}-e$ connected. We have
$M/e_{1}-e=(M-e)/e_{1}$; see [17, Proposition 3.1.26]. As $M$ is parallel
irreducible, $\overline{M/e}$ is connected by Proposition 2.5(iv). So by
Proposition 2.3(iv), $M-e$ is separable (recall that $\beta(M)=1$). Now since
$(M-e)/e_{1}$ is connected, we must have $M-e=\\{e_{1}\\}\oplus M^{\prime}$
with $M^{\prime}$ connected. It follows that $M-\\{e_{1},e\\}$ is connected,
and hence $M-e_{1}$ has at most 2 connected components. We will show that this
is impossible. Indeed, one sees from the proof of Claim 3 (i.e.
$h_{1}^{\prime\prime}=h_{1}$) that $h^{\prime\prime}_{r-3}=h_{r-2}$. This
implies $h^{\prime}_{r-2}=h_{r-2}-h^{\prime\prime}_{r-3}=0$. So by Proposition
2.3(iii), $M-e_{1}$ must have at least 3 connected components, a
contradiction.

Now we may use induction hypothesis to conclude that $M/e_{1}\cong U_{r-1,r}$.
Since $M$ is connected, it follows that $M\cong U_{r,r+1}$.

(iv)$\Rightarrow$(i): We may assume that $M$ is connected. The cases $M$ is a
coloop or $M\cong U_{m,m+1}$ are trivial. Now assume that
$M=P(M^{\prime},U_{m,m+1})$ is the parallel connection of two matroids
$M^{\prime}$ and $U_{m,m+1}$ relative to their only common point $e$, in which
$M^{\prime}$ admits an order $<$ on its ground set such that
$BC(M^{\prime},<)$ is a complete intersection. Then we have the following
description for the set of circuits of $M$:

(2) $\mathcal{C}(M)=\mathcal{C}(M^{\prime})\cup\\{C_{m+1}\\}\cup\\{C\cup
C_{m+1}-e:C\in\mathcal{C}(M^{\prime})\\},$

where $C_{m+1}$ is the unique circuit of the matroid $U_{m,m+1}$. Since
$BC(M^{\prime},<)$ is a complete intersection, the set
$\mathbf{m}(BC(M^{\prime},<))$ of minimal broken circuits of $(M^{\prime},<)$
consists of disjoint elements; see [13, Theorem 4.1]. Now let $\prec$ be an
extension of $<$ to the ground set of $M$ such that
$e=\min_{\prec}(U_{m,m+1})$. Then it can be easily deduced from (2) that
$\mathbf{m}(BC(M,\prec))=\mathbf{m}(BC(M^{\prime},<))\cup\\{C_{m+1}-e\\}$.
Hence $\mathbf{m}(BC(M,\prec))$ also consists of disjoint elements, from which
it follows that the broken circuit complex $BC(M,\prec)$ is a complete
intersection, again by [13, Theorem 4.1]. ∎

Finally, we prove Theorem 1.3.

###### Proof of Theorem 1.3.

(i)$\Rightarrow$(ii) is trivial. (iii)$\Rightarrow$(i) is also trivial because
for any ordering $<$ of $\mathcal{A}$ the Stanley–Reisner ideal of the broken
circuit complex $BC(M(\mathcal{A}),<)$ is an initial ideal of the Orlik–Terao
ideal of $\mathcal{A}$; see [18, Theorem 4].

(ii)$\Rightarrow$(iii): When $\mathbf{C}(\mathcal{A})$ is Gorenstein, it is
well-known that its $h$-vector is symmetric; see, e.g., [25, p. 51]. Since
this is also the $h$-vector of broken circuit complexes of $M(\mathcal{A})$,
Theorem 1.2 applies. ∎

## References

  * [1] A. Björner, _The homology and shellability of matroids and geometric lattices_. In _Matroid Applications_ , 226–283, Encyclopedia Math. Appl. 40, Cambridge University Press, Cambridge, 1992.
  * [2] A. Björner and G. Ziegler, _Broken circuit complexes: factorizations and generalizations_. J. Combin. Theory, Series B 51 (1991), no. 1, 96–126.
  * [3] T. Brylawski, _A combinatorial model for series-parallel networks_. Trans. Amer. Math. Soc. 154 (1971), 1–22.
  * [4] T. Brylawski, _The broken-circuit complex_. Trans. Amer. Math. Soc. 234 (1977), 417–433.
  * [5] T. Brylawski, _Connected matroids with the smallest Whitney numbers_. Discrete Math. 18 (1977), no. 3, 243–252.
  * [6] T. Brylawski and J. Oxley, _The broken-circuit complex: its structure and factorizations_. European J. Combin. 2 (1981), no. 2, 107–121.
  * [7] T. Brylawski and J. Oxley, _The Tutte polynomial and its applications_. In _Matroid Applications_ , 123–225, Encyclopedia Math. Appl. 40, Cambridge University Press, Cambridge, 1992.
  * [8] R. C. Cowsik and M. V. Nori, _On the fibres of blowing up_. J. Indian Math. Soc. (N.S.) 40 (1976), no. 1-4, 217–222.
  * [9] H. Crapo, _A higher invariant for matroids_. J. Combinatorial Theory 2 (1967), 406–417.
  * [10] G. Denham, M. Garrousian and S. Tohaneanu, _Modular decomposition of the Orlik–Terao algebra of a hyperplane arrangement_. To appear in Annals of Combinatorics.
  * [11] G. A. Dirac, _A property of 4-chromatic graphs and some remarks on critical graphs_. J. London Math. Soc. 27 (1952), 85–92.
  * [12] M. Hochster, _Cohen–Macaulay rings, combinatorics, and simplicial complexes_. In _Ring Theory II_ (Proc. Second Oklahoma Conference), 171–223, Dekker, New York, 1977.
  * [13] D. V. Le and T. Römer, _Broken circuit complexes and hyperplane arrangements_. J. Algebraic Combin. 38 (2013), no. 4, 989–1016.
  * [14] N. C. Minh and N. V. Trung, _Cohen–Macaulayness of powers of two-dimensional squarefree monomial ideals_. J. Algebra 322 (2009), no. 12, 4219–4227.
  * [15] P. Orlik and H. Terao, _Arrangements of Hyperplanes_. Grundlehren Math. Wiss., Bd. 300, Springer-Verlag, Berlin-Heidelberg-New York, 1992.
  * [16] P. Orlik and H. Terao, _Commutative algebras for arrangements_. Nagoya Math. J. 134 (1994), 65–73.
  * [17] J. Oxley, _Matroid theory, Second Edition_. Oxford Graduate Texts in Mathematics, vol. 21, Oxford University Press, Oxford, 2011.
  * [18] N. Proudfoot and D. Speyer, _A broken circuit ring_. Beiträge Algebra Geom. 47 (2006), 161–166.
  * [19] J. S. Provan, _Decompositions, shellings, and diameters of simplicial complexes and convex polyhedra_. Thesis, Cornell University, Ithaca, NY, 1977.
  * [20] H. Schenck, _Resonance varieties via blowups of $\mathbb{P}^{2}$ and scrolls_, Int. Math. Res. Not. 20 (2011), 4756–4778.
  * [21] H. Schenck, _Hyperplane Arrangements: Computations and Conjectures_. Arrangements of hyperplanes - Sapporo 2009, 323–358, Adv. Stud. Pure Math., 62, Math. Soc. Japan, Tokyo, 2012.
  * [22] H. Schenck and S. Tohaneanu, _The Orlik–Terao algebra and $2$-formality_. Math. Res. Lett. 16 (2009), 171–182.
  * [23] R. P. Stanley, _Hilbert functions of graded algebras_. Adv. Math. 28 (1978), 57–83.
  * [24] R. P. Stanley, _On the Hilbert function of a graded Cohen–Macaulay domain_. J. Pure Appl. Algebra 73 (1991), no. 3, 307–314.
  * [25] R. P. Stanley, _Combinatorics and Commutative Algebra, Second Edition_. Progress in Mathematics, vol. 41, Birkhäuser, 1996.
  * [26] E. Stokes, _The $h$-vectors of matroids and the arithmetic degree of squarefree strongly stable ideals_. Ph.D. thesis, University of Kentucky, 2008.
  * [27] E. Swartz, _$g$ -elements of matroid complexes_. J. Combin. Theory Ser. B 88 (2003), no. 2, 369–375.
  * [28] N. Terai and N. V. Trung, _Cohen–Macaulayness of large powers of Stanley–Reisner ideals_. Adv. Math. 229 (2012), no. 2, 711–730.
  * [29] N. Terai and K. Yoshida, _Locally complete intersection Stanley–Reisner ideals_. Illinois J. Math. 53 (2009), 413–429.
  * [30] H. Terao, _Algebras generated by reciprocals of linear forms_. J. Algebra 250 (2002), 549–558.
  * [31] H. Whitney, _A logical expansion in mathematics_. Bull. Amer. Math. Soc. 38 (1932), 572–579.
  * [32] H. Wilf, _Which polynomials are chromatic?_. Proc. 1973 Rome International Colloq. Combinatorial Theory I, pp. 247–257, Accademia Nazionale dei Lincei, Rome, 1976.
  * [33] T. Zaslavsky, _The Möbius function and the characteristic polynomial_. In _Combinatorial geometries_ , 114–138, Encyclopedia Math. Appl. 29, Cambridge University Press, Cambridge, 1987.


# Leveraging Video Coding Knowledge for Deep Video Enhancement

Thong Bach
<EMAIL_ADDRESS>Thuong Nguyen Canh
<EMAIL_ADDRESS>Van-Quang Nguyen
<EMAIL_ADDRESS>
###### Abstract

Recent advancements in deep learning techniques have significantly improved
the quality of compressed videos. However, previous approaches have not fully
exploited the motion characteristics of compressed videos, such as the drastic
change in motion between video contents and the hierarchical coding structure
of the compressed video. This study proposes a novel framework that leverages
the low-delay configuration of video compression to enhance the existing
state-of-the-art method, BasicVSR++. We incorporate a context-adaptive video
fusion method to enhance the final quality of compressed videos. The proposed
approach has been evaluated in the NTIRE22 challenge, a benchmark for video
restoration and enhancement, and achieved improvements in both quantitative
metrics and visual quality compared to the previous method.

##  1 Introduction

With the increasing demand for high-quality video transmission over the
Internet, video compression has become essential to efficiently transmit
videos over limited bandwidth. It has driven the development of video
compression standards such as H.265/HEVC [1], and beyond. However, compressed
videos suffer unavoidable compression artifacts. As a result, there is a
growing interest in the research community in enhancing the quality of
compressed videos.

Several studies have proposed methods to improve the quality of individual
frames in videos [2, 3] as well as leveraging temporal information between
frames [4, 5, 6, 7, 8, 9]. Most existing methods focus on the architecture
design of the model, which typically involves (i) designing the backbone
extraction module using CNNs or Transformers, (ii) designing the propagation
module to effectively capture information flow between frames, and (iii)
designing the enhancement module as a post-processing step to improve the
quality of the output video. However, there is often a little emphasis on
incorporating prior knowledge from video content such as motion information as
well as the compression algorithm. This represents an untapped potential for
improving the overall quality of the video compression process.

In this study, we propose several methods to enhance the performance of
BasicVSR++, a state-of-the-art video super-resolution method [10]. Our
approach begins by examining BasicVSR++’s performance with varying numbers of
input frames, taking into account the motion information of the content. As
compressed video uses the HEVC low-delay configuration, the first frame (also
known as the Intra frame) has significantly higher quality than the others. To
take advantage of this, we train a separate network called Intra frame
BasicVSR++ to improve the quality of the first frame. Finally, we introduce an
adaptive mechanism that combines multiple reconstructed instances with
different input sequence lengths to obtain the final enhanced output.

The experiments demonstrate that the proposed framework not only leverages the
low-delay configuration of video compression but also incorporates context-
adaptive video fusion to enhance the final quality of compressed videos. These
results demonstrate the potential of incorporating domain-specific knowledge
into deep learning models for advancing the state-of-the-art in compressed
video quality enhancement.

Figure 1:  Low-delay configuration of HEVC with a group of picture size of 4,
where the configuration has only one intra frame at the 0-th index and a
repeated group structure for 4 frames. Full references are only available from
the 13-th frame.

##  2 Performance Analysis of BasicVSR++

Figure 2: Performance variation of BasicVSR++ with respect to the number of
input frames. The per-frame PSNR difference between the outputs of trimmed
videos with different gaps ($90,122,154,186$) and the output of the original
video is shown as $\Delta$ PSNR. The first and second rows display the results
with start frame $0$ and $32$, respectively.

BasicVSR++ [10] is a state-of-the-art video super-resolution method that
enhances video quality through a combination of frame propagation and
alignment techniques. While BasicVSR++ has shown impressive results in
enhancing video quality, it does not take into account the unique
characteristics of compressed video.

In a compressed video [1], a frame can be an intra or inter frame. Intra frame
compression only uses information from the current image, while inter-frame
compression utilizes information from previously encoded frames to reduce
redundancy. The NTIRE22 challenge encoded video using a low-delay
configuration [1], as shown in Figure 1, with a group of pictures of 4. This
results in the compressed video having only one intra frame at a significantly
higher quality than other inter-frames. As a result, the quality of frames in
the compressed video is highly varied, but BasicVSR++ does not take this into
account. In practice, the entire video is fed as input to the model during
testing, which may not be the optimal choice.

We investigate the effect of varying input video frame lengths on the
performance of BasicVSR++. As in Fig. 2-a, the pre-trained network’s
performance varies significantly depending on the frame length, with shorter
frame length demonstrating higher performance for the first few frames.
Interestingly, we also noticed that this performance phenomenon did not occur
when the input of the trimmed video started from the 32nd frame, i.e., when
the intra frame is not included. This finding suggests that the network is
better able to exploit the high performance of the intra frame with a smaller
frame length. For later frames, our experiments demonstrated that using the
full frame length performed better thanks to the temporal dependency in both
backward and forward directions. In contrast, trimmed video inputs have
limited backward and forward dependency, resulting in lower performance for
later frames.

This observation suggests that the optimal choice of input frame length may
vary depending on the temporal characteristics of the video content. The use
of shorter frame lengths may be more effective for the early frames, while
longer frame lengths may be more suitable for later frames that rely on both
backward and forward dependencies. Additionally, the effectiveness of the
backward and forward dependencies may be limited to a certain temporal range,
as seen in Video 208 at shift 32. Examining 40 test sequences 201-240 in the
NTIRE22 dataset, we observe that the first 64 frames of enhanced outputs with
the video inputs trimmed at the length of 186 yields the best performance.

Figure 3: Performance of the fine-tuned Intra frame BasicVSR++ without the
Adaptive Context-Aware Fusion mechanism.

##  3 Proposed Method

###  3.1 Intra frame BasicVSR++

f

Figure 4: General framework of the adaptive context-aware mechanism for video
fusion. The mechanism detects static video content using the gradient of the
average frame.

To further leverage the superior quality of the intra frame, we introduce a
new network called Intra rame BasicVSR++. In order to do so, we created a new
intra frame video dataset by cutting the original videos into multiple non-
overlapping 30-frame segments and encoding them using HEVC with the same low-
delay configuration as the NTIRE22 dataset. The resulting dataset contains
videos with only one intra frame and multiple inter-frames. We then divided
the compressed videos into training and testing datasets for network training.
By utilizing this configuration, we ensured that the intra frame is of higher
quality than the inter-frames, reflecting the reality of compressed videos.

During training, we fine-tuned the Intra frame BasicVSR++ network by using the
intra frame as the first frame of each segment. This allowed the network to
learn to enhance the high-quality intra frame more effectively, resulting in a
more accurate and efficient network. In this way, the Intra frame BasicVSR++
network is designed to specifically target the improvement of the first
frame’s quality, while BasicVSR++ improves the quality of the entire video.

As shown in Fig.3, we observed that the Intra frame BasicVSR++ network
improves the performance of the intra frame in most cases, such as videos 201
and 202. However, this improvement is not universal and may not always hold
for sequences with high frame rates and slow motion, such as video 226. This
may be due to the limited amount of information in the intra frame in such
cases, and the network may struggle to extract and propagate useful features.
Nonetheless, these results suggest that the intra frame BasicVSR++ network can
be an effective tool in improving the performance of compressed video
enhancement for most types of content.

###  3.2 Adaptive Context-Aware Fusion

From our analysis, it is evident that BasicVSR++ benefits from adaptive input
frame length and can be further improved by Intra frame BasicVSR++. However,
it is also important to note that this improvement is not applicable to all
types of video content. In order to address this issue, we propose a heuristic
that separates cases where the frame rate is high and there is slow motion.
This is achieved by comparing the gradient of the average frame with a given
threshold. By using this threshold, we can determine if the video contains a
high frame rate and slow motion, and take appropriate measures to optimize the
final performance. Firstly, the average frame is obtained as follows:

$\bar{f}=\sum_{i}^{i\times m<N}f_{i\times m},$ (1)

where $m$ is a scaling factor that is proportional to the input video frame
rate. Specifically, we set $m=4$ for videos with a frame rate less than 30fps
and $m=8$ for those with a frame rate greater than 30fps. Next, we compare the
gradient of the average frame to a given threshold:

$\nabla(\bar{f})=||\nabla_{x}(\bar{f})||+||\nabla_{y}(\bar{f})||<\tau,$ (2)

where $\nabla_{x}$ and $\nabla_{y}$ denote the gradient in the horizontal and
vertical directions of a given frame $f$, and $\tau$ is a threshold of value
2300. The value of $\tau$ can be normalized based on the number of pixels, as
shown in Fig. 4.

We propose a novel video fusion mechanism called Adaptive Context-Aware
Fusion, as shown in Fig. 4. The method involves enhancing an input video using
three different approaches: full BasicVSR++, short BasicVSR++ with the first
154 frames, and the first 122 frames by Intra frame BasicVSR++. Depending on
the current content of the video, an adaptive fusion technique is performed to
select either the first frame from short BasicVSR++ or Intra frame BasicVSR++.
For the subsequent 63 frames, we select frames from the short BasicVSR++ set,
and the remaining frames are extracted from the full BasicVSR++ set.

###  3.3 Loss Function

In order to fine-tune the BasicVSR++ and Intra frame BasicVSR++ networks, we
used a weighted sum of three loss components: (i) Charbonnier loss [11], (ii)
total variation (TV) loss, and (iii) temporal gradient (TG) loss. The temporal
gradient loss captures the difference between the ground truth and the output
temporal sequence by calculating the loss between two consecutive frames in
each sequence. To calculate the temporal gradient loss, we subtract two
consecutive frames in each sequence to obtain the temporal gradient sequence.

The hyperparameters of the loss weights were optimized using grid search to
find the best values. The final loss function is given by:

$\displaystyle\mathcal{L}_{\mathrm{Final}}=\mathcal{L}_{\mathrm{Char}}+1e\times
10^{-3}*\mathcal{L}_{\mathrm{TG}}$ (3) $\displaystyle\ +1e\times
10^{-4}*\mathcal{L}_{\mathrm{TV}},$

where $L_{Char}$ is the Charbonnier loss, $L_{TV}$ is the total variation
loss, $L_{TG}$ is the temporal gradient loss.

##  4 Experiments

###  4.1 Datasets

For the NTIRE 2022 Challenge, we used the original LDV dataset [12] that
consists of 240 videos as our primary training set. To increase our training
data, we also utilized the LDV 2.0 dataset, which contains an additional 90
videos. We split the LDV 2.0 videos into six sets, each containing 15 videos.
Two of these sets were used as our validation and test sets, respectively. In
splitting the videos, we aimed to maintain the diversity of the videos in each
set, in terms of content, frame rate, and other factors, as similar as
possible. All videos in the LDV and LDV 2.0 datasets, as well as the splits
for the NTIRE 2021 and NTIRE 2022 Challenges, are publicly available at
https://github.com/RenYang-home/LDV_dataset.

###  4.2 Training Details

We employed the Adam optimizer [13] with a learning rate of $2\times 10^{-5}$
and utilized the Cosine Restart scheduler [14] with a period of 10,000
iterations. To ensure a stable optimization process, we linearly increased the
learning rate for the first 10% of iterations.

####  4.2.1 Fine-tuning BasicVSR++

Due to computational limitations, we fine-tuned only the upsample layer of the
pre-trained BasicVSR++ network. We found that increasing the input frame
length from 30 to 60 frames led to a 0.03 dB improvement in model performance.
The network was fine-tuned for 50,000 iterations.

####  4.2.2 Fine-tuning Intra frame BasicVSR++

We create a new dataset is created by trimming the original videos into
multiple segments, each of which consists of 30 frames without overlapping
frames. The video segments are encoded using HEVC at a low-delay profile
resulting in approximately 16,000 training samples. For the Intra frame
BasicVSR++ network, only the segments from the first 200 videos are used to
train the model, and the last 40 videos are used for testing.

Table 1: The PSNR results on the test set for our method and the compared
method, BasicVSR++. Values are given in dB.

Method \Video ID | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | Avg.
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
BasicVSR++ | 33.73 | 32.42 | 31.22 | 36.75 | 32.16 | 30.57 | 30.41 | 33.99 | 31.68 | 36.20 | 27.06 | 23.93 | 30.24 | 32.94 | 31.20 | 31.63
Ours (OCL-VCE) | 33.81 | 32.54 | 31.30 | 36.82 | 32.28 | 30.59 | 30.47 | 34.07 | 31.75 | 36.28 | 27.08 | 24.00 | 30.37 | 33.01 | 31.25 | 31.71

###  4.3 Ensembling with Test Time Augmentation

In our study, we perform Ensembling with test time augmentation (TTA) by
generating eight input variations by flipping and rotating the input sequences
in the spatial dimension. We then use our proposed framework shown in Fig. 4
to enhance each variation, and post-process the corresponding output by
flipping and rotating it back to its original location. Finally, we obtain the
final output by averaging the outputs of all variations. This approach helps
to reduce the impact of input variability and improve the overall performance
of the model.

###  4.4 Experimental Results

We participated in the NTIRE22 challenge as team OCL-VCE and submitted our
results on the test set. In Table 1, we present the performance of our method
and BasicVSR++ on the test set. Our method achieved a higher PSNR score of
31.71 dB compared to 31.63 dB obtained by BasicVSR++.

We further evaluated our framework on the validation set, which consists of 10
videos. Using the pre-trained BasicVSR++ and Intra frame BasicVSR++ models and
applying test-time augmentation (TTA), our framework achieved a PSNR score of
32.12 dB, surpassing the 31.84 dB score achieved without TTA.

In addition, without finetuning, with and without TTA, our framework achieved
a PSNR score of 32.02 dB and 31.86 dB, respectively, improving the baseline by
0.06 dB and 0.02 dB, respectively. Our results demonstrate the effectiveness
of our proposed Adaptive Context-Aware Fusion framework in enhancing the
quality of low-delay compressed videos.

##  5 Conclusion

In conclusion, this paper proposes a novel method that leverages the unique
characteristics of low-delay video compression algorithms to improve the
quality of compressed videos using deep learning techniques. By incorporating
this prior knowledge into the state-of-the-art method, BasicVSR++, we achieve
a significant improvement in performance over existing methods. Our
experimental results on the NTIRE22 challenge validate the effectiveness of
our proposed method. This work underscores the importance of incorporating
video compression knowledge into deep learning models to further enhance their
performance and enable real-world applications.

## References

  * [1] G. J. Sullivan, J.-R. Ohm, W.-J. Han, and T. Wiegand, “Overview of the high efficiency video coding (hevc) standard,” _IEEE Transactions on Circuits and Systems for Video Technology_ , vol. 22, no. 12, pp. 1649–1668, 2012\.
  * [2] R. Yang, M. Xu, T. Liu, Z. Wang, and Z. Guan, “Enhancing quality for hevc compressed videos,” _IEEE Transactions on Circuits and Systems for Video Technology_ , vol. 29, no. 7, pp. 2039–2054, 2018.
  * [3] T. Wang, M. Chen, and H. Chao, “A novel deep learning-based method of improving coding efficiency from the decoder-end for hevc,” in _2017 Data Compression Conference (DCC)_. IEEE, 2017, pp. 410–419.
  * [4] R. Yang, X. Sun, M. Xu, and W. Zeng, “Quality-gated convolutional lstm for enhancing compressed video,” in _2019 IEEE International Conference on Multimedia and Expo (ICME)_. IEEE, 2019, pp. 532–537.
  * [5] Z. Guan, Q. Xing, M. Xu, R. Yang, T. Liu, and Z. Wang, “Mfqe 2.0: A new approach for multi-frame quality enhancement on compressed video,” _IEEE transactions on pattern analysis and machine intelligence_ , vol. 43, no. 3, pp. 949–963, 2019.
  * [6] J. Deng, L. Wang, S. Pu, and C. Zhuo, “Spatio-temporal deformable convolution for compressed video quality enhancement,” in _Proceedings of the AAAI conference on artificial intelligence_ , vol. 34, no. 07, 2020, pp. 10 696–10 703.
  * [7] R. Yang, F. Mentzer, L. V. Gool, and R. Timofte, “Learning for video compression with hierarchical quality and recurrent enhancement,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2020, pp. 6628–6637.
  * [8] J. Wang, X. Deng, M. Xu, C. Chen, and Y. Song, “Multi-level wavelet-based generative adversarial network for perceptual quality enhancement of compressed video,” in _European Conference on Computer Vision_. Springer, 2020, pp. 405–421.
  * [9] Y. Huo, Q. Lian, S. Yang, and J. Jiang, “A recurrent video quality enhancement framework with multi-granularity frame-fusion and frame difference based attention,” _Neurocomputing_ , vol. 431, pp. 34–46, 2021.
  * [10] K. C. Chan, S. Zhou, X. Xu, and C. C. Loy, “Basicvsr++: Improving video super-resolution with enhanced propagation and alignment,” _arXiv preprint arXiv:2104.13371_ , 2021.
  * [11] W.-S. Lai, J.-B. Huang, N. Ahuja, and M.-H. Yang, “Fast and accurate image super-resolution with deep laplacian pyramid networks,” _IEEE transactions on pattern analysis and machine intelligence_ , vol. 41, no. 11, pp. 2599–2613, 2018.
  * [12] R. Yang, “Ntire 2021 challenge on quality enhancement of compressed video: Dataset and study,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2021, pp. 667–676.
  * [13] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” _arXiv preprint arXiv:1412.6980_ , 2014.
  * [14] I. Loshchilov and F. Hutter, “Sgdr: Stochastic gradient descent with warm restarts,” _arXiv preprint arXiv:1608.03983_ , 2016.