# Hadamard products of hypersurfaces

Cristiano Bocci, Enrico Carlini  Cristiano Bocci
Department of Information Engineering and Mathematics, University of Siena
Via Roma, 56 Siena, Italy<EMAIL_ADDRESS>Enrico Carlini
Department of Mathematical Sciences, Politecnico di Torino
Corso Duca degli Abruzzi 24, Turin, Italy<EMAIL_ADDRESS>
###### Abstract.

In this paper we, first, characterize hypersurfaces for which their Hadamard
product is still a hypersurface. Then we pass to study hypersurfaces and, more
generally, varieties which are idempotent under Hadamard powers.

##  1\. Introduction

According to the definition in [7, 8], the Hadamard product between projective
varieties $V,W\subset{\mathbb{P}}^{n}$, is the closure of the image of the
rational map

$V\times
W\dashrightarrow{\mathbb{P}}^{n},\quad([a_{0}:\dots:a_{n}],[b_{0}:\dots:b_{n}])\mapsto[a_{0}b_{0}:a_{1}b_{1}:\ldots:a_{n}b_{n}].$

For any projective variety $V$, we may consider its Hadamard square $V^{\star
2}=V\star V$ and its higher Hadamard powers $V^{\star r}=V\star
V^{\star(r-1)}$.

In [7], the authors use this definition to describe the algebraic variety
associated to the restricted Boltzmann machine, which is the undirected
graphical model for binary random variables specified by the bipartite graph
$K_{r,n}$. This variety is the $r-$th Hadamard power of the first secant
variety of $({\mathbb{P}}^{1})^{n}$. Note that [8] concerns the case
$r=2,n=4$.

Hadamard products and powers are well-connected to other operations of
varieties. They are the multiplicative analogs of joins and secant varieties,
and in tropical geometry, tropicalized Hadamard products equal Minkowski sums.
It is natural to study properties of this new operation, and see its effects
on various varieties.

For this reasons, in the last years, the Hadamard product of projective
varieties has been widely studied from the point of view of Projective
Geometry. The main problem in this setting is the behaviour of the Hadamard
product between varieties with many zero. The paper [2], where Hadamard
product of linear space is studied, can be considered the first step in this
direction. Successively, the first author, with Calussi, Fatabbi e Lorenzini
([3]) address the Hadamard product of linear varieties not necessarily in
general position, obtaining, in $\mathbb{P}^{2}$ a complete description of the
possible outcomes while, in $\mathbb{P}^{3}$, under suitable conditions (which
can be prove to be generic), they show that, for two set $V$ and $V^{\prime}$
of collinear points, $V\star V^{\prime}$ consists of $|V||V^{\prime}|$ points
on the two different rulings of a non-degenerate quadric. Then, in [4], they
address the Hadamard product of not necessarily generic linear varieties and
show that the Hilbert function of the Hadamard product $V\star W$ of two
varieties, with $\dim(V),\dim(W)\leq 1$, is the product of the Hilbert
functions of the original varieties $V$ and $W$. Moreover, they show that the
Hadamard product of two generic linear varieties $V$ and $W$ is projectively
equivalent to a Segre embedding. In [5], the second author, with Calussi,
Fatabbi e Lorenzini consider generic degenerate subvarieties
$V_{i}\subset{\mathbb{P}}^{n}$ and compute dimension and degree formulas for
the Hadamard product of the varieties $V_{i}$.

The construction of star configurations of points, via Hadamard product,
described in [2], found a generalization in [6] where the authors introduce a
new construction using the Hadamard product to present star configurations of
codimension $c$ of ${\mathbb{P}}^{n}$ and which they called Hadamard star
configurations. Successively, Bahmani Jafarloo and Calussi introduce a more
general type of Hadamard star configuration; any star configuration
constructed by their approach is called a weak Hadamard star configuration. In
[1] they classify weak Hadamard star configurations and, in the case $c=n$,
they investigate the existence of a (weak) Hadamard star configuration which
is apolar to the generic homogeneous polynomials of degree $d$.

In [9], the Hadamard product is studied for projective varieties and, in
particular, the authors consider Hadamard products of varieties of matrices
with fixed rank also due to their connection with problems related to
algebraic statistics and quantum information. The authors introduce the notion
of Hadamard decomposition and Hadamard rank of a matrix (multiplicative
versions of the well-studied additive decomposition of tensors and tensor
ranks). One of the result in [9] is the following

######  Proposition 1.1.

Let $V\subset{\mathbb{P}}^{n}$ be a projective variety generated by an ideal
with a minimal set of generators of the type
$f_{\alpha,\beta}=X^{\alpha}-X^{\beta}$. with
$\alpha,\beta\in{\mathbb{N}}^{n+1}$. Then

$V^{\star 2}=V.$

In this paper, starting from the previous result, we are interested to study
which varieties are idempotent under Hadamard power. We focus, in particular
in the case of hypersurfaces. Since, in general, the Hadamard product of two
hypersurfaces $V$ and $W$ is the whole ambient space, we first study which
conditions on $V$ and $W$ are required such that $V\star W$ is a hypersurface.
We obtain a necessary and sufficient condition which is stated in Propositions
4.5 and 4.7. Successively we pass to study Hadamard powers of a hypersuface
$V$ obtaining a sufficient and necessary condition for which $V^{\star r}=V$.
This results leads to Theorem 6.2 which generalizes the result of Proposition
1.1 to $V^{\star r}=V$.

The paper is organized in the following way.

In Section 2 we recall the definitions of Hadamard product of varieties and
Hadamard powers. In Section 3 we define the concept of Hadamard
transformation, $f^{\star P}$ of a polynomial $f$ for a point $P$ and we prove
some related results leading to Theorem 3.5 which prove the connection between
the ideals of $V$ and $P\star V$.

In Section 4 we study the Hamadard product of hypersurfaces, considering also
the case of union of coordinate hyperplanes. Proposition 4.5 shows that if
$C,D\subset{\mathbb{P}}^{n}$ are binomial hypersurfaces with “similar”
defining equations, then $C\star D$ is a binomial hypersurface. Proposition
4.7 show that this condition is also necessary.

The case of Hamadard powers of hypersurfaces, treated in Section 5, show
similar results to the case of the Hamadard product of hypersurfaces. In this
case the binomial equation of the hypersurface has an extra condition on the
coefficients leading us to introduce the concept of “binomial hypersurfaces of
type $(t,\epsilon)$”.

The generalization to any power of the result in [9] is described in Section
6.

Acknowledgments: the authors are members of GNSAGA of INDAM. The second author
was supported by MIUR grant Dipartimenti di Eccellenza 2018-2022
(E11G18000350001).

##  2\. Basic definitions

We recall now the definitions of Hadamard product of varieties and Hadamard
powers, as introduced in [2]. We work over the field of complex numbers
$\mathbb{C}$.

######  Definition 2.1.

Let $H_{i}\subset{\mathbb{P}}^{n},i=0,\ldots,n$, be the hyperplane $x_{i}=0$
and set

$\Delta_{i}=\bigcup_{0\leq j_{1}<\ldots<j_{n-i}\leq n}H_{j_{1}}\cap\ldots\cap
H_{j_{n-i}}.$

In other words, $\Delta_{i}$ is the $i-$dimensional variety of points having
at most $i+1$ non-zero coordinates. Thus $\Delta_{0}$ is the set of
coordinates points and $\Delta_{n-1}$ is the union of the coordinate
hyperplanes. Note that elements of $\Delta_{i}$ have at least $n-i$ zero
coordinates. We have the following chain of inclusions:

(2.1)
$\Delta_{0}=\\{[1:0:\ldots:0],\ldots,[0:\ldots:0:1]\\}\subset\Delta_{1}\subset\ldots\subset\Delta_{n-1}\subset\Delta_{n}={\mathbb{P}}^{n}.$

######  Definition 2.2.

Let $p,q\in{\mathbb{P}}^{n}$ be two points of coordinates respectively
$[a_{0}:a_{1}:\ldots:a_{n}]$ and $[b_{0}:b_{1}:\ldots:b_{n}]$. If
$a_{i}b_{i}\not=0$ for some $i$, their Hadamard product $p\star q$ of $p$ and
$q$, is defined as

$p\star q=[a_{0}b_{0}:a_{1}b_{1}:\ldots:a_{n}b_{n}].$

If $a_{i}b_{i}=0$ for all $i=0,\dots,n$ then we say $p\star q$ is not defined.

Given varieties $V$ and $W$ in ${\mathbb{P}}^{n}$, their Hadamard product
$V\star W$ is defined as

$V\star W=\overline{\\{p\star q:p\in V,q\in W,p\star q\mbox{ is defined}\\}}.$

######  Definition 2.3.

Given a positive integer $r$ and a variety $V\subset{\mathbb{P}}^{n}$, the
r-th Hadamard power of $V$ is

$V^{\star r}=V\star V^{\star(r-1)},$

where $V^{\star 0}=[1:\dots:1]$.

Note that $V\star W$ is a variety such that $\dim(V\star
W)\leq\dim(V)+\dim(W)$ and that $V\star W$ can be empty even if neither $V$
nor $W$ is empty. Moreover, $\dim(V^{\star r})\leq r\dim(V)$ and the $r$-th
Hadamard power cannot be empty if $V$ is not empty ([2]).

Note that if $V,W$ are irreducible, then $V\star W$ is irreducible.

For the rest of paper we denote by $Z(f)$ (respectively $Z(f_{1},\dots,f_{t})$
and $Z(J)$) the zero set of a polynomial $f$ (respectively, of a set of
polynomials $f_{1},\dots,f_{t}$ and of an ideal $J$).

##  3\. Preliminary results

Let $R$ be the ring ${\mathbb{C}}[x_{0},x_{1},\dots,x_{n}]$. Given a vector of
nonnegative integer $I=(i_{0},\dots,i_{n})$, we denote by $X^{I}$ the monomial
$x_{0}^{i_{0}}x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}$. Similarly, if $P$ is a point
of ${\mathbb{P}}^{n}$ of coordinates $[p_{0}:p_{1}:\cdots:p_{n}]$, we denote
by $P^{I}$ the monomial $X^{I}$ evaluated in $P$, that is
$p_{0}^{i_{0}}p_{1}^{i_{1}}\cdots p_{n}^{i_{n}}$. Moreover, if $P$ is a point
of ${\mathbb{P}}^{n}\setminus\Delta_{n-1}$ of coordinates
$[p_{0}:p_{1}:\cdots:p_{n}]$, we denote by $\frac{1}{P}$ the point of
coordinates $[\frac{1}{p_{0}}:\frac{1}{p_{1}}:\cdots:\frac{1}{p_{n}}]$.

######  Definition 3.1.

Let $f\in{\mathbb{C}}[x_{0},x_{1},\dots,x_{n}]$ be a homogenous polynomial, of
degree $d$, of the form $f=\sum_{|I|=d}a_{I}X^{I}$ and consider a point
$P\in{\mathbb{P}}^{n}\setminus\Delta_{n-1}$. The Hadamard transformation of
$f$ by $P$ is the polynomial

(3.1) $f^{\star P}=\sum_{|I|=d}\frac{a_{I}}{P^{I}}X^{I}.$

######  Lemma 3.2.

If $P\in{\mathbb{P}}^{n}\setminus\Delta_{n-1}$, one has

$(f^{\star P})^{\star\frac{1}{P}}=(f^{\star\frac{1}{P}})^{\star P}=f.$

###### Proof.

Since $f^{\star\frac{1}{P}}=\sum_{|I|=d}a_{I}P^{I}X^{I}$, one has

$(f^{\star
P})^{\star\frac{1}{P}}=(\sum_{|I|=d}\frac{a_{I}}{P^{I}}X^{I})^{\star\frac{1}{P}}=\sum_{|I|=d}\frac{a_{I}}{P^{I}}P^{I}X^{I}=\sum_{|I|=d}a_{I}X^{I}=f$

and

$(f^{\star\frac{1}{P}})^{\star P}=(\sum_{|I|=d}a_{I}P^{I}X^{I})^{\star
P}=\sum_{|I|=d}\frac{a_{I}P^{I}}{P^{I}}X^{I}=\sum_{|I|=d}a_{I}X^{I}=f$

∎

######  Lemma 3.3.

Let $Q\in{\mathbb{P}}^{n}$ and $P\in{\mathbb{P}}^{n}\setminus\Delta_{n-1}$.
Then $f(Q)=0$ if and only if $f^{\star P}(P\star Q)=0$.

###### Proof.

The statement easily follows from

(3.2)
$\begin{split}f(Q)&=(\sum_{|I|=d}a_{I}Q^{I})=\sum_{|I|=d}\frac{a_{I}P^{I}}{P^{I}}Q^{I}=\\\
&=\sum_{|I|=d}\frac{a_{I}}{P^{I}}(P^{I}Q^{I})=\sum_{|I|=d}\frac{a_{I}}{P^{I}}(P\star
Q)^{I}=\\\ &=f^{\star P}(P\star Q).\end{split}$

∎

For $P\in{\mathbb{P}}^{n}\setminus\Delta_{n-1}$ and $Q\in{\mathbb{P}}^{n}$
denote by $\frac{Q}{P}$ the Hadamard product $Q\star\frac{1}{P}$.

######  Lemma 3.4.

If $Q\in P\star V$ then $\frac{Q}{P}\in V$.

###### Proof.

If $Q\in P\star V$ then $\frac{Q}{P}\in\frac{1}{P}\star P\star V$. By the
associativity of Hadamard product one has $\frac{1}{P}\star P\star V=V$, and
the claim follows. ∎

######  Theorem 3.5.

Let $V\subset{\mathbb{P}}^{n}$ be a variety and consider a point
$P\in{\mathbb{P}}^{n}\setminus\Delta_{n-1}$. If
$f_{1},\dots,f_{s}\subset{\mathbb{C}}[x_{0},\dots,x_{n}]$ is a generating set
for $I(V)$, that is $I(V)=\langle f_{1},\dots,f_{s}\rangle$, then
$f_{1}^{\star P},\dots,f_{s}^{\star P}$ is a generating set for $I(P\star V)$.

Moreover, if $f_{1},\dots,f_{s}$ is a Groebner bases for $I(V)$, then
$f_{1}^{\star P},\dots,f_{s}^{\star P}$ is a Groebner bases for $I(P\star V)$.

###### Proof.

We first show that the polynomials $f_{1}^{\star P},\dots,f_{s}^{\star P}$
vanish on $P\star V$. If $Q\in P\star V$ then $Q=P\star C$ for some $C\in V$,
then by proof of Lemma 3.3 one has

$f_{i}^{\star P}(Q)=f_{i}^{\star P}(P\star C)=f_{i}(C)=0,\mbox{ for all
}i=1,\dots,s.$

Suppose now that $g\in I(P\star V)$ is different from $f_{i}$ for
$i=1,\dots,s$. Then $g(Q)=0$ for all $Q\in P\star V$ and, by Lemma 3.3 and
3.4, one has

$g^{\star\frac{1}{P}}(\frac{Q}{P})=0\mbox{ for all }\frac{Q}{P}\in V$

that is $g^{\star\frac{1}{P}}\in I(V)$. Hence, by hypothesis
$g^{\star\frac{1}{P}}=\sum_{i}\alpha_{i}f_{i}$ and by Lemma 3.2

$g=(g^{\star\frac{1}{P}})^{\star P}=(\sum_{i}\alpha_{i}f_{i})^{\star
P}=\sum_{i}(\alpha_{i})^{\star P}f_{i}^{\star P}$

and thus $f_{1}^{\star P},\dots,f_{s}^{\star P}$ is a generating set for
$I(P\star V)$.

Suppose now $f_{1},\dots,f_{s}$ is a Groebner bases for $I(V)$, that is

$\langle LT(I(V))\rangle=\langle LT(f_{1}),\dots,LT(f_{s})\rangle.$

Since $LT(f_{i})$ and $LT(f_{i}^{\star P})$ differs only by a constant, for
all $i=1,\dots,s$, one has

$\langle LT(f_{1}),\dots,LT(f_{s})\rangle=\langle LT(f_{1}^{\star
P}),\dots,LT(f_{s}^{\star P})\rangle.$

Hence it is enough to show that $\langle LT(I(V))\rangle=\langle LT(I(P\star
V))\rangle$. The monomial $LM(g)$, for all $g$ in $I(V)\setminus\\{0\\}$ span
$\langle LM(g):g\in I(V)\setminus\\{0\\}\rangle$ and since $LM(g)$ and $LT(g)$
differ only by a constant one has

$A:=\langle LM(g):g\in I(V)\setminus\\{0\\}\rangle=\langle LT(g):g\in
I(V)\setminus\\{0\\}\rangle=\langle LT(I(V))\rangle$

and similarly

$B:=\langle LM(g):g\in I(P\star V)\setminus\\{0\\}\rangle=\langle LT(g):g\in
I(P\star V)\setminus\\{0\\}\rangle=\langle LT(I(P\star V))\rangle$

Hence the proof is complete if we show $A=B$.

Let $h\in A$, then there exists $g\in I(V)$ such that $h=LM(g)$. Hence
$h=\sum\alpha_{i}f_{i}$ and clearly $h^{\star P}=\sum\alpha_{i}^{\star
P}f_{i}^{\star P}$ and moreover $LM(h^{\star P})=LM(h)$, since $h^{\star P}$
and $h$ differ only by a constant, from which one has $g\in B$.

Conversely if $h\in B$, then there exists $g\in I(P\star V)$ such that
$h=LM(g)$. Thus $h=\sum\alpha_{i}f_{i}^{\star P}$ and, by Lemma 3.2,
$h^{\star\frac{1}{P}}=\sum\alpha_{i}^{\star\frac{1}{P}}f_{i}$ with, again,
$LM(h^{\star P})=LM(h)$, from which one has $g\in A$. ∎

######  Remark 3.6.

It is easy to prove that if $f_{1},\dots,f_{s}$ is a minimal Groebner bases
for $I(V)$, then $f_{1}^{\star P},\dots,f_{s}^{\star P}$ is a minimal Groebner
bases for $I(P\star V)$.

##  4\. Hadamard products of hypersurfaces

In general the Hadamard product of two hypersurfaces is not a hypersurface but
the whole ambient space. However, there are hypersurfaces for which this
happens, as expressed by the following results.

First of all we analyze some pathological cases.

######  Lemma 4.1.

Let $H_{i}$ be any coordinate hyperplane, for $i=0,\dots,n$. Then

  * i)

$H_{i_{1}}\star\cdots\star H_{i_{t}}$ is the linear subspace
$Z(x_{i_{j}}\,;\,j=1,\dots,t)$.

  * ii)

$H_{i}\star C=H_{i}$ for any hypersurface $C$ different from a coordinate
hyperplane.

###### Proof.

For $i)$ notice that, since the hyperplane $H_{i_{j}}$ is the set of points
with $x_{i_{j}}=0$, the product $H_{i_{1}}\star\cdots\star H_{i_{t}}$ consists
of all points with $x_{i_{j}}$=0, for $j=1,\dots,t$, i.e. the desired linear
space. Similarly, for $ii)$, since the hyperplane $H_{i}$ is the set of points
with $x_{i}=0$, the product $H_{i}\star C$ is contained in $H_{i}$=0. To see
that $H_{i}\star C=H_{i}$ it is enough to apply [2, Lemma 2.12] which yields
that the tangent space to $H_{i}\star C=H_{i}$ in $P\star Q$ is

$Q\star T_{P}(H_{i})$

for generic points $P$ and $Q$. Note that $T_{P}(H_{i})=H_{i}$ and that, since
$C$ is not contained in a coordinate hyperplane, $Q\star H_{i}=H_{i}$.

∎

Before to show the results for the Hadamard product of hypersurface, we need
the following definition.

######  Definition 4.2.

An irreducible hypersurface $C\subset{\mathbb{P}}^{n}$ is called binomial
hypersurface if the equation defining $C$ is of the form

$\alpha_{1}X^{I_{1}}-\alpha_{2}X^{I_{2}}=0.$

######  Remark 4.3.

The condition of irreducibility of $C$ in the previous definition forces
$I_{1}$ and $I_{2}$ to be coprime (Lemma 4.30 in [10]).

######  Remark 4.4.

From the definition of the binomial hypersurface $C$, if $I_{1}+I_{2}$ has not
zero entries (that is, all variables are involved in the binomial defining
$C$), then $C$ does not contain points in $\Delta_{n-1}\setminus\Delta_{n-2}$
since such points contain only one zero coordinate and so the equation on $C$
does not vanish on them. If $I_{1}+I_{2}$ has zero entries, say, for example
$(I_{1}+I_{2})_{i}=0$, then $C$ contains points $P$ with $P_{i}=0$. However,
in this case, it is still possible to apply the Hadamard transformation by
such $P$ to the polynomial of $C$, since $P_{i}$ will be not involved in the
formula (3.1).

On the other side, the hypersurface $C$, of equation
$\alpha_{1}X^{I_{1}}-\alpha_{2}X^{I_{2}}=0$, contains all subspaces of
equations

$\begin{cases}x_{i_{1}}=0\\\ x_{i_{2}}=0\\\ \vdots\\\ x_{i_{t}}=0\\\
x_{j_{1}}=0\\\ x_{j_{2}}=0\\\ \vdots\\\ x_{j_{s}}=0\\\ \end{cases}$

where $\emptyset\not=\\{i_{1},i_{2},\dots,i_{t}\\}\subset I_{1}$ and
$\emptyset\not=\\{j_{1},j_{2},\dots,j_{s}\\}\subset I_{2}$. For any such
subspace $W\subset C\subset\Delta_{n-2}$ and a point $P\in W$, the Hadamard
product $P\star C$ is exactly $W$ if the only zero entries of $P$ are in
coordinates $i_{1},i_{2},\dots,i_{t},j_{1},j_{2},\dots,j_{s}$, while it is
strictly contained in $W$ if $P$ has extra zero coordinates. In both case,
however, one has $P\star C\subset C$.

######  Proposition 4.5.

If $C,D\subset{\mathbb{P}}^{n}$ are the following binomial hypersurfaces

$C=Z(\alpha_{1}X^{I_{1}}-\alpha_{2}X^{I_{2}})\mbox{ and
}D=Z(\beta_{1}X^{I_{1}}-\beta_{2}X^{I_{2}}),$

then $C\star D$ is the binomial hypersurface

$C\star D=Z(\alpha_{1}\beta_{1}X^{I_{1}}-\alpha_{2}\beta_{2}X^{I_{2}}).$

###### Proof.

Let $P\in C\setminus\Delta_{n-1}$. Since $P\in C$ one has

(4.1) $P^{I_{1}}=\frac{\alpha_{2}}{\alpha_{1}}P^{I_{2}}.$

Moreover, by Theorem 3.5, the equation defining $P\star D$ is

$\frac{\beta_{1}}{P^{I_{1}}}X^{I_{1}}-\frac{\beta_{2}}{P^{I_{2}}}X^{I_{2}}=0$

Using (4.1) in the previous equation, we get the desired equation for $C\star
D$.

If $P\in C\cap\Delta_{n-1}$ then, by Remark 4.4, $P$ has a zero entry in a
coordinate not involved in $\alpha_{1}X^{I_{1}}-\alpha_{2}X^{I_{2}}$ or $P\in
C\cap\Delta_{n-2}$. In the first case we can still apply the first part of the
proof. Im the second case, $P$, and hence also $P\star D$ are contained in a
linear space $W\subset C,D$. Looking at the equations of $C$, $D$ and $C\star
D$, again by Remark 4.4, one has $P\star D\in
Z(\alpha_{1}\beta_{1}X^{I_{1}}-\alpha_{2}\beta_{2}X^{I_{2}})$.

∎

######  Remark 4.6.

Theorem 2.5 in [3] states that if $H$ and $K$ are two hyperplanes in
${\mathbb{P}}^{n}$ of equation respectively $a_{i}x_{i}+a_{j}x_{j}=0$ and
$b_{i}x_{i}+b_{j}x_{j}=0$, then their Hadamard product $H\star K$ is the
hyperplane of equation $a_{i}b_{i}x_{i}-a_{j}b_{j}x_{j}=0$.

Observe that this result is a special case of Proposition 4.5 just taking as
$I_{1}$ and $I_{2}$ respectively the $i-$th and the $j-$th coordinate vectors
and

$\alpha_{1}=a_{i},\alpha_{2}=-a_{j}\mbox{ and
}\beta_{1}=b_{i},\beta_{2}=-b_{j}.$

The following result shows that also the converse of Proposition 4.5 is true.

######  Proposition 4.7.

Let $C,D$ be irreducible hypersurfaces not contained in $\Delta_{n-1}$. If
$C\star D$ is a hypersurface, then $C$ and $D$ are binomials, such that

$C=Z(\alpha_{1}X^{I_{1}}-\alpha_{2}X^{I_{2}})\mbox{ and
}D=Z(\beta_{1}X^{I_{1}}-\beta_{2}X^{I_{2}}).$

Moreover, $C\star
D=Z(\alpha_{1}\beta_{1}X^{I_{1}}-\alpha_{2}\beta_{2}X^{I_{2}})$.

###### Proof.

Let $E=C\star D$ and suppose that

$\begin{array}[]{rcl}C&=&Z(\alpha_{1}X^{I_{1}}+\alpha_{2}X^{I_{2}}+\cdots+\alpha_{s}X^{I_{s}})\\\
D&=&Z(\beta_{1}X^{J_{1}}+\beta_{2}X^{J_{2}}+\cdots+\beta_{r}X^{J_{r}})\\\
E&=&Z(\gamma_{1}X^{K_{1}}+\gamma_{2}X^{K_{2}}+\cdots+\gamma_{t}X^{K_{t}}).\end{array}$

By hypothesis $C\star Q\subseteq E$ for all $Q\in D$, but, since $D$ is not
contained in $\Delta_{n-1}$, there are points $Q\in D$ for which $C\star Q=E$
(see [2]). Thus, by Theorem 3.5,

$Z\left(\alpha_{1}X^{I_{1}}+\alpha_{2}X^{I_{2}}+\cdots+\alpha_{s}X^{I_{s}})^{\star
Q}\right)=E$

and hence $s=t$ and $I_{i}=K_{i}$ for $i=1,\dots,t$.

The same argument applied to $P\star D=E$, for $P\in C$ gives $r=t$ and
$J_{i}=K_{i}$ for $i=1,\dots,t$.

Consider now a point $P\in C\setminus\Delta_{n-1}$. Again by Theorem 3.5 one
has that the equation defining $P\star D$ is

$\frac{\beta_{1}}{P^{I_{1}}}X^{I_{1}}+\frac{\beta_{2}}{P^{I_{2}}}X^{I_{2}}+\cdots+\frac{\beta_{s}}{P^{I_{s}}}X^{I_{s}}=0$

and since $P\star D=E$ one has

$\frac{\beta_{i}}{P^{I_{i}}}=\lambda\gamma_{i},\mbox{ for }i=1,\dots,s$

for a suitable choice of $\lambda$. Hence the points of $C$ satisfies

$\frac{\gamma_{1}}{\beta_{1}}X^{I_{1}}-\frac{\gamma_{i}}{\beta_{i}}X^{I_{i}}=0,\mbox{
for }i=2,\dots,s$

Thus the ideal

$\langle\frac{\gamma_{1}}{\beta_{1}}X^{I_{1}}-\frac{\gamma_{2}}{\beta_{2}}X^{I_{2}},\dots,\frac{\gamma_{1}}{\beta_{1}}X^{I_{1}}-\frac{\gamma_{s}}{\beta_{s}}X^{I_{s}}\rangle$

is contained in the ideal
$\langle\alpha_{1}X^{I_{1}}+\alpha_{2}X^{I_{2}}+\cdots+\alpha_{s}X^{I_{s}}\rangle$,
but this is possible only if $s=2$ and

(4.2)
$\frac{\gamma_{1}}{\beta_{1}}=\rho\alpha_{1},\,\frac{\gamma_{2}}{\beta_{2}}=-\rho\alpha_{2}.$

It follows, possibly changing the signs, that

$\begin{array}[]{rcl}C&=&Z(\alpha_{1}X^{I_{1}}-\alpha_{2}X^{I_{2}})\\\
D&=&Z(\beta_{1}X^{I_{1}}-\beta_{2}X^{I_{2}})\end{array}$

and, using (4.2) that

$E=Z(\alpha_{1}\beta_{1}X^{I_{1}}-\alpha_{2}\beta_{2}X^{I_{2}}).$

∎

##  5\. Hadamard powers of hypersurfaces

We study now which hypersurfaces are idempotent under Hadamard powers. The
first easy case to study concerns union, eventually with multiplicities, of
coordinate hyperplanes $H_{i}$.

######  Lemma 5.1.

Let $C\subset{\mathbb{P}}^{n}$ be the reducible hypersurfaces
$a_{0}H_{0}+a_{1}H_{1}+\cdots+a_{n}H_{n}$. Then $C^{\star t}=C$ for all $t\geq
1$.

###### Proof.

Since $H_{i}^{\star t}=H_{i}$, for every $t$ and $H_{i}\star H_{j}=H_{i}\cap
H_{j}$ (by part ii of Lemma 4.1), expanding all terms in
$(a_{0}H_{0}+a_{1}H_{1}+\cdots+a_{n}H_{n})^{\star t}$ we get the desired
equality. ∎

######  Definition 5.2.

Let $C\subset{\mathbb{P}}^{n}$ be a binomial hypersurface. We say that $C$ is
of type $(t,\epsilon))$ if

$\alpha_{1}=1\mbox{ and }\alpha_{2}=\xi^{\epsilon}$

where $\xi$ is a primitive $(t-1)-$th root of unity and $1\leq\epsilon\leq
t-1$.

######  Proposition 5.3.

If $C\subset{\mathbb{P}}^{n}$ be a binomial hypersurface of type
$(t,\epsilon)$, then $C^{\star t}=C$. Moreover, if $gcd(t-1,\epsilon)=1$ then
$t$ is the minimal exponent for which the previous equality holds.

###### Proof.

By hypothesis the equation of $C$ is
$F(x_{0},\dots,x_{n})=X^{I_{1}}-\xi^{\epsilon}X^{I_{2}}$. Consider $t-1$
points $P_{i}\in C\setminus\Delta_{n-1}$. Each point satisfies
$P_{i}^{I_{1}}=\xi^{\epsilon}P_{i}^{I_{2}}$ and

(5.1) $\begin{split}(P_{1}\star\cdots\star
P_{t-1})^{I_{1}}&=P_{1}^{I_{1}}\cdots P_{t-1}^{I_{1}}=\\\
&=\xi^{\epsilon}P_{1}^{I_{2}}\cdots\xi^{\epsilon}P_{t-1}^{I_{2}}=\\\
&=(\xi^{\epsilon})^{t-1}P_{1}^{I_{2}}\cdots P_{t-1}^{I_{2}}=\\\
&=P_{1}^{I_{2}}\cdots P_{t-1}^{I_{2}}=\\\ &=(P_{1}\star\cdots\star
P_{t-1})^{I_{2}}\end{split}$

By Theorem 3.5, the equation for $P_{1}\star\cdots\star P_{t-1}\star C$ is

$F^{\star(P_{1}\star\cdots\star
P_{t-1})}(x_{0},\dots,x_{n})=\frac{X^{I_{1}}}{(P_{1}\star\cdots\star
P_{t-1})^{I_{1}}}-\frac{\xi^{\epsilon}X^{I_{2}}}{(P_{1}\star\cdots\star
P_{t-1})^{I_{2}}}$

and, by (5.1), one has that $F^{\star(P_{1}\star\cdots\star
P_{t-1})}(x_{0},\dots,x_{n})$ and $F(x_{0},\dots,x_{n})$ define the same
hypersurface, hence $P_{1}\star\cdots\star P_{t-1}\star C=C$. Since this holds
for every choice of $P_{1},\dots,P_{t-1}\in C\setminus\Delta_{n-2}$ and, by
Remark 4.4, $P_{1}\star\cdots\star P_{t-1}\star C\subset C$ if
$P_{1}\star\cdots\star P_{t-1}\in C\cap\Delta_{n-2}$, one has $C^{\star t}=C$.

Suppose now that $gcd(t-1,\epsilon)=\alpha>1$. Since
$\xi^{\epsilon}=e^{\frac{2\pi\epsilon i}{t-1}}$ and assuming that
$t-1=\alpha(r-1)$ and $\epsilon=\alpha\theta$ one has
$\xi^{\epsilon}=e^{\frac{2\pi\alpha\theta i}{\alpha(r-1)}}=e^{\frac{2\pi\theta
i}{r-1}}$. Hence $\xi^{\epsilon}$ is the power of a primitive $(r-1)-$th root
of unity and, by the first part of the proof, we get $C^{\star r}=C$.

∎

The following result proves that being binomial hypersurface of type
$(t,\epsilon)$ is also a necessary condition.

######  Proposition 5.4.

Let $C\subset{\mathbb{P}}^{n}$ be an irreducible hypersurface not contained in
$\Delta_{n-1}$. If $C^{\star t}=C$ then $C$ is a binomial hypersurface of type
$(t,\epsilon)$. In particular, if $gdc(t-1,\epsilon)=1$ then $t$ is the
minimal exponent for which $C^{\star t}=C$ holds.

###### Proof.

Let $C$ be a hypersurface satisfying $C^{\star t}=C$. Hence, by dimensional
reasons, each successive Hadamard product in

$C\star(C\star(C\star(\cdots(C\star C))))$

is a hypersurface. In particular, by Proposition 4.7, applied to $C\star C$,
one has that $C$ must be binomial of equation
$\alpha_{1}X^{I_{1}}-\alpha_{2}X^{I_{2}}=0$. Then by Corollary 4.7 we get that
the equation of $C^{\star t}$ is
$\alpha_{1}^{t}X^{I_{1}}-\alpha_{2}^{t}X^{I_{2}}=0$.

Finally, the condition $C^{\star t}=C$ implies that
$\alpha_{i}^{t}=\lambda\alpha_{i}$, for $i=1,2$ and a suitable nonzero
$\lambda$. Thus we get

$\frac{\alpha_{2}}{\alpha_{1}}\left[\left(\frac{\alpha_{2}}{\alpha_{1}}\right)^{t-1}-1\right]=0.$

Excluding the trivial solution $\alpha_{2}=0$, we get that
$\frac{\alpha_{2}}{\alpha_{1}}$ is a $(t-1)-$th root of unity, showing that
$C$ is binomial of type $(t,\epsilon)$.

If $gdc(t-1,\epsilon)=1$, then, by Proposition 5.3, we get the minimality of
$t$.

∎

We end this section with an analogue results for reducible hypersurfaces.

######  Proposition 5.5.

Let $C_{1}+\cdots+C_{s}$ be a reducible hypersurface such that
$C_{i}\not\subset\Delta_{n-1}$ for each component $C_{i}$, $i=1,\dots,s$. Then
$(C_{1}+\cdots+C_{s})^{\star r}=C_{1}+\cdots+C_{s}$ if and only if

  * 1.

each component $C_{i}$ is a binomial hypersurfaces of type $(t,\epsilon_{i})$
and equation $X^{I_{1}}-\xi^{\epsilon_{i}}X^{I_{2}}$, for $i=1,\dots,s$;

  * 2.

each product $\xi^{d_{1}\epsilon_{1}}\cdots\xi^{d_{s}\epsilon_{s}}$, with
$d_{1}+d_{2}+\cdots+d_{s}=r$, is equal to $\xi^{\epsilon_{j}}$ for some
$j=1,\dots,s$.

###### Proof.

Assume 1. and 2. holds. Any terms in $(C_{1}+\cdots+C_{s})^{\star r}$ has the
form $C_{1}^{\star d_{1}}\star C_{2}^{\star d_{2}}\star\cdots\star
C_{s}^{\star d_{s}}$, with $d_{1}+d_{2}+\cdots+d_{s}=r$. Condition 2, together
with Proposition 4.5, applied to the binomial hypersurfaces $C_{i}$’s, of type
$(t,\epsilon_{i})$, implies that $C_{1}^{\star d_{1}}\star C_{2}^{\star
d_{2}}\star\cdots\star C_{s}^{\star d_{s}}=C_{j}$ for some $j=1,\dots,s$.

For the other direction, the condition $(C_{1}+\cdots+C_{s})^{\star
r}=C_{1}+\cdots+C_{s}$ implies that $C_{1}^{\star d_{1}}\star C_{2}^{\star
d_{2}}\star\cdots\star C_{s}^{\star d_{s}}$, with
$d_{1}+d_{2}+\cdots+d_{s}=r$, is equal to $C_{j}$ for some $j=0,\dots,s$. In
particular, if we consider $d_{i}=r$ we get $C_{i}^{\star r}=C_{j}$ for some
$j=1,\dots,s$ and for all $i=1,\dots s$. Applying repeatedly Proposition 4.7
(starting from $C_{i}\star C_{i}^{\star r-1}$) it follows that $C_{i}$ is a
binomial hypersurfaces for $i=1,\dots,s$ and, moreover, that all these
components are defined by the same binomial equation, with different
coefficients. Let assume that $C_{i}=Z(X^{I_{1}}-\alpha_{i}X^{I_{2}})$, for
$i=1,\dots,s$. Hence, by the previous consideration we have

(5.2)
$\begin{array}[]{ll}\alpha_{1}^{d_{1}}\cdots\alpha_{1}^{d_{s}}=\alpha_{j}&\mbox{
for some }j=0,\dots s,\\\ &\mbox{ with }d_{1}+\cdots+d_{s}=r\end{array}$

We claim that $|\alpha_{i}|=1$ for $i=1,\dots,s$. To this aim, assume,
eventually reordering the components, that

$|\alpha_{1}|\leq|\alpha_{2}|\leq\cdots\leq|\alpha_{s}|.$

If there exists an index $j$ such that $|\alpha_{j}|>1$, then
$|\alpha_{s}|>1$. Taking the the Hadamard power $C_{s}^{\star r}$ we get, by
Proposition 4.5, a hypersurface defined by the polynomial
$X^{I_{1}}-\alpha_{s}^{r}X^{I_{2}}$. Since $|\alpha_{s}^{r}|>|\alpha_{s}|$,
condition (5.2) is not satisfied and we get a contradiction. Similarly, if
there exists a index $j$ such that $|\alpha_{j}|<1$, then $|\alpha_{1}|<1$.
Taking the the Hadamard power $C_{1}^{\star r}$ we get a hypersurface defined
by the polynomial $X^{I_{1}}-\alpha_{1}^{r}X^{I_{2}}$. Since
$|\alpha_{1}^{r}|<|\alpha_{1}|$, condition (5.2) is not satisfied and, again,
we get a contradiction.

Since $\alpha_{i}$ lies in the unitary circle, we can write
$\alpha_{i}=\eta_{i}^{\rho_{i}}$ where $\eta_{i}$ is a $t_{i}-$th root of
unity, and $1\leq\rho_{i}\leq t_{i}$, for $i=1,\dots,s$.

Let $t-1=lcm(t_{1},t_{2},\dots,t_{s})$, then we can see each $\alpha_{i}$ as a
$(t-1)-$th root of unity and write $\alpha_{i}=\xi^{\epsilon_{i}}$, where
$\xi$ is a primitive $(t-1)$-th root of unity and $1\leq\epsilon_{i}\leq t-1$,
for $i=1,\dots,s$. Hence each component $C_{i}$ is a binomial hypersurfaces,
of type $(t,\epsilon_{i})$, $i=1,\dots,s$, and condition 1. is verified.

Condition 2. now follows directly substituting $\alpha_{i}=\xi^{\epsilon_{i}}$
in condition (5.2). ∎

######  Example 5.6.

Let $\xi$ be the $6-$th root of unity $\frac{2\pi}{6}$ and define the
hypersurfaces

$\mathcal{C}_{j}=Z(h_{j})$

where $h_{j}=X^{I_{1}}-\xi^{j}X^{I_{2}}$, $j=1,\dots 6$, and $X^{I_{1}}$ and
$X^{I_{2}}$ are coprime monomials of degree $d$.

According to Proposition 5.3 we have the following Hadamard multiplication
table

$\begin{array}[]{c|c|c|c|c|c|c|}&\mathcal{C}_{1}&\mathcal{C}_{2}&\mathcal{C}_{3}&\mathcal{C}_{4}&\mathcal{C}_{5}&\mathcal{C}_{6}\\\
\hline\cr\mathcal{C}_{1}&\mathcal{C}_{2}&\mathcal{C}_{3}&\mathcal{C}_{4}&\mathcal{C}_{5}&\mathcal{C}_{6}&\mathcal{C}_{1}\\\
\hline\cr\mathcal{C}_{2}&\mathcal{C}_{3}&\mathcal{C}_{4}&\mathcal{C}_{5}&\mathcal{C}_{6}&\mathcal{C}_{1}&\mathcal{C}_{2}\\\
\hline\cr\mathcal{C}_{3}&\mathcal{C}_{4}&\mathcal{C}_{5}&\mathcal{C}_{6}&\mathcal{C}_{1}&\mathcal{C}_{2}&\mathcal{C}_{3}\\\
\hline\cr\mathcal{C}_{4}&\mathcal{C}_{5}&\mathcal{C}_{6}&\mathcal{C}_{1}&\mathcal{C}_{2}&\mathcal{C}_{3}&\mathcal{C}_{4}\\\
\hline\cr\mathcal{C}_{5}&\mathcal{C}_{6}&\mathcal{C}_{1}&\mathcal{C}_{2}&\mathcal{C}_{3}&\mathcal{C}_{4}&\mathcal{C}_{5}\\\
\hline\cr\mathcal{C}_{6}&\mathcal{C}_{1}&\mathcal{C}_{2}&\mathcal{C}_{3}&\mathcal{C}_{4}&\mathcal{C}_{5}&\mathcal{C}_{6}\\\
\hline\cr\end{array}$

Consider $C=\mathcal{C}_{1}+\mathcal{C}_{3}+\mathcal{C}_{5}$. If we compute
$C^{\star 2}$, using the previous table, we get

$(\mathcal{C}_{1}+\mathcal{C}_{3}+\mathcal{C}_{5})^{\star
2}=\mathcal{C}_{2}+\mathcal{C}_{4}+\mathcal{C}_{6}.$

Then, computing $(\mathcal{C}_{1}+\mathcal{C}_{3}+\mathcal{C}_{5})^{\star 3}$
we get

$\begin{split}(\mathcal{C}_{1}+\mathcal{C}_{3}+\mathcal{C}_{5})^{\star
3}&=(\mathcal{C}_{2}+\mathcal{C}_{4}+\mathcal{C}_{6})\star(\mathcal{C}_{1}+\mathcal{C}_{3}+\mathcal{C}_{5})=\\\
&=\mathcal{C}_{1}+\mathcal{C}_{3}+\mathcal{C}_{5}.\end{split}$

Notice that Conditions 1. and 2. are satisfied for
$\mathcal{C}_{1},\mathcal{C}_{3},\mathcal{C}_{5}$ and $r=3$.

##  6\. On Hadamard powers of varieties

The result presentend in this section, generalizes, to the case of higher
powers, the sufficient condition to have $V^{\star 2}=2$, in [9] and stated in
Proposition 1.1 in the introduction of this paper.

Similarly to the case of hypersurfaces, we define a specific class of
varieties.

######  Definition 6.1.

Let $C\subset{\mathbb{P}}^{n}$ be a binomial variety. We say that $C$ is of
type $[(t_{1},\epsilon_{1}),\dots,(t_{s},\epsilon_{s})]$ if the ideal of $C$
is generated by

$X^{I_{1,1}}-\xi_{1}^{\epsilon_{1}}X^{I_{1,2}},\dots,X^{I_{s,1}}-\xi_{s}^{\epsilon_{s}}X^{I_{s,2}},$

where $\xi_{i}$ is a primitive $(t_{i}-1)-$th root of unity and
$1\leq\epsilon_{i}\leq t_{i}-1$ for $i=1,\dots,s$.

######  Theorem 6.2.

Let $C\subset{\mathbb{P}}^{n}$ be a binomial variety of type
$[(t_{1},\epsilon_{1}),\dots,(t_{s},\epsilon_{s})]$ with

(6.1)
$lcm\left(\frac{t_{1}}{gcd(t_{1},\epsilon_{1})},\frac{t_{2}}{gcd(t_{2},\epsilon_{2})}\dots,\frac{t_{s}}{gcd(t_{s},\epsilon_{s})}\right)=t-1$

Then $C^{\star t}=C$ and $t$ is the minimal exponent for which the previous
equality holds.

###### Proof.

By hypothesis the ideal of $C$ is generated by
$F_{j}(x_{0},x_{1},x_{2})=X^{I_{j,1}}-\xi_{j}^{\epsilon_{j}}X^{I_{j,2}}$ for
$j=1,\dots,s$. Consider $t-1$ points $P_{i}\in C\setminus\Delta_{n-1}$. Each
point satisfies $P_{i}^{I_{j,1}}=\xi_{j}^{\epsilon_{j}}P_{i}^{I_{j,2}}$ and

(6.2) $\begin{split}(P_{1}\star\cdots\star
P_{t-1})^{I_{j,1}}&=P_{1}^{I_{j,1}}\cdots P_{t-1}^{I_{j,1}}=\\\
&=(\xi_{j}^{\epsilon_{j}})^{t-1}P_{1}^{I_{j,2}}\cdots P_{t-1}^{I_{j,2}}=\\\
&=P_{1}^{I_{j,2}}\cdots P_{t-1}^{I_{j,2}}=\\\ &=(P_{1}\star\cdots\star
P_{t-1})^{I_{j,2}}\end{split}$

where the third equality follows from the fact that $t-1$ is the least
integers such that $1=(\xi_{j}^{\epsilon_{j}})^{t-1}$, for all $j=1,\dots s$
since (6.1) holds.

By Theorem 3.5, the equations for $P_{1}\star\cdots\star P_{t-1}\star C$ are

$F_{j}^{\star(P_{1}\star\cdots\star
P_{t-1})}(x_{0},\dots,x_{n})=\frac{X^{I_{j,1}}}{(P_{1}\star\cdots\star
P_{t-1})^{I_{j,1}}}-\frac{\xi_{j}^{\epsilon_{j}}X^{I_{j,2}}}{(P_{1}\star\cdots\star
P_{t-1})^{I_{j,2}}}$

and, by (6.2), one has that $F_{j}^{\star(P_{1}\star\cdots\star
P_{t-1})}(x_{0},\dots,x_{n})$ and $F_{j}(x_{0},\dots,x_{n})$ are equal up to a
multiplicative constant, hence $P_{1}\star\cdots\star P_{t-1}\star C=C$. Since
this holds for every choice of $P_{1},\dots,P_{t-1}\in C\setminus\Delta_{n-2}$
and, by Remark 4.4, $P_{1}\star\cdots\star P_{t-1}\star C\subset C$ if
$P_{1}\star\cdots\star P_{t-1}\in C\cap\Delta_{n-2}$, one has $C^{\star t}=C$.
∎

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