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# Centers of Multilinear Forms and Applications

Hua-Lin Huang, Huajun Lu, Yu Ye and Chi Zhang  Huang and Lu: School of
Mathematical Sciences, Huaqiao University, Quanzhou 362021, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>Ye: School of Mathematical
Sciences, University of Science and Technology of China, Hefei 230026, China
<EMAIL_ADDRESS>Zhang: Department of Mathematics, Northeastern University,
Shenyang 110819, China<EMAIL_ADDRESS>
###### Abstract.

In this paper we study the center algebras of multilinear forms. It is shown
that the center of a nondegenerate multilinear form is a finite dimensional
commutative algebra and can be effectively applied to its direct sum
decompositions. As an application of the algebraic structure of centers, we
also show that almost all multilinear forms are absolutely indecomposable. The
theory of centers can be extended to multilinear maps and be applied to their
symmetric equivalence. Moreover, with a help of the results of symmetric
equivalence, we are able to provide a linear algebraic proof of a well known
Torelli type result which says that two complex homogeneous polynomials with
the same Jacobian ideal are linearly equivalent.

###### Key words and phrases:

multilinear form, tensor, direct sum, congruence

###### 2010 Mathematics Subject Classification:

15A69, 11E76, 14J70

Supported by NSFC 11911530172, 11971181 and 11971449.

##  1\. Introduction

Let $V$ be a vector space over a field $\mathbbm{k}$ of dimension $n.$ Let
$d\geq 3$ be a positive integer. A $d$-linear form on $V$ is a multilinear
mapping $\Theta:V^{d}=V\times\cdots\times V\to\mathbbm{k}$ and is denoted by
$(V,\Theta)$ or $\Theta$ for short. Take a basis $e_{1},e_{2},\dots,e_{n}$ of
$V$ and set $a_{i_{1}i_{2}\cdots
i_{d}}=\Theta(e_{i_{1}},e_{i_{2}},\dots,e_{i_{d}}).$ The resulting tensor
$A=(a_{i_{1}i_{2}\cdots i_{d}})_{1\leq i_{1},i_{2},\dots,i_{d}\leq n}$ is
called the associated tensor of $(V,\Theta)$ under the basis
$e_{1},e_{2},\dots,e_{n}.$ A fundamental problem in invariant theory and
multilinear algebra is finding canonical forms for multilinear forms under
base change, or equivalently, canonical forms of tensors under congruence by
invertible matrices.

Unlike bilinear forms, it seems hopeless to find complete representatives for
$d$-linear forms, see for example [3, 7]. One of our main concerns is direct
sum decompositions of multilinear forms, that is to find whether there exist
nonzero subspaces $V_{1},V_{2},\dots,V_{m}$ of $(V,\Theta)$ such that
$V=V_{1}\oplus V_{2}\oplus\cdots\oplus V_{m}$ and
$\Theta(v_{1},\cdots,v_{d})=0$ unless all the $v_{i}$’s are in the same
$V_{j}$ for some $j.$ In terms of tensors, this is equivalent to finding an
invertible matrix $P$ such that the congruent tensor $AP^{d}$ is block
diagonal. This is a natural problem as direct sum decompositions provide
dimension reduction for multilinear forms.

In [9, 10], we investigated direct sum decompositions of symmetric multilinear
forms via Harrison’s theory of centers [8]. The authors showed that the
problem can be boiled down to some standard tasks of linear algebra,
specifically the computations of eigenvalues and eigenvectors. The main aim of
the present paper is to extend [9, 10] to the situation of multilinear forms.

We generalize the key notion of centers as follows.

######  Definition 1.1.

Given a $d$-linear form $(V,\Theta),$ set

(1.1)
$Z(V,\Theta)\colon=\left\\{\phi\in\operatorname{End}(V)\left|\begin{array}[]{c}\Theta(v_{1},\cdots,\phi(v_{i}),\cdots,v_{j},\cdots
v_{d})\\\ =\Theta(v_{1},\cdots,v_{i},\cdots,\phi(v_{j}),\cdots,v_{d}),\\\
\forall 1\leq i,j\leq d,\ \ \forall v_{1},\cdots,v_{d}\in
V\end{array}\right.\right\\}$

and call it the center of $(V,\Theta).$

Elements of centers for multilinear forms were also defined and applied to
direct sum decomposition in [4], where they were called self-adjoint linear
mappings. However, the algebraic structure of all centers, or self-adjoint
elements were not considered therein.

The centers of multilinear forms enjoy the same properties as those of
symmetric multilinear forms or equivalently homogeneous polynomials, cf. [8,
9, 10].

######  Theorem 1.2.

Suppose $(V,\Theta)$ is a nondegenerate $d$-linear form. Then

  * (1)

The center $Z(V,\Theta)$ is a commutative algebra.

  * (2)

There is a one-to-one correspondence between direct sum decompositions of
$(V,\Theta)$ and complete sets of orthogonal idempotents of $Z(V,\Theta).$

  * (3)

The decomposition of $(V,\Theta)$ into a direct sum of indecomposable
$d$-linear forms is unique up to permutation of indecomposable summands.

As a consequence, we have a simple algorithm for direct sum decompositions of
arbitrary multilinear forms which is equivalent to the classical eigenvalue
problem of matrices, see [9, Algorithm 3.12].

Let $T_{d,n}$ be the set of all $d$-linear forms on a $n$-dimensional linear
$\mathbbm{k}$-space. If a multilinear form is not a direct sum, then we say it
is indecomposable. It is clear by Theorem 1.2 that $(V,\Theta)\in T_{d,n}$ is
indecomposable if and only if $Z(V,\Theta)$ is a local algebra. In particular,
if $(V,\Theta)$ is central, i.e. $Z(V,\Theta)\cong\mathbbm{k},$ then
$(V,\Theta)$ is absolutely indecomposable. It was already noticed in [4,
Remark 10] that multilinear forms are more likely indecomposable. We confirm
this with a help of the center theory. In fact, we show in terms of elementary
algebraic geometry that almost all multilinear forms are absolutely
indecomposable.

######  Theorem 1.3.

The set of all central $d$-linear forms is Zariski open and dense in
$T_{d,n}.$

When we consider simultaneously several $d$-multilinear forms
$\Theta_{i}:V\times\cdots\times V\to\mathbbm{k}$ for $i=1,\cdots,r$, it is
convenient to consider the multilinear map
$(\Theta_{1},\cdots,\Theta_{r}):V\times\cdots\times V\to\mathbbm{k}^{r}$.
Centers of multilinear forms can be naturally extended to multilinear maps.
Motivated by [2, 4], we focus on direct sum decomposition and symmetric
equivalence of multilinear maps. We show that the center theory is highly
effective in dealing with these problems. Interestingly enough, the results of
symmetric equivalence can be applied to provide a simple linear algebraic
proof for a well known Torelli type result of Donagi [5, Proposition 1.1].

######  Theorem 1.4.

Suppose the field $\mathbbm{k}$ is algebraically closed and its characteristic
is zero or greater than $d$. If $f$ and $g$ are two homogeneous polynomials of
degree $d$ with the same Jacobian ideal, then they are related by an
invertible linear transformation.

We conclude the introduction with a brief outline of the paper. In Section 2
we investigate the algebraic structure of centers of multilinear forms with
applications to their direct sum decompositions. In Section 3 the theory of
centers is extended to multilinear maps and is applied to their symmetric
equivalence. Throughout we assume that $d>2$ is an integer, $\mathbbm{k}$ is a
field of characteristic $0$ or $>d.$ The results are presented in terms of
multilinear forms. We leave the equivalent version for tensors to the
interested reader.

##  2\. Centers and direct sum decompositions of multilinear forms

In this section, we consider the center algebras of multilinear forms with
applications to direct sum decompositions. The main aim is to prove Theorems
1.2 and 1.3.

First of all, we recall some concepts.

######  Definition 2.1.

Let $(V,\Theta)$ be a $d$-linear form. If there exist nonzero subspaces
$V_{1},\cdots,V_{s}\ (s\geq 2)$ of $(V,\Theta)$ such that
$V=V_{1}\oplus\cdots\oplus V_{s}$ and $\Theta(v_{1},\cdots,v_{d})=0$ unless
all the $v_{i}$’s are in the same $V_{k}$ for some $k$, then $\Theta$ is
called the (inner) direct sum of $\Theta_{1},\cdots,\Theta_{s}$, where
$\Theta_{i}=\Theta_{|V_{i}}$ is the restriction of $\Theta$ to $V_{i}$ for
$1\leq i\leq s$ and we denote it by
$(V,\Theta)=(V_{1},\Theta_{1})\oplus\cdots\oplus(V_{s},\Theta_{s})$. We call
$(V,\Theta)$ decomposable if it is a direct sum. Otherwise, we call
$(V,\Theta)$ indecomposable.

Similar to the symmetric case [8] there is no harm in assuming that the
$d$-linear form $(V,\Theta)$ is nondegenerate, that is $u=0$ is the only
solution to the following linear equations

(2.1)
$\Theta(u,v_{1},\cdots,v_{d-1})=\Theta(v_{1},u,\cdots,v_{d-1})=\cdots=\Theta(v_{1},\cdots,v_{d-1},u)=0,\quad\forall
v_{1},\cdots,v_{d-1}\in V.$

For an arbitrary $d$-linear form $(V,\Theta),$ let $V_{0}$ be the solution
space of the previous equations (2.1) and take a subspace $V_{1}$ of $V$ such
that $V=V_{1}\oplus V_{0},$ then
$(V,\Theta)=(V_{1},\Theta_{1})\oplus(V_{0},\Theta_{0}).$ In particular,
$(V_{1},\Theta_{1})$ is nondegenerate and $(V_{0},\Theta_{0})$ is a zero form.
Note that $V_{0}$ is uniquely determined by $\Theta,$ see also [4].

Now we are ready to prove Theorem 1.2.

Proof of Theorem 1.2 (1) Let us show that $Z(V,\Theta)$ is a commutative
subalgebra of $\operatorname{End}(V).$ It is obvious that $Z(V,\Theta)$ is
closed under linear combinations. Choose two arbitrary $\phi,\psi\in
Z(V,\Theta)$ and we want to show $\phi\circ\psi\in Z(V,\Theta).$ According to
the definition of centers, for all $v_{1},\cdots,v_{d}\in V$ we have

$\displaystyle\Theta(v_{1},\cdots,\phi\circ\psi(v_{i}),\cdots,v_{j},\cdots
v_{d})$ $\displaystyle=$
$\displaystyle\Theta(v_{1},\cdots,\psi(v_{i}),\cdots,v_{j},\cdots,\phi(v_{d}))$
$\displaystyle=$
$\displaystyle\Theta(v_{1},\cdots,v_{i},\cdots,\psi(v_{j}),\cdots,\phi(v_{d}))$
$\displaystyle=$
$\displaystyle\Theta(v_{1},\cdots,v_{i},\cdots,\phi\circ\psi(v_{j}),\cdots,v_{d}).$

Hence we have $\phi\circ\psi\in Z(V,\Theta)$. Similarly we show the
commutativity of $Z(V,\Theta).$

$\displaystyle\Theta(v_{1},\cdots,\phi\circ\psi(v_{i}),\cdots,v_{j},\cdots
v_{d})$ $\displaystyle=$
$\displaystyle\Theta(v_{1},\cdots,\psi(v_{i}),\cdots,\phi(v_{j}),\cdots
v_{d})$ $\displaystyle=$
$\displaystyle\Theta(v_{1},\cdots,v_{i},\cdots,\phi(v_{j}),\cdots,\psi(v_{d}))$
$\displaystyle=$
$\displaystyle\Theta(v_{1},\cdots,\phi(v_{i}),\cdots,v_{j},\cdots,\psi(v_{d}))$
$\displaystyle=$
$\displaystyle\Theta(v_{1},\cdots,\psi\circ\phi(v_{i}),\cdots,v_{j},\cdots
v_{d}).$

We conclude that
$\Theta(v_{1},\cdots,[\phi\circ\psi-\psi\circ\phi](v_{i}),\cdots,v_{j},\cdots
v_{d})=0,\forall v_{1},\cdots,v_{d}\in V$. As $(V,\Theta)$ is nondegenerate,
it follows that $\phi\circ\psi-\psi\circ\phi=0$, that is,
$\phi\circ\psi=\psi\circ\phi$.

(2) Suppose there exists a direct sum decomposition
$(V,\Theta)=(V_{1},\Theta_{1})\oplus\cdots\oplus(V_{s},\Theta_{s}).$ For
$1\leq i\leq s$, let $e_{i}:V\twoheadrightarrow V_{i}\hookrightarrow V$ be the
composition of the canonical projection $V\twoheadrightarrow V_{i}$ and the
embedding $V_{i}\hookrightarrow V$. Then it is obvious that $e_{i}^{2}=e_{i},\
e_{i}e_{j}=0,\ \forall i\neq j,$ and by definition it is direct to verify that
each $e_{i}\in Z(V,\Theta).$ In other words, $e_{1},\dots,e_{s}$ are a
complete set of orthogonal idempotents of $Z(V,\Theta).$

Conversely, suppose $e_{1},\dots,e_{s}$ are a complete set of orthogonal
idempotents of $Z(V,\Theta).$ Let $V_{i}=e_{i}V$ and
$\Theta_{i}=\Theta_{|V_{i}}.$ Then it is not hard to verify that
$(V_{1},\Theta_{1})\oplus\cdots\oplus(V_{s},\Theta_{s})$ is a direct sum
decomposition of $(V,\Theta).$ Indeed, assume $v_{1},\dots,v_{s}$ are taken
from the subspaces $V_{i}$’s and $v_{j}\in V_{j},\ v_{k}\in V_{k}$ with $j<k,$
then

$\displaystyle\Theta(v_{1},\cdots,v_{j},\cdots,v_{k},\cdots,v_{s})$
$\displaystyle=$
$\displaystyle\Theta(v_{1},\cdots,e_{j}v_{j},\cdots,e_{k}v_{k},\cdots,v_{s})$
$\displaystyle=$
$\displaystyle\Theta(v_{1},\cdots,v_{j},\cdots,e_{j}e_{k}v_{k},\cdots,v_{s})$
$\displaystyle=$ $\displaystyle 0.$

(3) It suffices to prove that $Z(V,\Theta)$ has a unique complete set of
primitive orthogonal idempotents disregarding their order thanks to (2).
Suppose $1=e_{1}+\cdots+e_{s}=f_{1}+\cdots+f_{t}$ where all $e_{i}$’s and
$f_{j}$’s are primitive orthogonal idempotents. Then for any fixed $i,$
$e_{i}=e_{i}(f_{1}+\cdots+f_{t})=e_{i}f_{1}+\cdots+e_{i}f_{t}.$ Since
$(e_{i}f_{j})^{2}=e_{i}^{2}f_{j}^{2}=e_{i}f_{j}$ and $e_{i}$ is primitive,
$e_{i}=e_{i}f_{j}$ for some certain $j.$ Similarly, $f_{j}=f_{j}e_{k}$ for
some certain $k.$ We claim that $i=k$ and so $e_{i}=f_{j}.$ Otherwise, if
$i\neq k,$ then $e_{i}=e_{i}f_{j}=e_{i}f_{j}e_{k}=e_{i}e_{k}f_{j}=0.$ This is
absurd. Then we are done.

######  Remarks 2.2.

Keep the assumption that $(V,\Theta)$ is a nondegenerate $d$-linear form.

  1. (1)

$(V,\Theta)$ is indecomposable if and only if $Z(V,\Theta)$ is a local
algebra.

  2. (2)

The uniqueness of direct sum decomposition of multilinear forms were dealt
with by other approaches in [8, Proposition 2.3] (the symmetric case) and [4,
Theorem 9]. The treatment via centers seems much more convenient.

  3. (3)

The algorithm of direct sum decomposition of symmetric multilinear forms
proposed by the authors [10, Algorithm 3.12] can be extended verbatim to the
present situation.

Finally, we consider the algebraic structure of the center of a general
$d$-linear form. This will provide important information for $d$-linear forms.
Our main concern is whether a general $d$-linear form is decomposable or not.
It was already noticed in [4, Remark 10] that multilinear forms are more
likely indecomposable. We confirm this in terms of elementary algebraic
geometry with a help of the center theory. It is shown that almost all
multilinear forms have trivial center, namely the center is isomorphic to the
ground field. Such multilinear forms are called central. Clearly, a central
multilinear form is indecomposable by item (1) of Remarks 2.2. The relevant
result for symmetric multilinear forms was proved in [10, Theorem 3.2], where
the same idea can be extended to the present situation.

Explicit examples of central symmetric $d$-linear forms were constructed in
[10]. Here we construct some examples of central $d$-linear forms which are
not symmetric. Assume $V$ is an $n$-dimensional $\mathbbm{k}$-space with a
basis $e_{1},\cdots,e_{n}.$ Let $A=(a_{i_{1}i_{2}\cdots i_{d}})_{1\leq
i_{1},i_{2},\dots,i_{d}\leq n}$ be the associated tensor of $(V,\Theta)$ under
the basis $e_{1},e_{2},\dots,e_{n}.$ Then we have

(2.2) $\displaystyle Z(V,\Theta)\cong\\{X\in k^{n\times n}\mid
X^{T}A_{i_{1}\cdots\underline{i_{k}}\cdots\underline{i_{l}}\cdots
i_{d}}=A_{i_{1}\cdots\underline{i_{k}}\cdots\underline{i_{l}}\cdots
i_{d}}X,\forall 1\leq i_{1},\cdots,i_{d}\leq n\\},$

where $A_{i_{1}\cdots\underline{i_{k}}\cdots\underline{i_{l}}\cdots i_{d}}$
denotes the $n\times n$ matrix $(a_{i_{1}\cdots
i_{k-1},i,i_{k+1},\cdots,i_{l-1},j,i_{l+1},\cdots i_{d}})_{1\leq i,j\leq n}.$

######  Example 2.3.

We construct a $d$-linear form with trivial center for each $d\geq 3$ and
$n\geq 2$.

If $n=2$, let $\left(a_{ij}\right)_{2\times
2}=\left(\begin{array}[]{cc}1&-1\\\ 1&0\end{array}\right)$. Let $\Theta$ be
the $d$-linear form such that $a_{i_{1}i_{2}i_{3}\cdots i_{d}}=a_{i_{1}i_{2}}$
for all $1\leq i_{1},\cdots,i_{d}\leq 2$. An easy calculation shows that
$Z(V,\Theta)\cong\mathbbm{k}$.

If $n\geq 3$, let $p_{1},\cdots,p_{n},q_{1}\cdots,q_{n}$ be $2n$ nonzero
elements of $\mathbbm{k}$ such that
$\frac{p_{j}}{p_{i}}\neq\frac{q_{j}}{q_{i}}$ whenever $i\neq j$. Let
$A_{1}=(a^{(1)}_{ij})($resp. $A_{2}=(a^{(2)}_{ij}))$ be the diagonal $n\times
n$ matrix with the $a^{(1)}_{ii}=p_{i}$ $($resp. $a^{(2)}_{ii}=q_{i})$ for
$1\leq i\leq n$. Let $A_{3}=(a^{(3)}_{ij})$ be the matrix with
$a^{(3)}_{ij}=1$ for all $1\leq i,j\leq n$. Let $\Theta$ be the $d$-linear
form such that $a_{1i_{2}i_{3}\cdots
i_{d}}=a^{(1)}_{i_{2}i_{3}},a_{2i_{2}i_{3}\cdots
i_{d}}=a^{(2)}_{i_{2}i_{3}},a_{3i_{2}i_{3}\cdots
i_{d}}=a^{(3)}_{i_{2}i_{3}},a_{ii_{2}i_{3}\cdots i_{d}}=0$ for all $4\leq
i\leq n,1\leq i_{2},i_{3},\cdots,i_{d}\leq n$. Suppose $X=(x_{ij})\in
Z(V,\Theta)$, then we have $X^{T}A_{i}=A_{i}X$ for $i=1,2,3$. Consequently, we
have $p_{i}x_{ij}=p_{j}x_{ji}$ and $q_{i}x_{ij}=q_{j}x_{ji}$ for all $1\leq
i,j\leq n$. As $\frac{p_{j}}{p_{i}}\neq\frac{q_{j}}{q_{i}}$ whenever $i\neq
j$, we conclude that $X$ must be a diagonal matrix. Moreover $X$ must be a
mutilple of identity matrix since $X^{T}A_{3}=A_{3}X$. Therefore we have
$Z(V,\Theta)\cong\mathbbm{k}$.

Proof of Theorem 1.3 Let $C$ be the set of all central $d$-linear forms.
Clearly, $C$ is not empty by Example 2.3. See also [10] for many other
examples of central $d$-linear forms which are symmetric. The center of
$(V,\Theta)$ is the solution space to a system of linear equations on
$x_{ij}$’s: $X^{T}A_{i_{1}\cdots\underline{i_{k}}\cdots\underline{i_{l}}\cdots
i_{d}}=A_{i_{1}\cdots\underline{i_{k}}\cdots\underline{i_{l}}\cdots i_{d}}X$
for all possible index $i_{1}\cdots i_{d}$, where we use the same notations as
in Equation (2.2). The $d$-linear form $(V,\Theta)$ is central if and only if
the rank of the coefficient matrix, denoted by $B,$ of the linear system (2.2)
is equal to $n^{2}-1$. Hence $C$ is the union of all the principal open sets
defined by the $(n^{2}-1)$-minors of $B$ regarding all $a_{i_{1}\cdots
i_{d}}$’s as indeterminates. Consequently, $C$ is a nonempty Zariski open set
of $T_{d,n}$ and so is dense.

##  3\. Centers and symmetric equivalence of $d$-linear maps

This section is motivated by [2, 4]. We extend the theory of centers to
multilinear maps and apply it to the problems of direct sum decomposition and
symmetric equivalence.

We begin by recalling some notions introduced in [2, 4].

######  Definition 3.1.

Let $V$ and $T$ be two finite dimensional vector spaces over $\mathbbm{k}$.

  1. (1)

A $d$-linear map on $V$ with target $T$ is a multilinear map $\Theta:V^{d}\to
T$ and is denoted by $(V,T,\Theta)$ or $\Theta$ for short.

  2. (2)

The $d$-linear map $(V,T,\Theta)$ is called nondegenerate if $u=0$ is the only
solution to the linear equations
$\Theta(u,v_{1},\cdots,v_{d-1})=\Theta(v_{1},u,\cdots,v_{d-1})=\cdots=\Theta(v_{1},\cdots,v_{d-1},u)=0$
for all $v_{1},\cdots,v_{d-1}\in V.$

  3. (3)

Suppose there exist nonzero subspaces $V_{1}$ and $V_{2}$ such that
$V=V_{1}\oplus V_{2}$ and
$\Theta(v_{1}^{\prime}+v_{1}^{\prime\prime},\cdots,v_{d}^{\prime}+v_{d}^{\prime\prime})=\Theta(v_{1}^{\prime},\cdots,v_{d}^{\prime})+\Theta(v_{1}^{\prime\prime},\cdots,v_{d}^{\prime\prime})$
for all $v_{1}^{\prime},\cdots,v_{d}^{\prime}\in V_{1}$ and
$v_{1}^{\prime\prime},\cdots,v_{d}^{\prime\prime}\in V_{2}$, then $\Theta$ is
called the (inner) direct sum of $\Theta_{|V_{1}}$ and $\Theta_{|V_{2}}$. If
this is the case, then $(V,T,\Theta)$ is called decomposable. Otherwise,
$(V,T,\Theta)$ is called indecomposable.

######  Definition 3.2.

Let $(U,S,\Delta)$ and $(V,T,\Theta)$ be two $d$-linear maps.

  1. (1)

$(U,S,\Delta)$ and $(V,T,\Theta)$ are called symmetrically equivalent, denoted
by $(U,S,\Delta)\simeq_{s}(V,T,\Theta),$ if there exist linear bijections
$\phi_{1},\cdots,\phi_{d}:U\to V$ and $\psi:S\to T$ such that

$\psi\Delta(u_{1},\cdots,u_{d})=\Theta(\phi_{\sigma_{1}}(u_{1}),\cdots,\phi_{\sigma_{d}}(u_{d}))$

for all $u_{1},\cdots,u_{d}\in U$ and each reordering
$\sigma_{1},\cdots,\sigma_{d}$ of $1,\cdots,d$.

  2. (2)

$(U,S,\Delta)$ and $(V,T,\Theta)$ are called isomorphic, denoted by
$(U,S,\Delta)\cong(V,T,\Theta),$ if there exist linear bijections $\phi:U\to
V$ and $\psi:S\to T$ such that

$\psi\Delta(u_{1},\cdots,u_{d})=\Theta(\phi(u_{1}),\cdots,\phi(u_{d}))$

for all $u_{1},\cdots,u_{d}\in U$.

  3. (3)

Suppose $S=T$. The $($outer$)$ direct sum of $(U,T,\Delta)$ and $(V,T,\Theta)$
is the multilinear map $\Delta\oplus\Theta:(U\oplus V)^{d}\to T$ defined by

$\Delta\oplus\Theta(u_{1}+v_{1},\dots,u_{d}+v_{d})=\Delta(u_{1},\dots,u_{d})+\Theta(v_{1},\dots,v_{d})$

for all $u_{1},\dots,u_{d}\in U$ and $v_{1},\dots,v_{d}\in V.$

The notion of centers for multilinear forms can be naturally extended to
multilinear maps.

######  Definition 3.3.

For a $d$-linear map $(V,T,\Theta)$, its center is defined as

(3.1)
$Z(V,T,\Theta)\colon=\left\\{\phi\in\operatorname{End}(V)\left|\begin{array}[]{c}\Theta(v_{1},\cdots,\phi(v_{i}),\cdots,v_{j},\cdots
v_{d})\\\ =\Theta(v_{1},\cdots,v_{i},\cdots,\phi(v_{j}),\cdots,v_{d}),\\\
\forall 1\leq i,j\leq d,\ \ \forall v_{1},\cdots,v_{d}\in
V\end{array}\right.\right\\}.$

If $Z(V,T,\Theta)\cong\mathbbm{k},$ then we say the $d$-linear map
$(V,T,\Theta)$ is central.

It is not hard to see that Theorems 1.2 and 1.3, as well as their proofs, can
be generalized almost verbatim to multilinear maps.

######  Theorem 3.4.

Let $M_{V,d,T}$ denote the affine space of $d$-linear maps on $V$ with target
$T$ and $C_{V,d,T}\subset M_{V,d,T}$ the subset of central $d$-linear maps.
Suppose $(V,T,\Theta)\in M_{V,d,T}$ is nondegenerate.

  * (1)

The center $Z(V,T,\Theta)$ is a commutative algebra.

  * (2)

There is a one-to-one correspondence between direct sum decompositions of
$(V,T,\Theta)$ and complete sets of orthogonal idempotents of $Z(V,T,\Theta).$

  * (3)

The decomposition of $(V,T,\Theta)$ into a direct sum of indecomposable
$d$-linear maps is unique up to permutation of indecomposable direct summands.

  * (4)

$C_{V,d,T}$ is a Zariski open and dense subset of $M_{V,d,T}.$

Now we investigate symmetric equivalence of $d$-linear maps via their centers.

######  Proposition 3.5.

Suppose $(U,S,\Delta)$ and $(V,T,\Theta)$ are nondegenerate $d$-linear maps.
If $(U,S,\Delta)\simeq_{s}(V,T,\Theta),$ then $Z(U,S,\Delta)$ is isomorphic to
$Z(V,T,\Theta).$

###### Proof.

Let $\psi:S\to T$ and $\phi_{1},\cdots,\phi_{d}:U\to V$ be the linear
bijections such that
$\psi\Delta(u_{1},\cdots,u_{d})=\Theta(\phi_{\sigma_{1}}(u_{1}),\cdots,\phi_{\sigma_{d}}(u_{d}))$
for all $u_{1},\cdots,u_{d}\in U$ and each reordering
$\sigma_{1},\cdots,\sigma_{d}$ of $1,\cdots,d$. In particular, for fixed
bijections $\phi_{k}$ and $\phi_{l}$ we have

$\displaystyle\Delta(u_{1},\cdots,\phi_{k}^{-1}\phi_{l}(u_{i}),\cdots,u_{j},\cdots,u_{d})$
$\displaystyle=$
$\displaystyle\psi^{-1}\Theta(\phi_{\sigma_{1}}(u_{1}),\cdots,\phi_{k}(\phi^{-1}_{k}\phi_{l}(u_{i})),\cdots,\phi_{l}(u_{j}),\cdots,\
\phi_{\sigma_{d}}(u_{d}))\ \
(\phi_{\sigma_{i}}=\phi_{k},\phi_{\sigma_{j}}=\phi_{l})$ $\displaystyle=$
$\displaystyle\psi^{-1}\Theta(\phi_{\sigma_{1}}(u_{1}),\cdots,\phi_{l}(u_{i}),\cdots,\phi_{l}(u_{j}),\cdots,\phi_{\sigma_{d}}(u_{d}))$
$\displaystyle=$
$\displaystyle\psi^{-1}\Theta(\phi_{\sigma_{1}}(u_{1}),\cdots,\phi_{l}(u_{i}),\cdots,\phi_{k}(\phi_{k}^{-1}\phi_{l}(u_{j})),\cdots,\phi_{\sigma_{d}}(u_{d}))$
$\displaystyle=$
$\displaystyle\Delta(u_{1},\cdots,u_{i},\cdots,\phi^{-1}_{k}\phi_{l}(u_{j}),\cdots,u_{d})\
\ \ \ $

Therefore we show that $\phi_{k}^{-1}\phi_{l}\in Z(U,S,\Delta)$ for all
possible pairs $(k,l)$. Similarly, we can show that $\phi_{k}\phi_{l}^{-1}\in
Z(V,T,\Theta)$ for all possible pairs $(k,l)$.

For any $\phi\in Z(U,S,\Delta)$, let us show that
$\phi_{k}\phi\phi^{-1}_{k}=\phi_{l}\phi\phi^{-1}_{l}$ for all possible pairs
$(k,l)$ and $\phi_{k}\phi\phi^{-1}_{k}\in Z(V,T,\Theta)$ as follows.

$\displaystyle\Theta(v_{1},\cdots,\phi_{k}\phi\phi_{k}^{-1}(v_{i}),\cdots,v_{j},\cdots,v_{d})$
$\displaystyle=$
$\displaystyle\psi\Delta(\phi^{-1}_{\sigma_{1}}(v_{1}),\cdots,\phi^{-1}_{l}\phi_{k}\phi\phi^{-1}_{k}(v_{i}),\cdots,\phi^{-1}_{k}(v_{j}),\cdots,\phi^{-1}_{\sigma_{d}}(v_{d}))\
\ (\phi^{-1}_{\sigma_{i}}=\phi^{-1}_{l},\phi^{-1}_{\sigma_{j}}=\phi^{-1}_{k})$
$\displaystyle=$
$\displaystyle\psi\Delta(\phi^{-1}_{\sigma_{1}}(v_{1}),\cdots,\phi\phi^{-1}_{l}\phi_{k}\phi^{-1}_{k}(v_{i}),\cdots,\phi^{-1}_{k}(v_{j}),\cdots,\phi^{-1}_{\sigma_{d}}(v_{d}))\
(\mathrm{the\ commutativity\ of}Z(U,S,\Delta))$ $\displaystyle=$
$\displaystyle\psi\Delta(\phi^{-1}_{\sigma_{1}}(v_{1}),\cdots,\phi\phi^{-1}_{l}(v_{i}),\cdots,\phi^{-1}_{k}(v_{j}),\cdots,\phi^{-1}_{\sigma_{d}}(v_{d}))$
$\displaystyle=$
$\displaystyle\Theta(v_{1},\cdots,\phi_{l}\phi\phi_{l}^{-1}(v_{i}),\cdots,v_{j},\cdots,v_{d}).$

As $\Theta$ is nondegenerate, we have
$\phi_{k}\phi\phi_{k}^{-1}=\phi_{l}\phi\phi^{-1}_{l}$ for all possible pairs
$(k,l)$.

Similarly, we also have

$\displaystyle\Theta(v_{1},\cdots,\phi_{k}\phi\phi_{k}^{-1}(v_{i}),\cdots,v_{j},\cdots,v_{d})$
$\displaystyle=$
$\displaystyle\psi\Delta(\phi^{-1}_{\sigma_{1}}(v_{1}),\cdots,\phi^{-1}_{k}\phi_{k}\phi\phi^{-1}_{k}(v_{i}),\cdots,\phi^{-1}_{l}(v_{j}),\cdots,\phi^{-1}_{\sigma_{d}}(v_{d}))\
\ (\phi^{-1}_{\sigma_{i}}=\phi^{-1}_{k},\phi^{-1}_{\sigma_{j}}=\phi^{-1}_{l})$
$\displaystyle=$
$\displaystyle\psi\Delta(\phi^{-1}_{\sigma_{1}}(v_{1}),\cdots,\phi\phi^{-1}_{k}(v_{i}),\cdots,\phi^{-1}_{l}(v_{j}),\cdots,\phi^{-1}_{\sigma_{d}}(v_{d}))$
$\displaystyle=$
$\displaystyle\psi\Delta(\phi^{-1}_{\sigma_{1}}(v_{1}),\cdots,\phi^{-1}_{k}(v_{i}),\cdots,\phi\phi^{-1}_{l}(v_{j}),\cdots,\phi^{-1}_{\sigma_{d}}(v_{d}))$
$\displaystyle=$
$\displaystyle\Theta(v_{1},\cdots,v_{i},\cdots,\phi_{l}\phi\phi_{l}^{-1}(v_{j}),\cdots,v_{d})$
$\displaystyle=$
$\displaystyle\Theta(v_{1},\cdots,v_{i},\cdots,\phi_{k}\phi\phi_{k}^{-1}(v_{j}),\cdots,v_{d}).$

Therefore we conclude $\phi_{k}\phi\phi_{k}^{-1}\in Z(V,T,\Theta)$ for all
$1\leq k\leq d$. Finally, we can construct the isomorphism
$\Psi:Z(U,S,\Delta)\to Z(V,T,\Theta)$ by
$\Psi(\phi)=\phi_{1}\phi\phi_{1}^{-1}$. ∎

Before stating the main results on symmetric equivalence of multilinear forms,
we need some technical preparations particularly in commutative algebra, see
for instance [1, 6]. A multilinear map $(V,T,\Theta)$ over $\mathbbm{k}$ is
called absolutely indecomposable if it remains indecomposable after any field
extension of $\mathbbm{k}.$ A central multilinear map is obviously absolutely
indecomposable.

######  Lemma 3.6.

Suppose the characteristic of $\mathbbm{k}$ is zero or coprime to $d$. Let $A$
be a commutative finite local $\mathbbm{k}$-algebra with the maximal ideal
$m$. Let $K=A/m$ be its residue field. Then we have
$A^{\times}/(A^{\times})^{d}\cong K^{\times}/(K^{\times})^{d}$, where
$A^{\times}$ (resp. $K^{\times}$) is the group of units of $A$ (resp. $K$).
Moreover, if $K/\mathbbm{k}$ is purely inseparable, then
$K^{\times}/(K^{\times})^{d}\cong\mathbbm{k}^{\times}/(\mathbbm{k}^{\times})^{d}$.

###### Proof.

As $A$ is local, we have an exact sequence
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1+m\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$.
After tensor with $\mathbb{Z}/d\mathbb{Z}$, we obtain the exact sequence

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
39.07072pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1+m/(1+m)^{d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
45.66818pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 73.52345pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
53.29709pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 73.52345pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{A^{\times}/(A^{\times})^{d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces{\hbox{\kern
136.22983pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 155.60204pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
135.37567pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 155.60204pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{K^{\times}/(K^{\times})^{d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
220.87088pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
220.87088pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{1}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Since the characteristic of $k$ is zero or coprime to $d$, for each $a\in 1+m$
the equation $X^{d}-a=0$ always has a solution in $A$ by Hensel Lemma [6].
Therefore each element of $1+m$ is a $d$-th power, and we have
$A^{\times}/(A^{\times})^{d}\cong(K)^{\times}/(K^{\times})^{d}$.

The canonical morphism $\mathbbm{k}\rightarrow A\rightarrow K$ induces the map
$\phi:\mathbbm{k}^{\times}/(\mathbbm{k}^{\times})^{d}\to
K^{\times}/(K^{\times})^{d}$. If $K/\mathbbm{k}$ is purely inseparable, we
first show that $\phi$ is surjective. Let $p=\mathrm{char}\mathbbm{k}$, then
for each $a\in K$, there exits a $p$-th power $p^{n}$ such that
$b=a^{p^{n}}\in\mathbbm{k}$ as $K/\mathbbm{k}$ is purely separable. As
$(p,d)=1$, there exists two integers $\alpha,\beta$ such that
$p^{n}\alpha+d\beta=1$. Therefore,
$a=a^{p^{n}\alpha+d\beta}=b^{\alpha}a^{d\beta}\in\mathbbm{k}^{\times}(K^{\times})^{d}$
and $\phi$ is surjective. Next if there exits an element $c\in\mathbbm{k}$
such that $c=u^{d}$ for some $u\in K$. Then $u$ is a root of the separable
polynomial $T^{d}-c$ and $\mathbbm{k}(u)/\mathbbm{k}$ is a separable
subextension of $K/\mathbbm{k}$. However, as $K/\mathbbm{k}$ is purely
inseparable, we must have $\mathbbm{k}(u)=\mathbbm{k}$, that is,
$u\in\mathbbm{k}$. Hence we show that $\phi$ is an isomorphism and finish the
proof. ∎

######  Proposition 3.7.

A multilinear map $(V,T,\Theta)$ is absolutely indecomposable if and only if
$Z(V,T,\Theta)$ is local and its residue field is purely inseparable over
$\mathbbm{k}$.

###### Proof.

A form $\Theta$ is absolutely indecomposable if and only if
$Z(V,T,\Theta)\otimes k^{\prime}$ is local for any field extension
$k^{\prime}/\mathbbm{k}$ by Theorem 3.4. Let $m$ be the maximal ideal of
$Z(V,T,\Theta)$ and $K=Z(V,T,\Theta)/m$. As $m$ is nilpotent,
$Z(V,T,\Theta)\otimes k^{\prime}$ is local if and only if $K\otimes
k^{\prime}$ is local. Therefore it is enough to show that $K/\mathbbm{k}$ is
purely inseparable if and only if $K\otimes k^{\prime}$ is local for any field
extension $k^{\prime}/\mathbbm{k}$. If $K/\mathbbm{k}$ is purely inseparable,
then $p=\mathrm{char}(\mathbbm{k})>0$ and some $p$-th power of any element of
$K\otimes k^{\prime}$ belongs to $\mathbbm{k}\otimes k^{\prime}=k^{\prime}$.
Hence the element of $K\otimes k^{\prime}$ is either nilpotent or invertible,
consequently $K\otimes k^{\prime}$ is local. If $K/\mathbbm{k}$ is not purely
inseparable, then there exists a maximal separable subextension
$M/\mathbbm{k}$ of degree $r>1$. Let $\mathbbm{k}^{alg}$ be the algebraic
closure of $\mathbbm{k}$, and we have $K\otimes\mathbbm{k}^{alg}\cong
K\otimes_{M}(M\otimes_{\mathbbm{k}}\mathbbm{k}^{alg})\cong(K\otimes_{M}\mathbbm{k}^{alg})^{r}$
which is not local. ∎

######  Theorem 3.8.

Let $(U,S,\Delta)$ and $(V,T,\Theta)$ be two $d$-linear maps.

  * (1)

Suppose $\Delta\simeq_{s}\Theta.$ Let
$\Delta=\Delta_{0}\oplus\Delta_{1}\oplus\cdots\oplus\Delta_{r}($resp.
$\Theta=\Theta_{0}\oplus\Theta_{1}\oplus\cdots\oplus\Theta_{s})$ be the
decomposition of $\Delta($resp. $\Theta)$ as direct sum of zero maps and
indecomposable $d$-linear maps where $\Delta_{0}$ and $\Theta_{0}$ are zero
maps and the other $\Delta_{i}$’s and $\Theta_{i}$’s are indecomposable. Then
we have $r=s$, and $\Delta_{i}\simeq\Theta_{i}$ for each $i$ after suitable
reordering of $\Theta_{i}$’s.

  * (2)

Suppose the characteristic of $\mathbbm{k}$ is zero or coprime to $d.$ Let
$\Delta$ and $\Theta$ be two absolutely indecomposable $d$-linear maps. Then
$\Delta\simeq_{s}\Theta$ if and only if $\Delta\cong a\Theta$ for some
$a\in\mathbbm{k}^{*}$.

  * (3)

Suppose $\mathbbm{k}$ is algebraically closed and its characteristic is zero
or coprime to $d$. Then $\Delta\simeq_{s}\Theta$ if and only if
$\Delta\cong\Theta.$

###### Proof.

(1) Let $U_{0}=\\{u\in
U:\Theta(u,v_{1},\cdots,v_{d-1})=\Theta(v_{1},u,\cdots,v_{d-1})=\cdots=\Theta(v_{1},\cdots,v_{d-1},u)=0,\quad\forall
v_{1},\cdots,v_{d-1}\in V\\}$. Let $\overline{U}=U/U_{0}$ and define the
$d$-linear map
$\overline{\Delta}:\overline{U}\times\cdots\times\overline{U}\to S$ by
$\overline{\Delta}(\overline{u_{1}},\cdots,\overline{u_{n}})=\Delta(u_{1},\cdots,u_{d})$
where $u_{i}\in U$ is the lifting of $\overline{u_{i}}$ for each $1\leq i\leq
n$. Then $\overline{\Delta}$ is nondegenerate by construction. Choose a
subspace $U^{\prime}$ of $U$ such that $U=U_{0}\oplus U^{\prime}$ and let
$\Delta^{\prime}=\Delta_{|U^{\prime}}$. It is easy to verify that we have an
inner direct sum decomposition
$(U,S,\Delta)=(U_{0},S,0)\oplus(U^{\prime},S,\Delta^{\prime})$. The natural
morphism $U^{\prime}\to U\to\overline{U}$ gives an isomorphism between
$(U^{\prime},S,\Delta^{\prime})$ and $(\overline{U},S,\overline{\Delta})$ and
we have an outer direct sum decomposition
$(U,S,\Delta)\cong(U_{0},S,0)\oplus(\overline{U},S,\overline{\Delta})$.
Similarly, we construct $V_{0},(\overline{V}=V/V_{0},T,\overline{\Theta})$ and
$(V,T,\Theta)\cong(V_{0},T,0)\oplus(\overline{V},T,\overline{\Theta})$.

Let $\psi:S\to T$ and $\phi_{1},\cdots,\phi_{d}:U\to V$ be the linear
bijections such that
$\psi\Delta(u_{1},\cdots,u_{d})=\Theta(\phi_{\sigma_{1}}(u_{1}),\cdots,\phi_{\sigma_{d}}(u_{d}))$
for all $u_{1},\cdots,u_{d}\in U$ and each reordering
$\sigma_{1},\cdots,\sigma_{d}$ of $1,\cdots,d$. Then we have
$\phi_{i}(U_{0})\subset V_{0}$ and $\phi^{-1}_{i}(V_{0})\subset U_{0}$ for
each $1\leq i\leq d$ by their definitions. Thus $\phi_{i}$ induces the
isomorphisms between $U_{0}$ (resp. $\overline{U}$) and $V_{0}$ (resp.
$\overline{V}$) for each $1\leq i\leq d$. Let $\overline{\phi_{i}}$ be induced
isomorphism between $\overline{U}$ and $\overline{V}$, then
$\overline{\phi_{1}},\cdots,\overline{\phi_{d}}$ gives an symmetrical
equivalence between $\overline{\Delta}$ and $\overline{\Theta}$. As $U_{0}$
and $V_{0}$ are isomorphic as vector space, the zero $d$-linear maps over them
are also isomorphic. Therefore we can reduce the proposition to the
nondegenerate case, that is, we assume $\Delta$ and $\Theta$ are
nondegenerate.

By Theorem 3.4, we can uniquely decompose
$(U,S,\Delta)=(U_{1},S,\Delta_{1})\oplus\cdots\oplus(U_{r},S,\Delta_{r})$ as
an inner direct sum of indecomposable summands such that each $\Delta_{i}$ is
correspondent to a primitive idempotent $e_{i}$ of the center $Z(U,S,\Delta)$
and $U_{i}=e_{i}(U)$. As $\Delta$ and $\Theta$ are symmetrically equivalent,
we have $Z(U,S,\Delta)\cong Z(V,T,\Theta)$ by sending each $\phi\in
Z(U,S,\Delta)$ to $\phi_{1}\phi\phi_{1}^{-1}$ by Proposition 3.5. By Theorem
3.4 again, we obtain the unique decomposition
$(V,T,\Theta)=(V_{1},T,\Theta_{1})\oplus\cdots\oplus(V_{r},T,\Theta_{r})$ as
an inner direct sum of indecomposable summands such that
$V_{i}=\phi_{1}e_{i}\phi_{1}^{-1}(V)$ for each $i$. Moreover the restrictions
of $\phi_{k}$’s on $U_{i}$ give an symmetric equivalence between $\Delta_{i}$
and $\Theta_{i}$ for each $i$ since
$\phi_{k}(U_{i})=\phi_{k}e_{i}(U)=\phi_{k}e_{i}\phi_{k}^{-1}\phi_{k}(U)=\phi_{1}e_{i}\phi_{1}^{-1}(V)=V_{i}$
by the proof of Proposition 3.5.

(2) If $\Delta\cong a\Theta$, it is obvious that $\Delta\simeq_{s}\Theta$.
Conversely, if $\Delta\simeq_{s}\Theta$, we have $\phi_{i}\phi_{j}^{-1}\in
Z(V,T,\Theta)$ for all $1\leq i,j\leq d$ which was shown in the proof of
Proposition 3.5. For each $i$, we can write $\phi_{i}=a_{i}\phi_{1}$ for some
$a_{i}\in Z(V,T,\Theta)^{\times}$. As $(V,T,\Theta)$ is absolutely
indecomposable, its center $Z(V,T,\Theta)$ is a commutative finite local
algebra with purely inseparable residue field by Proposition 3.7. By Lemma
3.6, there exists a scale $a\in\mathbbm{k}^{\times}$ such that the product
$a^{-1}\cdot a_{1}\cdots a_{d}$ has a $d$-th root $b\in Z(V,T,\Theta)$. Since

$\displaystyle\psi\Delta(u_{1},\cdots,u_{d})$ $\displaystyle=$
$\displaystyle\Theta(\phi_{1}(u_{1}),\cdots,\phi_{d}(u_{d}))$ $\displaystyle=$
$\displaystyle\Theta(a_{1}\phi_{1}(u_{1}),\cdots,a_{d}\phi_{1}(u_{d}))$
$\displaystyle=$ $\displaystyle\Theta(a_{1}\cdots
a_{d}\phi_{1}(u_{1}),\phi_{1}(u_{2}),\cdots,\phi_{1}(u_{d}))$ $\displaystyle=$
$\displaystyle a\Theta(a^{-1}\cdot a_{1}\cdots
a_{d}\phi_{1}(u_{1}),\phi_{1}(u_{2}),\cdots,\phi_{1}(u_{d}))$ $\displaystyle=$
$\displaystyle
a\Theta(b^{d}\phi_{1}(u_{1}),\phi_{1}(u_{2}),\cdots,\phi_{1}(u_{d}))$
$\displaystyle=$ $\displaystyle
a\Theta(b\phi_{1}(u_{1}),b\phi_{1}(u_{2}),\cdots,b\phi_{1}(u_{d}))$

we have $\Delta\cong a\Theta$.

(3)The isomorphism of $d$-linear maps clearly implies they are also
symmetrically equivalent, and we only need to show the converse. Suppose
$(U,S,\Delta)\simeq_{s}(V,T,\Theta)$, by item (1) we have the decompositions
$(U,S,\Delta)\cong(U_{0},S,0)\oplus(U_{1},S,\Delta_{1})\oplus\cdots\oplus(U_{r},S,\Delta_{r})$
and
$(V,T,\Theta)\cong(V_{0},T,0)\oplus(V_{1},T,\Theta_{1})\oplus\cdots\oplus(V_{r},T,\Theta_{r})$
such that all $\Delta_{i}$’s and $\Theta_{i}$’s are indecomposable and
$\Delta_{i}\simeq_{s}\Theta_{i}$ for each $i$. As $\mathbbm{k}$ is
algebraically closed, $\Delta_{i}$ and $\Theta_{i}$ are also absolutely
indecomposable. Then for each $i$, we have $\Delta_{i}\cong
a_{i}\Theta_{i}\cong\Theta_{i}$ for some $a_{i}\in\mathbbm{k}^{*}$ by item
(2), and consequently we prove $\Delta\cong\Theta$. ∎

As direct consequences, we recover easily the related main results of [2, 4].

######  Corollary 3.9.

  * (1)

Two $d$-linear maps over the complex field $\mathbbm{C}$ are isomorphic if and
only if they are symmetrically equivalent to each other.

  * (2)

When $d$ is a positive odd integer, two $d$-linear maps over the real field
$\mathbbm{R}$ are isomorphic if and only if they are symmetrically equivalent
to each other.

  * (3)

Assume $d$ is a positive even integer. If two $d$-linear maps $(U,S,\Delta)$
and $(V,T,\Theta)$ over $\mathbbm{R}$ are symmetrically equivalent, then there
exist the decompositions $\Delta\cong\Delta_{1}\oplus\Delta_{2}$ and
$\Theta\cong\Theta_{1}\oplus\Theta_{2}$ such that $\Delta_{1}$ is isomorphic
to $\Theta_{1}$ and $\Delta_{2}$ is isomorphic to $-\Theta_{2}$.

###### Proof.

We only prove (3) and the others are similar. We first prove the statement in
the case of indecomposable forms. Suppose that two $d$-linear maps
$(U,S,\Delta)$ and $(V,T,\Theta)$ over $\mathbbm{R}$ are indecomposable and
symmetrically equivalent. Let $\psi:S\to T$ and $\phi_{1},\cdots,\phi_{d}:U\to
V$ be the linear bijections such that
$\psi\Delta(u_{1},\cdots,u_{d})=\Theta(\phi_{\sigma_{1}}(u_{1}),\cdots,\phi_{\sigma_{d}}(u_{d}))$
for all $u_{1},\cdots,u_{d}\in U$ and each reordering
$\sigma_{1},\cdots,\sigma_{d}$ of $1,\cdots,d$. For each $i$, we can write
$\phi_{i}=a_{i}\phi_{1}$ for some $a_{i}\in Z(V,T,\Theta)^{\times}$ by
Proposition 3.5. As $\Theta$ is indecomposable, the center $Z(V,T,\Theta)$ is
local and its residue field is $\mathbbm{R}$ or $\mathbbm{C}$. If the residue
field is $\mathbbm{C}$, then each $a_{i}$ is a $d$-th power in $Z(V,T,\Theta)$
by Lemma 3.6. The same argument as Theorem 3.8 (2) shows that
$\Delta\cong\Theta$. If the residue field is $\mathbbm{R}$, then $a_{i}$ or
$-a_{i}$ is a $d$-th power, and we similarly have $\Delta\cong\Theta$ or
$\Delta\cong-\Theta$.

In general, we have the decompositions
$(U,S,\Delta)\cong(U_{0},S,0)\oplus(U_{1},S,\Delta^{\prime}_{1})\oplus\cdots\oplus(U_{r},S,\Delta^{\prime}_{r})$
and
$(V,T,\Theta)\cong(V_{0},T,0)\oplus(V_{1},T,\Theta^{\prime}_{1})\oplus\cdots\oplus(V_{r},T,\Theta^{\prime}_{r})$
such that all $\Delta^{\prime}_{i}$’s and $\Theta^{\prime}_{i}$’s are
indecomposable and $\Delta_{i}^{\prime}\simeq_{s}\Theta^{\prime}_{i}$ for each
$i$. By the previous argument on indecomposable forms, we have
$\Delta_{i}^{\prime}\cong\Theta_{i}^{\prime}$ or
$\Delta_{i}^{\prime}\cong-\Theta_{i}^{\prime}$. Let $\Delta_{1}$ be the direct
sum of $\Delta_{0}$ and all $\Delta_{i}^{\prime}$ ’s such that
$\Delta_{i}\cong\Theta_{i}^{\prime}$, and let $\Delta_{2}$ be the direct sum
of all $\Delta_{i}^{\prime}$ ’s such that
$\Delta_{i}\cong-\Theta_{i}^{\prime}$. Similarly define $\Theta_{1}$ and
$\Theta_{2}$. Combining together, we have
$\Delta\cong\Delta_{1}\oplus\Delta_{2}$ and
$\Theta\cong\Theta_{1}\oplus\Theta_{2}$ with $\Delta_{1}\cong\Theta_{1}$ and
$\Delta_{2}\cong-\Theta_{2}$. ∎

Finally, we apply Theorem 3.8 to provide a linear algebraic proof for Theorem
1.4, the well known Torelli type result of Donagi [5, Proposition 1.1]. Let
$f(x_{1},\cdots,x_{n})\in\mathbbm{k}[x_{1},\cdots,x_{n}]$ be a homogeneous
polynomial of degree $d$. Let $J(f)$ be its Jacobian ideal generated by
$\frac{\partial f}{\partial x_{1}},\cdots,\frac{\partial f}{\partial x_{n}}$.
The classical Torelli problem concerns about whether $J(f)$ determines $f.$

Homogeneous polynomials are naturally associated to symmetric multilinear
forms and symmetric tensors, see e.g. [9, 10]. Write $f$ in the symmetric way

$f(x_{1},\cdots,x_{n})=\sum_{1\leq i_{1},\cdots,i_{d}\leq n}a_{i_{1}\cdots
i_{d}}x_{i_{1}}\cdots x_{i_{d}}$

where the $a_{i_{1}\cdots i_{d}}$’s are symmetric with respect to their
indices. Let $V=\mathbbm{k}^{n}$ with a basis $e_{1},\dots,e_{n}.$ Define the
symmetric $d$-linear form $\Theta\colon V\times\cdots\times
V\longrightarrow\mathbbm{k}$ by

$\Theta(e_{i_{1}},\cdots,e_{i_{d}})=a_{i_{1}\cdots i_{d}},\quad\forall 1\leq
i_{1},\cdots,i_{d}\leq n.$

The pair $(V,\Theta)$ is called the associated symmetric $d$-linear form of
$f$ under the basis $e_{1},\cdots,e_{n}.$ The homogeneous polynomial $f$ and
the symmetric multilinear form $(V,\Theta)$ is explicitly related as

$f(x_{1},\cdots,x_{n})=\Theta\left(\sum_{1\leq i\leq n}x_{i}e_{i},\ \dots,\
\sum_{1\leq i\leq n}x_{i}e_{i}\right).$

For each $1\leq i\leq n$, let $\Theta_{i}$ be the $(d-1)$-linear form on $V$
such that

$\Theta_{i}(v_{1},\cdots,v_{d-1})=\Theta(e_{i},v_{1},\cdots,v_{d-1}),\forall
v_{1},\cdots,v_{d-1}\in V.$

It is well known that

(3.2) $\frac{1}{d}\frac{\partial f}{\partial
x_{i}}=\Theta_{i}\left(\sum_{1\leq i\leq n}x_{i}e_{i},\ \cdots,\ \sum_{1\leq
i\leq n}x_{i}e_{i}\right).$

In other words, $\Theta_{i}$ is associated to $\frac{1}{d}\frac{\partial
f}{\partial x_{i}}$ under the basis $e_{1},\dots,e_{n}.$ Recall that centers
can also be equivalently defined in terms of homogeneous polynomials, see [8,
9]. Let $H$ be the Hessian matrix $(\frac{\partial^{2}f}{\partial
x_{i}\partial x_{j}})_{1\leq i,\ j\leq n}$ of $f$ and define its center $Z(f)$
as

$Z(f)=\\{X\in\mathbbm{k}^{n\times n}\mid(HX)^{T}=HX\\}.$

It is clear that $Z(V,\Theta)\cong Z(f)$.

Proof of Theorem 1.4 Let $\Theta$ and $\Delta$ be the $d$-linear forms
associated to $f$ and $g$ respectively. Let $E(f)$ be the vector space spanned
by $\frac{\partial f}{\partial x_{1}},\cdots,\frac{\partial f}{\partial
x_{n}}$ which is the $(d-1)$-th part of $J(f)$. If $\operatorname{dim}J<n$,
then $f$ is degenerate and let $V_{0}$ be the subspace consisting of the
vectors along which the partial derivative of $f$ is zero and choose another
subspace $V_{1}$ such that $V=V_{0}\oplus V_{1}$. Then we have a decomposition
$(V,\Theta)=(V_{0},0)\oplus(V_{1},\Theta|_{V_{1}})$ by Theorem 3.8. As
$J(f)=J(g)$ and thanks to Equation $\eqref{der}$, we have a similar direct sum
decomposition $(V,\Delta)=(V_{0},0)\oplus(V_{1},\Delta|_{V_{1}})$. Moreover
the restrictions of $f$ and $g$ on $V_{1}$ also have the same Jacobian ideal.
Therefore, we reduce the theorem to the nondegenerate case.

As $J(f)=J(g)$ and $\operatorname{dim}E(f)=n$, there exists a matrix
$A=(a_{ij})_{n\times n}\in GL_{n}(\mathbbm{k})$ such that

(3.3) $(\frac{\partial g}{\partial x_{1}},\cdots,\frac{\partial g}{\partial
x_{n}})=(\frac{\partial f}{\partial x_{1}},\cdots,\frac{\partial f}{\partial
x_{n}})A.$

Define $\phi\in\operatorname{End}(V)$ by
$\phi(e_{i})=\sum\limits_{j=1}^{n}a_{ji}e_{j}$ for all $1\leq i\leq n.$ Note
by [10, Lemma 3.10] that $A\in Z(f),$ hence $\phi\in Z(V,\Theta).$ In
addition, for all $1\leq i_{1},\dots,i_{d}\leq n$ we have

$\displaystyle\Theta(\phi(e_{i_{1}}),e_{i_{2}},\cdots,e_{i_{d}})$
$\displaystyle=$
$\displaystyle\Theta(\sum\limits_{j=1}^{n}a_{ji_{1}}e_{j},e_{i_{2}},\cdots,e_{i_{d}})$
$\displaystyle=$
$\displaystyle\sum\limits_{j=1}^{n}a_{ji_{1}}\Theta_{j}(e_{i_{2}},\cdots,e_{i_{d}})$
$\displaystyle=$
$\displaystyle\Delta_{i_{1}}(e_{i_{2}},\cdots,e_{i_{d}})\quad(by\ \eqref{der}\
\&\ \eqref{jac})$ $\displaystyle=$
$\displaystyle\Delta(e_{i_{1}},e_{i_{2}},\cdots,e_{i_{d}}).$

It follows that $\phi,\operatorname{Id},\dots,\operatorname{Id}$ make a
symmetric equivalence between $\Theta$ and $\Delta.$ Then by item (3) of
Theorem 3.8, $f$ and $g$ are equivalent up to an invertible linear
transformation.

## References

  * [1] M. Atiyah, I. Macdonald, Introduction to commutative algebra, Addison-Wesley, Reading, MA, 1969.
  * [2] G. Belitskii, V. Futorny, M. Muzychuk, V. Sergeichuk, Congruence of matrix spaces, matrix tuples, and multilinear maps, Linear Alg. Appl. 609 (2021) 317–331.
  * [3] G. Belitskii, V. Sergeichuk, Complexity of matrix problems, Linear Alg. Appl. 361 (2003) 203–222.
  * [4] G. Belitskii, V. Sergeichuk, Congruence of multilinear forms, Linear Alg. Appl. 418 (2006) 751–762.
  * [5] R. Donagi, Generic Torelli for projective hypersurfaces, Compositio Math. 50 (1983), no. 2-3, 325–353.
  * [6] D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, No. 150, Springer-verlag, 1994.
  * [7] V. Futorny, J. Grochow, V. Sergeichuk, Wildness for tensors, Linear Alg. Appl. 566 (2019) 212–244.
  * [8] D. Harrison, A Grothendieck ring of higher degree forms, J. Algebra 35 (1975) 123–138.
  * [9] H.-L. Huang, H. Lu, Y. Ye, C. Zhang, Diagonalizable higher degree forms and symmetric tensors, Linear Alg. Appl. 613 (2021) 151–169.
  * [10] H.-L. Huang, H. Lu, Y. Ye, C. Zhang, On centers and direct sum decompositions of higher degree forms, Linear Multilinear Algebra, DOI: 10.1080/03081087.2021.1985057.


columns=fixed, numbers=left, stepnumber=1, numbersep=6pt, numberstyle=,
xleftmargin=11pt, framextopmargin=1pt, aboveskip=0pt, identifierstyle=,
stringstyle=, keywords=[1]platform, architecture, cpu, os,"passthru-
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11institutetext:  University of Neuchâtel, Switzerland, 11email:
<EMAIL_ADDRESS>
# Understanding Cryptocoins Trends Correlations

Pasquale De Rosa  0000-0001-9726-7075 Valerio Schiavoni  0000-0003-1493-6603

###### Abstract

Crypto-coins (also known as cryptocurrencies) are tradable digital assets.
Notable examples include Bitcoin, Ether and Litecoin. Ownerships of
cryptocoins are registered on distributed ledgers (, blockchains). Secure
encryption techniques guarantee the security of the transactions (transfers of
coins across owners), registered into the ledger. Cryptocoins are exchanged
for specific trading prices. While history has shown the extreme volatility of
such trading prices across all different sets of crypto-assets, it remains
unclear what and if there are tight relations between the trading prices of
different cryptocoins. Major coin exchanges (, Coinbase) provide trend
correlation indicators to coin owners, suggesting possible acquisitions or
sells. However, these correlations remain largely unvalidated.

In this paper, we shed lights on the trend correlations across a large variety
of cryptocoins, by investigating their coin-price correlation trends over a
period of two years. Our experimental results suggest strong correlation
patterns between main coins (Ethereum, Bitcoin) and alt-coins. We believe our
study can support forecasting techniques for time-series modeling in the
context of crypto-coins. We release our dataset and code to reproduce our
analysis to the research community.

###### Keywords:

cryptocoins correlations

##  1 Introduction

Cryptocurrencies, also known as crypto-coins, are tradable digital assets,
backed by secure encryption techniques to ensure the security of transactions
(typically, the transfer of coins across wallets). Notable examples include
Bitcoin [14], Ether (the native cryptocurrency of the Ethereum blockchain [8])
or Litecoin (used in a fork of the original Bitcoin network). Nowadays there
exists thousands of cryptocurrencies (CoinMarketCap [3] lists 10039 coins as
of April 2022). Cryptocoins are designed to be traded as a form of digital
money: the first useful Bitcoin transaction was used by a peer-to-peer payment
between Satoshi Nakamoto (Bitcoin’s founder) and one of its early adopters,
and dates back to 2009.111https://www.blockchain.com/btc/block/170 Cryptocoins
are nowadays traded over online (centralized or decentralized) _exchanges_ ,
including Coinbase [2], Kraken [5], Binance [1], Uniswap [6], . With a current
estimated worldwide market-cap of 1.71 Trillion dollars, the cryptocoins
economy roughly match the GDP of South Korean in 2021 [4].

Figure 1: Normalized (min-max) average prices of the top-5 cryptocoins since
January 2020.

The ownership of cryptocoins is registered on distributed ledgers (,
blockchains), together with the corresponding transactions (transfers of coins
across wallets). Cryptocoins are exchanged (, sold, bought) for specific
trading prices. While it is beyond the scope of this work to understand the
exact nature of those prices, history has shown the extreme volatility of such
trading prices across all different sets of crypto-assets. For instance,
Figure 1 shows the normalized average daily prices of 5 popular cryptocoins (,
BTC, ETH, BNB, XRP, ADA) since January 2020.

It remains unclear what and if there are tight relations between the trading
prices of different cryptocoins. Major cryptocoin exchanges, in particular
given the enormous popularity that such digital assets have grown into the
large public, and further facilitated by the easy access to these markets via
mobile apps, started to provide _trend correlation_ indicators to coin/wallet
owners. Such correlation indicators can possibly drive end-users towards
acquisitions or sells. Coinbase, among the most popular cryptocoin exchanges,
indicates the price correlation as the tendency of other asset prices to
change at the same time as the asset shown on the page. In their case,
correlation is computed leveraging the Pearson correlation with USD order
books over the last 90 days.

However, the nature of such correlations, their intensity as well as the
evolution of the correlations through time, remain largely unvalidated.

The contributions of this work-in-progress paper are twofold. First, we
extract the trading prices, as well as other exchange metadata (, open and
closing price, market cap, volume), for the top-100 cryptocoins since the last
two years from a popular cryptocoin monitoring web-site. Second, we leverage
this dataset to carry out our preliminary study of the trend correlations
between and across crypto-coins. Specifically, we investigate daily, weekly
and monthly correlation patterns exhibited by two principal cryptocoins, BTC
and ETH, against the remaining set of _alt-coins_ in our dataset. Our analysis
show strong correlations between the observed trends. We will leverage these
observations in our future work, where we plan to exploit the observed
correlations to forecast the future trading trends and by considering the
problem of time-series forecasting applied to the crypto-coin market.

We follow an open science approach: our datasets will be released and made
available to the research and open-source community.

Roadmap. This paper is organized as follows. Section 2 provides background
materials on Bitcoin, Ethereum, as well as general notions of correlation
analysis. Section 3 describes our dataset, as well as our work-in-progress
analysis. We briefly cover related work in Section 4, before concluding and
presenting our future work in Section 5.

##  2 Background

Cryptocoins in a nutshell. Cryptocoins are digitally encrypted assets. They
were typically designed to replace fiat currencies and used mostly in peer-to-
peer networks. Depending on the incentive natures of the underlying
blockchain, cryptocoins (or token) are rewarded to nodes in the network. We
differentiate between three main types of cryptocoins: _(i)_ Bitcoin, _(ii)_
alt-coins, and _(iii)_ stable coins.222Some characterizations define stable
coins as sub-classes of alt-coins, together with secure tokens, utility
tokens, and more. We leave as future work to study in-depth the correlations
between such sub-types of alt-coins. Alt-coins are _alternative_ coins to
Bitcoin. Notable examples include Ether (ETH), Cardano (ADA), Litecoin (LTC),
or Ripple (XRP). A stablecoin is a class of cryptocurrencies that attempt to
offer price stability and are backed by a reserve asset, gold or the value of
the American dollar. Examples include USDT (Tether) and USDC.

Time series analysis. A time series is an $n$-tuple of observations collected
sequentially over time. Common examples of time series include trends of
interest rates and stock prices, daily high and low temperatures, the
electrical activity of the heart, . The purpose of time series analysis is
generally twofold: _(i)_ to understand the mechanisms and the inner dynamics
of an observed series, and _(ii)_ forecast the future values of the series
based on the historical ones. To analyze time series as sequences of random
variables (stochastic processes), it is common to assume their stationarity: a
time series is _stationary_ if the probability laws that govern its behavior
do not change, and its mean $\mu$ is constant over time. The Autoregressive
Moving Average is a state-of-the-art stationary time series modeling approach,
which combines an Autoregressive (AR) process of order $p$ and of a Moving
Average (MA) process of order $q$.

Real applications do not expose stationary trends. The Autoregressive
Integrated Moving Average model (ARIMA) differentiates a nonstationary process
a number $d$ of times, until it becomes stationary. It is common to observe
the presence of a seasonality in the trend of a time series, especially in
applications where cyclical tendencies are very common (like business or
economics). To handle periodical components, a common model is the Seasonal
Autoregressive Integrated Moving Average (SARIMA), that can be mapped to a
standard ARIMA model in absence of seasonality [10].

Figure 2: ML-based time series forecasting approaches: RNN, LSTM and GRU.

A significant progress in time series modeling was introduced by _temporally-
aware_ ML models, Recurrent Neural Networks (RNNs) (see Figure 2). In those,
the behaviour of hidden neurons is not only determined by the activations in
previous hidden layers, but also by the activations at earlier times. The
activation function for every hidden layer of a RNN is:
$h^{(t)}=f(h^{(t-1)},x^{(t)},\theta)$. There, the hidden layer at the time
$t$, $h^{(t)}$, is a function of the previous status, $h^{(t-1)}$, of the
current input $x^{(t)}$ and of the activation function adopted, $\theta$.

The training process of RNNs is usually complex, due to the _unstable gradient
problem_ : the gradient of the adopted cost function tends to get smaller or
bigger as it is propagated back through layers, resulting in a final vanishing
or exploding effect, respectively. RNNs are unable to model _long term
dependencies_ , lacking predictive ability when dealing with long sequences of
data. To solve this problem, more effective sequence models are adopted in
practical applications, such as Long Short-Term Memory (LSTM) and networks
based on the Gated Recurrent Unit (GRU). Such _gated RNN_ architectures allow
the network to accumulate information over a long time period, learning to
decide how to forget the old states once that information has been used and
processed [11].

##  3 Preliminary Evaluation

We describe here our experimental evaluation of the correlations between
cryptocoins. First we describe our dataset, and then we show several
correlation patterns.

Coin | Open | High | Low | Close | Volume | Market Cap
---|---|---|---|---|---|---
BTC | 29.39/19.55 | 30.18/20.07 | 28.50/18.92 | 29.42/19.53 | 39.63/20.41 | 550.41/368.54
ETH | 1.57/1.44 | 1.63/1.49 | 1.51/1.39 | 1.57/1.44 | 20.48/11.05 | 184.19/170.90
BNB | 0.20/0.21 | 0.21/0.22 | 0.19/0.20 | 0.20/0.21 | 1.63/1.91 | 33.18/35.39
XRP | 5.64e-4/3.94e-4 | 5.90e-4/4.17e-4 | 5.36e-4/3.68e-4 | 5.64e-4/3.94e-4 | 4.59/4.91 | 25.70/17.96
ADA | 7.97e-4/8.21e-4 | 8.33e-4/8.55e-4 | 7.60e-4/7.84e-4 | 7.99e-4/8.21e-4 | 2.29/2.77 | 25.67/26.77
Table 1: Mean/standard deviation for the top-5 cryptocoins since January 2020
(Open, High, Low and Close expressed in 1K US dollars, Volume and Market Cap
in 1B US dollars).

Dataset. We collected our dataset from CoinMarketCap [3], a leading aggregator
of cryptocurrency market data. It contains records (High, Low, Open, Close,
Volume and Market Capitalization) for 68 coins registered during a time frame
of 25 months, namely from 24.12.2019 to 24.01.2022. "High" and "Low" are the
highest and lowest prices reached by the asset during the considered time
frame; "Open" and "Close" the opening and closing market prices; "Volume" the
measure of how much it was traded in the last period. Finally, "Market
Capitalization" indicates the total market value of its circulating supply.
The dataset includes a total of 51884 observations. The resulting time series
for each coin trend includes 763 steps. Table 1 reports mean and standard
deviation for the gathered records and across the top-5 cryptocoins in our
dataset.

Figure 3: Trend of Bitcoin and Ethereum prices during the last 2 years.

The two major coins in terms of Volume and Market Capitalization are Bitcoin
(BTC) and Ethereum (ETH), that we selected as the benchmarks for our
subsequent study. The price trend of those cryptocoins over the past two years
(shown in Figure 3) showed on average a high positive correlation (with a
Pearson coefficient $\approx 0.9$).

Correlation Patterns. The aim of the present study is to identify and analyze
the presence of cross-correlation patterns in cryptocurrency trends. To do so,
we analyze the correlations of 66 alt-coins present in our dataset against BTC
and ETH, and for three different time frames: daily, weekly and monthly. For
weekly and monthly correlations we define the sequence segments adopting a
sliding window approach, where observations are grouped within a window that
slides across the data stream. The daily observations for each coin are
averaged over sliding partitions of 7 and 30 days respectively, and then the
correlations with other coins are computed on the resulting aggregated values.
Note that we postpone the study of _thumbing_ windows, where there is no
overlapping of data clusters, to future work. We represent those correlations,
averaged among all the studied variables (, High, Low, Open, Close, Volume and
Market Cap), as a series of "cross-correlograms" of coins (Figure 4). The
radius of each circle represents the strength of the relation (in terms of
Pearson coefficient) between each of the considered alt-coin and BTC (Figs.
4a/c/e) or ETH (Figs. 4b/d/f). The color identifies the sign of the
correlation (green if positive, red otherwise). The analysis of the cross-
correlogram clearly shows how the vast majority of considered alt-coins are
strongly correlated with and follows the same trend of the two market leaders.
Their average values of the Pearson coefficient very close to $1$. Not
surprisingly, the only visible exceptions are represented by the stablecoins
available in our dataset (, USDP, TUSD, DAI, BUSD, USDC, USDT), that are
pegged to the US dollar and follow standalone trends with total independence
from the rest of the coins in the market.

Figure 4: Daily, weekly and monthly cross-correlations between the alt-coins
and BTC/ETH.

##  4 Related Work

There exists studies which analyzed co-movement and cross-correlation
phenomena in cryptocurrency market trends. Similar to our study, Katsiampa
[12] investigated the volatility dynamics of the two major cryptocurrencies,
Bitcoin and Ethereum, finding evidence of interdependencies between the two
and price responsiveness to major news in the market. Aslanidis et al. [7]
showed that cryptocurrencies exhibit similar mean correlation among them, with
an unstable trend over time; in addition, the authors computed coins
correlation against more traditional assets, detecting an independent behavior
respect to other financial markets. In [13], Bitcoin is identified as the
leader in the cryptocurrency market using wavelet-based methods, showing how
other coins trends are dependent from BTC price movements: as a result,
Bitcoin price drops are immediately reflected in other cryptocurrency prices.
Finally, [9] studied the collective behaviour for the cryptocurrency market
discovering distinct and not time-persistent community structures
characterized by cross-correlation.

##  5 Conclusion and Future Work

Cryptocoins present very volatile trends on public exchanges. In this work-in-
progress paper, we presented our preliminary evaluation of the correlations
between BTC, Ether and 66 other alt-coins. Our analysis shows strong
correlations, suggesting alt-coins follow closely the trends of the two main
ones. Following this initial study, we will further investigate the cross-
correlation between the two market leaders and the alt-coins, in the
perspective to forecast their price trends by using the time-series techniques
from §2. We believe that our work could represent a significant starting point
for further analyses in co-movement behaviors within the cryptocoin markets
and in modeling and forecasting trends of the asset prices.

Metadata, analysis data, tools and code for reproducibility are available to
the research community at https://github.com/quapsale/cryptoanalytics/.

## References

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