Real Ghosts of Complex Hadamard Products

Abstract

For all integers $n\ge 1$ and $k\ge 2$, the Hadamard product $v_1★\cdots ★v_k$ of k elements of $K^{n+1}$ (with K being the complex numbers or real numbers) is the element $v\in K^{n+1}$ which is the coordinate-wise product of $v_1,\dots ,v_k$ (introduced by Cueto, Morton, and Sturmfels for a model in Algebraic Statistics). This product induces a rational map $h:{\mathbb{P}}^n{\left(K\right)}^k⤏{\mathbb{P}}^n\left(K\right)$. When $K=\mathbb{C}$, $k=2$ and $X_i\left(\mathbb{C}\right)\subset {\mathbb{P}}^n\left(\mathbb{C}\right)$, $i=1,2$ are irreducible, we prove four theorems for the case $dim{X}_2\left(\mathbb{C}\right)=1$, three of them with $X_2\left(\mathbb{C}\right)$ as a line. We discuss the existence (non-existence) of a cancellation law for ★-products and use the symmetry group of the Hadamard product. In the second part, we work over $\mathbb{R}$. Under mild assumptions, we prove that by knowing $X_1\left(\mathbb{R}\right)★\cdots ★X_k\left(\mathbb{R}\right)$, we know $X_1\left(\mathbb{C}\right)★\cdots ★X_k\left(\mathbb{C}\right)$. The opposite, i.e., taking and multiplying a set of complex entries that are invariant for the complex conjugation and then seeing what appears in the screen ${\mathbb{P}}^n\left(\mathbb{R}\right)$, very often provides real ghosts, i.e., images that do not come from a point of $X_1\left(\mathbb{R}\right)\times \cdots \times X_k\left(\mathbb{R}\right)$. We discuss a case in which we certify the existence of real ghosts as well as a few cases in which we certify the non-existence of these ghosts, and ask several open questions. We also provide a scenario in which ghosts are not a problem, where the Hadamard data are used to test whether the images cover the full screen.

1. Introduction

Suppose there are $k\ge 2$ channels, each providing packets of $(n+1)$-ples of real (or complex) numbers. Then, for each packet, some device makes the entry-wise multiplication of the k received packets. The screen sees a sequence of packets formed by $(n+1)$-ples of numbers (the results of the multiplications made by the device). In [1], the authors explained why this occurs in real-life applications and explained some of the key mathematical tools used to handle it. They introduced the name of Hadamard products of element of an $(n+1)$-dimensional (real or complex) vector space or of elements of n-dimensional projective spaces. Real-life applications can be found in Algebraic Statistics in the form of statistical models associated with the graphical models called Restricted Boltzmann Machines [1,2]. Recently, C. Bocci and E. Carlini published a monograph on Hadamard products [3], which interested readers may refer to along with the references therein for applications of Restricted Boltzmann Machines in machine learning. Among these references, we use [4,5,6,7,8,9] in the present paper. Of course, Hadamard products of matrices are much older, but these products do not occur in this paper.

This paper has two parts, which are connected by the notion of Hadamard products of elements of a projective space. The title only reflects the second part. The first part, Section 2 and Section 3, consists of “abstract Algebraic Geometry” over an algebraically closed field of characteristic 0; the interested reader will recognize several key features used in the applications over the real number field $\mathbb{R}$. In this part, we use the group of symmetries of the Hadamard product and show that it is a strong tool. In the second part, Section 4, we work over $\mathbb{R}$ with $\mathbb{C}$ as its algebraic closure. Supposing that the “picture” over $\mathbb{C}$ is known (for instance, by Algebraic Geometry) and that all of the packets come from k sets (or varieties) that are invariant for the action of the complex conjugation, we want to know whether it is possible to reconstruct the data from the real numbers among the complex numbers on the screen. We provide several counterexamples and explain some mitigation strategies.

Taking a field K and calling $x_0,\dots ,x_n$ the variables of the vector space $K^{n+1}$ and the homogeneous variables of the associated projective space ${\mathbb{P}}^n\left(K\right)$, the Hadamard product $★:K^{n+1}\times K^{n+1}\to K^{n+1}$ is the coordinate-wise product, i.e.,

$$(a_0,\dots ,a_n)★(b_0,\dots ,b_n):=(a_0{b}_0,\dots ,a_n{b}_n).$$

This map strongly depends on the choice of the coordinates. The ★-product induces a rational map $h:{\mathbb{P}}^n\times {\mathbb{P}}^n⤏{\mathbb{P}}^n$ the indeterminacy locus ${\mathcal{J}}_n$ of which is the set of all $([a_0:\cdots :a_n],[b_0:\cdots :b_n])\in {\mathbb{P}}^n\times {\mathbb{P}}^n$ such that $a_i{b}_i=0$ for all i. The set ${\mathcal{J}}_n$ is Zariski closed in ${\mathbb{P}}^n\times {\mathbb{P}}^n$. Taking irreducible algebraic varieties $X,Y\subset {\mathbb{P}}^n$ such that $X\times Y⊈{\mathcal{J}}_n$, the rational map $h_{X,Y}:X\times Y⤏{\mathbb{P}}^n$ that restricts h to $X\times Y$ is a morphism on its Zariski open subset $X\times Y\setminus (X\times Y)\cap {\mathcal{J}}_n$; hence, over any field K, we see the set $h_{X,Y}(X\left(K\right)\times Y\left(K\right)\setminus (X\left(K\right)\times Y\left(K\right))\cap {\mathcal{J}}_n))$ with our screen, which receives and sees only numbers in K. In Algebraic Geometry (even in parts with a strong applied flavor, as in [3]), when we are interested in, say, $K=\mathbb{R}$, we must first check the algebraic closure $\overline{K}$ and consider ★ over ${\mathbb{P}}^n\left(\overline{K}\right)\times {\mathbb{P}}^n\left(\overline{K}\right)$, where we call $X★Y$ the closure of $h_{X,Y}(X\left(\overline{K}\right)\times Y\left(\overline{K}\right)\setminus (X\left(\overline{K}\right)\times Y\left(\overline{K}\right)\cap {\mathcal{J}}_n))$ ([3], Def. 1.3). With this definition, $X★Y$ is an irreducible projective variety defined over K, and as such is defined by finitely many polynomial equations with coefficients in K. Knowing an upper bound for the degrees of these polynomials and knowing a large number of points of $X★Y\left(\overline{K}\right)$, we obtain polynomial equations defining $X★Y$ over $\overline{K}$ (and, if the points are in ${\mathbb{P}}^n\left(K\right)$, over K). Thus, the set $(X★Y)\left(K\right)$ of K points of the irreducible variety is well defined. It contains the set $h_{X,Y}(X\left(K\right)\times Y\left(K\right)\setminus (X\left(K\right)\times Y\left(K\right)\cap {\mathcal{J}}_n))$, although it is often far larger. The “ghosts” in the title arise in the case of $K=\mathbb{R}$ with $\overline{K}=\mathbb{C}$; obviously, $X★Y=Y★X$.

In this paper, we use the following notation (as in [3] and most references): for $i=0,\dots ,n$ we set $H_i:=\{x_i=0\}\subset {\mathbb{P}}^n$, ${\Delta }_{n-1}:={\cup }_{i=0}^n{H}_i$ and $U_n:={\mathbb{P}}^n\setminus {\Delta }_{n-1}$. A coordinate linear space is the intersection of finitely many coordinate hyperplanes $H_0,\dots ,H_n$. Because the coordinate system is fixed, our group of symmetries is not the linear group $GL(n+1,K)$ or its quotient $PGL(n+1,K)$ by the multiples $K^{*}{\mathrm{Id}}_{(n+1)\times (n+1)}$ of the identity matrix, but rather the diagonal group $D(n+1,K)$ (which is isomorphic as an algebraic group to the product of $n+1$ copies of the multiplicative group) or its quotient $PD(n+1,K)$ by its subgroup $K^{*}{\mathrm{Id}}_{(n+1)\times (n+1)}$. Recall that for all $P=[a_0:\cdots :a_n]\in U_n$, by definition we have $P^{-1}=[a_0^{-1}:\cdots :a_n^{-1}]$ ([3], §1.2). The group $PD(n+1,K)$ is the set of all transformations $P^{-1}$ for some $P\in U_n\left(K\right)$.

For any $W\subset {\mathbb{P}}^n$, let $\mathrm{DiagAut}\left(W\right)$ denote the set of all $f\in PD(n+1)$ such that $f\left(W\right)=W$. If $W=\left\{P\right\}$, then the connected group $\mathrm{DiagAut}\left(\right\{P\left\}\right)$ counts the number of zero-coordinates of P. In particular, $\mathrm{DiagAut}\left(\left\{P\right\}\right)=\left\{{\mathrm{Id}}_{{\mathbb{P}}^n}\right\}$ if and only if $P\in U_n$.

In the following Theorems 1–4, we assume that the base field is algebraically closed and of characteristic 0. Here, we only use the assumption of characteristic 0 in order to freely quote [3] in their proofs.

Theorem 1.

Let $W\subset {\mathbb{P}}^n$ be an integral projective variety not contained in a coordinate hyperplane. Set $m:=dimW$, $d:=deg\left(W\right)$, and $W=X★Y$ with $dimY=1$ and $dimX=m-1$. Set $a:=deg\left(X\right)$ and assume $dim\mathrm{DiagAut}\left(W\right)=0$. Then, $a\le d^2$.

To explain the importance of Theorem 1, we start with 5 positive integers n, x, y, a, and b such that $x<n$ and $y<n$. From the existence of the Chow variety of ${\mathbb{P}}^n$, there is a (not connected in general) quasi-projective family T of pairs $(X,Y)$ such that X is a closed and irreducible subvariety of dimension x and degree a and that Y is a closed and irreducible subvariety of dimension y and degree b. Hence, T also parameterizes (not one-to-one) all possible varieties $X★Y$. Now, assume $dimX★Y=dimX+dimY<n$ and set $d:=deg(X★Y)$. It is known that $d\le \left({\displaystyle \genfrac{}{}{0pt}{}{x+y}{x}}\right)ab$ (Remark 7); however, a priori it could be very low, even 1. If $y=1$, then Theorem 1 states that this is not the case under very mild assumptions, as $d\ge \sqrt{a}$.

Easy examples show that in general we may have $X★Y=X_1★Y_1$ with neither $X=X_1$ and $Y=Y_1$ nor $X=Y_1$ and $Y=X_1$; e.g., in the case with $n=2$, to obtain $X★Y={\mathbb{P}}^2$ it is sufficient that neither X nor Y be a coordinate line. Hence, reconstructing X and Y from $X★Y$ seems to be hopeless, except perhaps if we restrict the class of possible X and Y. We ask the following question.

Question 1: Assume $dimX★Y=dimX+dimY<n$, $dimY=1$, and that $X★Y=X★L$ for some line L. Is Y a line?

On the ★-decomposition of surfaces, we prove the following three theorems.

Theorem 2.

Let $W\subset {\mathbb{P}}^n$, $n\ge 4$ be an integral surface such that $W=X★Y$ for some curves X and Y. Assume $W\cap V=\varnothing$ for all codimension 3 coordinate linear spaces V. Then:

(a) The rational map $h_{X,Y}:X\times Y⤏X★Y$ is a morphism.

(b) $V_1\cap X=V_1\cap Y=\varnothing$ for all codimension 2 coordinate linear spaces $V_1$, i.e., for $i=0,\dots ,n$, each element of $[a_0:\cdots :a_n]\in (X\cup Y)\cap H_i$ satisfies $a_j\ne 0$ for all $j\ne i$.

Theorem 3.

Let $W=X★Y\subset {\mathbb{P}}^n$, $n\ge 4$ be a plane such that X and Y are integral curves $W\cap V=\varnothing$ for all codimension 3 coordinate linear spaces V. Then, X and Y are lines and $X\cap V_1=Y\cap V_1=\varnothing$ for all codimension 2 coordinate linear spaces $V_1$. Moreover, Y is uniquely determined by the pair $(W,X)$.

Conversely, take a line $L\subset {\mathbb{P}}^n$, $n\ge 3$ such that $L\cap V_1=\varnothing$ for all codimension 2 coordinate linear spaces $V_1$. Then, $L★L$ is a plane and $V\cap (L★L)=\varnothing$ for all codimension 3 coordinate linear subspaces V.

The following result is an example in which we have a “cancellation law” for the ★-product.

Theorem 4.

Take lines $L,R\subset {\mathbb{P}}^n$, $n\ge 3$ such that $L★L=L★R$ and $L\cap V=\varnothing$ for all codimension 2 coordinate linear spaces V. Then, $L=R$.

We think that cancellation results are not frequent; see Example 3 for a non-uniqueness related to Theorems 3 and 4. This example is a byproduct of the study of symmetries of the ★-product. See the end of Section 3 for the algebraically closed case (Remark 18) and Section 4 (Remark 19) for the case $K=\mathbb{R}$ with the Abelian group $PD(n+1,\mathbb{R})$.

In the rest of this introduction the main field that we use is the field $\mathbb{R}$ of real numbers; however, it is important to also use its algebraic closure $\mathbb{C}$. Let $\sigma :\mathbb{C}\to \mathbb{C}$ denote the complex conjugation; in addition, we call $\sigma$ the complex conjugation on ${\mathbb{C}}^{n+1}$, ${\mathbb{P}}^n\left(\mathbb{C}\right)$, and the subsets of ${\mathbb{P}}^n\left(\mathbb{C}\right)$. If $X\subset {\mathbb{P}}^n$ is an algebraic projective variety defined over $\mathbb{R}$, then $X\left(\mathbb{R}\right)=X\left(\mathbb{C}\right)\cap {\mathbb{P}}^n\left(\mathbb{R}\right)=\{a\in X\left(\mathbb{C}\right)\mid \sigma \left(a\right)=a\}$. We perform the construction for $k\ge 2$ screens. Fix an integer $k\ge 2$, and consider the map ${\left({\mathbb{C}}^{n+1}\right)}^k\to {\mathbb{C}}^{n+1}$ defined taking the entry-wise product of the k vectors. This induces a rational map $h:({\mathbb{P}}^n{\left(\mathbb{C}\right)}^k⤏{\mathbb{P}}^n\left(\mathbb{C}\right)$ in the following way. Let ${\mathcal{J}}_{n,k}$ denote the set of all $([p{\left(1\right)}_0:\cdots :p_n\left(1\right)],\dots ,[p_0\left(k\right):\cdots :p_n\left(k\right)])\in {\mathbb{P}}^n{\left(\mathbb{C}\right)}^k$ such that ${\prod }_{j=1}^k{p}_i\left(j\right))=0$ for all $i=0,\dots ,k$. The closed algebraic set ${\mathcal{J}}_{n,k}$ is the indeterminacy locus of h, and for all $([p_0\left(1\right):\cdots :p_n\left(1\right)],\dots ,[p_0\left(k\right):\cdots :p_n\left(k\right)])\in ({\mathbb{P}}^n{\left(\mathbb{C}\right)}^k\setminus {\mathcal{J}}_{n,k})$ we call $([p{\left(1\right)}_0:\cdots :p_n\left(1\right)]★\cdots ★[p_0\left(k\right):\cdots :p_n\left(k\right)]$ point $[{\prod }_{j=1}^k{p}_0\left(j\right):\cdots :{\prod }_{j=1}^k{p}_n\left(j\right)]$. This ★ product is invariant for the complex conjugations, and the ★ product of k real points (not in ${\mathcal{J}}_{n,k}$) is an element of ${\mathbb{P}}^n\left(\mathbb{R}\right)$. If $X_i\left(\mathbb{C}\right)\subseteq {\mathbb{P}}^n\left(\mathbb{C}\right)$ are irreducible subvarieties and $X_1\left(\mathbb{C}\right)\times \cdots \times X_k\left(\mathbb{C}\right)⊈{\mathcal{J}}_{n,k}$, by restricting h and then taking the Zariski closure we obtain the ★-products $X_1★\cdots ★X_k$ of k varieties $X_i\subset {\mathbb{P}}^n$, $1\le i\le k$. As the multiplication map of a field, the ★-products obey the commutative and associative laws. Here, it is important to distinguish the results we see on our screen, i.e., the set of all $P_1★\cdots ★P_k$, $P_i\in X_i\left(K\right)$ with K either $\mathbb{R}$ or $\mathbb{C}$, from its closure in the Zariski topology (which by definition is $X_1\left(\mathbb{C}\right)★\cdots ★X_k\left(\mathbb{C}\right)$ if $K=\mathbb{C}$) or Euclidean topology. The two closures coincide over $\mathbb{C}$, but do not coincide not if we work over $\mathbb{R}$. For any field K and any $X_i\subset {\mathbb{P}}^n$, $1\le i\le k$, let $X_1\left(K\right){★}^{\circ }\cdots {★}^{\circ }{X}_k\left(K\right)$ denote the set of all $P_1★\cdots ★P_k$ with $P_i\in X_i\left(K\right)$ for all i and $(P_1,\dots ,P_k)\notin {\mathcal{J}}_{n,k}$. We have $X_1\left(K\right){★}^{\circ }\cdots {★}^{\circ }{X}_k\left(K\right)\subseteq {\mathbb{P}}^n\left(K\right)$. If $K=\mathbb{C}$ (or is just algebraically closed), then $X_1\left(\mathbb{C}\right){★}^{\circ }\cdots {★}^{\circ }{X}_k\left(\mathbb{C}\right)$ is constructable, i.e., a finite union of algebraic varieties (Remark 5, i.e. ([10], Ex. II.3.18, Ex. II.3.19). If $K=\mathbb{R}$, then $X_1\left(\mathbb{R}\right){★}^{\circ }\cdots {★}^{\circ }{X}_k\left(\mathbb{R}\right)$ is semialgebraic and its closure in the Euclidean topology is semialgebraic ([11], Prop. 2.2.2 and 2.2.7). Under very natural conditions the real picture is sufficient to describe the complex one (Theorem 5).

A key problem is that our real screen ${\mathbb{P}}^n\left(\mathbb{R}\right)$ may see something much larger than $X_1\left(\mathbb{R}\right){★}^{\circ }\cdots {★}^{\circ }{X}_k\left(\mathbb{R}\right)$ (Examples 8 and 9). We call these real ghosts. We provide some examples in which they arise, and one (Example 9) in which it is certified that they must occur.

We write a subsection (full screen) explaining a different problem. The Hadamard data is a test for determining whether the images cover the full screen. We prove that for this problem ghosts are not an issue. We conclude the section with four mitigating suggestions.

In the Conclusions section, we add a question related to the ones discussed in Section 4. Section 4 explains why the question would certify the nonexistence of real ghosts in that particular setup.

Many thanks are due to the referees for their useful suggestions.

2. The Algebro-Geometric Part, I

In this section and the next one, our base field K is algebraically closed and of characteristic zero, the latter assumption being used only in order to freely quote [3].

We write $GL(n+1)$, $PGL(n+1)$, $D(n+1)$, and $PD(n+1)$ instead of $GL(n+1,K)$, $PGL(n+1,K)$, $D(n+1,K)$, and $PD(n+1,K)$.

For each $i=0,\dots ,n$, let $e_i=[a_0:\cdots :a_n]$ denote the coordinate point with $a_i=1$ and $a_j=0$ for all $j\ne i$. For each integer $x\in \{0,\dots ,n-1\}$, let ${\Delta }_x\subset {\mathbb{P}}^n$ denote the union of all x-dimensional coordinate linear subspaces. We have ${\Delta }_0=\{e_0,\dots ,e_n\}$.

Remark 1.

Take $P,Q\in U_n$. There is a unique $O\in U_n$ such that $P★O=Q$. With the notation from ([3], §1.2), we have $O=Q★P^{-1}$.

Remark 2.

Fix $i,j\in \{0,\dots ,n\}$ such that $i\ne j$ and take $O_i\in H_i\setminus H_i\cap {\Delta }_{n-2}$ and $O_j\in H_j\setminus H_j\cap {\Delta }_{n-2}$. For any $P=[p_0:\cdots :p_n]\in {\mathbb{P}}^n$, if we know $O_i$ and $O_i★P$, then we know $p_x$ for all $x\ne i$. If we also know $O_j$ and $O_j★P$, then we know P.

Remark 3.

Because $D(n+1)\cong {\left(K^{*}\right)}^{n+1}$ as an algebraic group, the algebraic group $PD(n+1)=D(n+1)/K^{*}{\mathrm{Id}}_{n+1,n+1}$ is isomorphic to ${\left(K^{*}\right)}^n$. Hence, all its connected algebraic subgroups are isomorphic (as algebraic groups) to a product of the multiplicative group.

Remark 4.

Take a connected one-dimensional algebraic subgroup $G\subset PD(n+1)$. Here, G is isomorphic to the multiplicative group. Hence, each orbit of G is either one point or an affine rational curve.

Remark 5.

Take irreducible varieties $X\subset {\mathbb{P}}^n$ and $Y\subset {\mathbb{P}}^n$ such that $X\times Y⊈{\mathcal{J}}_n$. This condition is satisfied if $X⊈{\Delta }_{n-1}$. Note that $X\times Y\setminus (X\times Y\cap {\mathcal{J}}_n)$ is a non-empty Zariski open subset of $X\times Y$. Because our base field is algebraically closed, the set $h(X\times Y\setminus (X\cap Y\cap {\mathcal{J}}_n))$ is a constructable subset of ${\mathbb{P}}^n$, i.e., it is a finite union of locally closed subsets of ${\mathbb{P}}^n$ for the Zariski topology ([10], Ex. II.3.18, Ex. II.3.19). Thus, its closure $X★Y$ in ${\mathbb{P}}^n$ is a projective variety of dimension $\le dimX+dimY$. Because $X\times Y$ is irreducible, $X★Y$ is irreducible.

Remark 6.

Take integral projective varieties $X\subset {\mathbb{P}}^n$, $Y\subset {\mathbb{P}}^n$ such that $X\times Y\cap {\mathcal{J}}_n=\varnothing$, i.e., such that the rational map $h_{X,Y}:X\times Y⤏X★Y$ is a morphism. Because $X\times Y$ is a projective variety, $h_{X,Y}(X\times Y)$ is closed ([10], Th. II.4.9), i.e., in the definition of $X★Y$ we do not need to take the closure.

Remark 7.

Let $X\subset {\mathbb{P}}^n$ and $Y\subset {\mathbb{P}}^n$ be integral and closed subvarieties. Set $x:=dimX$, $y:=dimY$, $a:=deg\left(X\right)$, and $b:=deg\left(B\right)$. Let $j:X\times Y\hookrightarrow {\mathbb{P}}^n\times {\mathbb{P}}^n$ denote the associated embedding and $\nu :{\mathbb{P}}^n\times {\mathbb{P}}^n\to {\mathbb{P}}^{n^2+2n}$ the Segre embedding. The set $\nu \left(j\right(X\times Y\left)\right)$ is an integral projective subvariety of dimension $x+y$ and degree $\left({\displaystyle \genfrac{}{}{0pt}{}{x+y}{x}}\right)ab$ ([3], Cor. 5.1).

Remark 8.

Take an integral surface $W\subset {\mathbb{P}}^n$, $n\ge 3$, $W⊈{\Delta }_{n-1}$ such that $W=X★Y$ for some irreducible curves X and Y. If $h_{X,Y}$ is a morphism, then $deg\left(W\right)$ divides $2deg\left(X\right)deg\left(Y\right)$ (Remark 7).

Lemma 1.

Let $W\subset {\mathbb{P}}^n$, $n\ge 3$ be an integral surface.

1 

If $\mathrm{DiagAut}\left(W\right)=1$, then W is a rational fibration.

2 

If $dim\mathrm{DiagAut}\left(W\right)>1$, then W is a toric surface.

Proof. 

Remark 4 provides the first assertion. Now, assume $dim\mathrm{DiagAut}\left(W\right)>1$. Because $PD(n+1)$ is Abelian, we find that W is a rational surface and that ${\left({\mathbb{C}}^{*}\right)}^2$ acts on it with an open orbit. Hence, W is a (possibly singular) toric surface with the embedding $W\subset {\mathbb{P}}^n$ being toric. □

Example 1.

Fix integral curves $C\subset {\mathbb{P}}^3$ and $D\subset {\mathbb{P}}^3$. Fix any $P\in C$ and $Q\in D$. Let $g,g^{\prime }$ be any element of $PGL\left(4\right)$ such that $g\left(P\right)=[0:0:1:1]$ and $g\prime \left(Q\right)=[1:1:0:0]$. Set $X:=g\left(C\right)$ and $Y:=g^{\prime }\left(D\right)$. Because $[0:0:1:1]★[1:1:0:0]$ is not defined, $h_{X,Y}$ is not a morphism.

★-Powers

In this subsection, we consider the ★-powers $X^{★i}$ of an integral variety $X\subset {\mathbb{P}}^n$. We always assume $X⊈{\Delta }_{n-1}$. This is not very restrictive, as X is irreducible and we only consider ★-power of X. Hence, if X is contained in the coordinate hyperplanes $H_{i_1}\cap \cdots \cap H_{i_s}$, $0\le i_1<\cdots <i_s\le n$, then it is sufficient to consider the variety as a subvariety of the $(n-s)$-dimensional projective space $H_{i_1}\cap \cdots \cap H_{i_s}$. Set $m:=dimX$. For each $i\ge 1$ set, $m_i:=dim{X}^{★i}$; hence, $m_1=m$. We have $m_i\le min\{im,n\}$ for all $i\ge 1$. For each $i\ge 2$, set $a_i:=m_i-m_{i-1}$.

Remark 9.

Take an irreducible variety $X\subset {\mathbb{P}}^n$ such that $X⊈{\Delta }_{n-1}$ and take any $P\in X\cap U_n$ and any variety $Y\subseteq {\mathbb{P}}^n$. Then, $Y★P$ is $PD(n+1)$-equivalent to Y. Hence, we have $X^{★i}⊈{\Delta }_{n-1}$ for all $i\ge 1$ and $a_i\ge 0$ for all $i\ge 2$.

The following two results (the first being the particular case $Y=\varnothing$ of the second) are proved as in ([12], part (1) of Prop. 2.1).

Proposition 1.

Let $X\subset {\mathbb{P}}^n$ be an integral variety. Assume $X⊈{\Delta }_{n-1}$ and set $a_i:=a_i\left(X\right)$. We have $a_{i+1}\le a_i$ for all $i\ge 2$.

Proof. 

Fix a general $(P_1,\dots ,P_{i+1})\in X^{i+1}$. Because $P_i$ is general, it is a smooth point of $X\cap U_n$. For any $E⊊\{1,\dots ,i+1\}$ and any $h\in \{1,\dots ,i+1\}\setminus E$, consider the ★-product $B_E$ of all $P_j$, $j\in E$. Set $E_{A,h}:=B_E★T_{P_h}X$. The set $U_{E,i}$ is an m-dimensional vector space containing the point $B_E★P_h$. By the Terracini Lemma for ★ products ([3], Lemma 1.6), $U_{E,h}$ is the tangent space of $B_E★X$ at $B_A★P_h$. Take A, $A^{\prime }$, and $A″$ with $\#A=i-1$, $\#A^{\prime }=i$, $\#A″=i+1$, and $A\subset A^{\prime }\subset A″$. We first take $E=A$ and $\left\{h\right\}:=A^{\prime }\setminus A$. We have $m_{i-1}+a_i=dim{E}_{A,h}$. Taking $E=A^{\prime }$ and $\left\{h\right\}:=A″\setminus A\prime$, we obtain $m_i+a_{i+1}=dim{E}_{A^{\prime },h}$. Thus, $E_{A^{\prime },h}$ is the linear span of three linear spaces containing the point $B_{A″}$, 1, L of dimension $m_{i-1}$ and 2 of dimension m with a certain order, say, $L^{\prime }$ and $L^{″}$. Because $L″\cap ⟨L\cup L^{\prime }⟩\supseteq L″\cap L$, we have $a_{i+1}\le a_i$. □

Proposition 2.

Fix integral varieties X, Y of ${\mathbb{P}}^n$ such that $X⊈{\Delta }_{n-1}$ and $Y⊈{\Delta }_{n-1}$. For all integers $i\ge 1$, set $c_i:=dim{X}^{★i}★Y$. Then, $c_{i+1}\le c_i$.

Proof. 

Fix a general $(Q,P_1,\dots ,P_{i+1})\in Y\times X^{i+1}$. Because Q is general, Y is smooth at Q. For any $E⊊\{1,\dots ,i+1\}$ and any $h\in \{1,\dots ,i+1\}\setminus E$, consider the ★-product $B_E$ of Q and all $P_j$, $j\in E$. Set $E_{A,h}:=T_{Q}Y★B_E★T_{P_h}X$, then repeat the proof of Proposition 1. □

If X is a binomial hypersurface of type $(t,ϵ)$, then $X^{★t}=X$ ([3], Prop. 6.4). Hence, there are irreducible varieties $X⊈{\Delta }_{n-1}$ such that $X^{★i}\ne {\mathbb{P}}^n$ for all $i>0$.

Definition 1.

Take an irreducible variety $X\subset {\mathbb{P}}^n$ such that $X⊈{\Delta }_{n-1}$. A point $O\in {\mathbb{P}}^n$ is said to be a strict ★-vertex of X if $O\in T_{P}X★Q$ for a general $(P,Q)\in (X_{\mathrm{reg}},X_{\mathrm{reg}})$. The ★-vertex $★\mathrm{Vert}\left(X\right)$ of X is the set of all its strict ★-vertices.

Remark 10.

A point O is a strict ★-vertex of X if and only if $O\in T_{P}X★Q$ for all $(P,Q)\in (X_{\mathrm{reg}},X_{\mathrm{reg}})$. The ★-vertex $★\mathrm{Vert}\left(X\right)$ of X is a linear subspace, as it is an intersection of linear subspaces $T_{P}X★Q$. Moreover, if X is not a linear space, then $dim★\mathrm{Vert}\left(X\right)<dimX$.

The following example taken from [3] shows the existence of $X⊈{\Delta }_{n-1}$ with $★\mathrm{Vert}\left(X\right)\ne \varnothing$.

Example 2.

Fix $i,j\in \{0,\dots ,n\}$ such that $i<j$. Let V be the binomial hyperplane $x_i-x_j=0$. We have $V★V=V$ and $V=T_{P}V$ for all $P\in V$ ([3], Cor. 2.2)). Thus, $★\mathrm{Vert}\left(V\right)=V$.

Remark 11.

Let $X\subset {\mathbb{P}}^n$ be an irreducible variety such that $★\mathrm{Vert}\left(X\right)\ne \varnothing$. The Terracini Lemma ([3], Lemma 1.6) provides $dim{X}^{★2}\le 2dimX-dim★\mathrm{Vert}\left(X\right)-1$.

Remark 12.

Let $V⊊{\mathbb{P}}^n$, $V\ne \varnothing$ be an irreducible variety such that $V⊈{\Delta }_{n-1}$. Let $I\left[V\right]\subseteq K[x_0,\dots ,x_n]$ denote the ideal generated by all homogeneous polynomials vanishing on V. Because V is irreducible, the ideal $I\left[V\right]$ is contained in the inessential ideal $(x_0,\dots ,x_n)$ and is saturated. The homogeneous ideal is prime ([13], Th. 8.5). Because $V⊈{\Delta }_{n-1}$ and V is irreducible, $V⊈H_i$ for any i. Because $I\left[V\right]$ is a prime ideal, $I\left[V\right]:\left(x_i\right)=I\left[V\right]$ for all i. Hence, the assumption $I\left[V\right]:(x_0\cdots x_n)=I\left[V\right]$ of ([4], Th. 5.3) is satisfied by the variety V.

Proposition 3.

Take an irreducible $X\subset {\mathbb{P}}^n$. If $i\ge 1$, $X^{★i}⊊{\mathbb{P}}^n$, and X is not contained in a binomial hypersurface, then $a_{i+1}>0$.

Proof. 

As we have assumed that X is not contained in a binomial hypersurface, $X⊈{\Delta }_{n-1}$. Remark 9 provides $a_{i+1}\ge 0$. Assume $a_{i+1}=0$ and fix a general $(P,Q)\in X\times X$. Because $X⊈{\Delta }_{n-1}$ and $(P,Q)$ is general, $P\notin {\Delta }_{n-1}$ and $Q\notin {\Delta }_{n-1}$. Thus, $X^{★i}★P$ and $X^{★i}★Q$ are $PD(n+1)$-equivalent. Because $a_i=0$, we obtain $X^{★i}★P=X^{★i}★Q$; hence, $X^{★i+1}=X^{★i}★O$ for all $O\in X\setminus X\cap {\Delta }_{n-1}$. Let Y denote the closure in ${\mathbb{P}}^n$ of the set of all $O\in {\mathbb{P}}^n\setminus {\Delta }_{n-1}$ such that $X^{★i}★O=X^{★i}★P$. By Remark 12 and ([3], Th. 7.4), i.e., by ([4], 5.3), Y is a binomial variety. Thus, X is contained in a binomial hypersurface, contradicting our assumption. □

Applying Proposition 3 several times, we obtain the following result.

Corollary 1.

Let $X⊈{\Delta }_{n-1}$ be an integral variety not contained in a binomial hypersurface. Then, $X^{★i}={\mathbb{P}}^n$ for all $i>n-dimX$.

Remark 13.

Take an integral variety $X\subset {\mathbb{P}}^n$ and set $x:=dimX$. Every codimension x linear subspace of ${\mathbb{P}}^n$ meets X. Hence, each coordinate linear subspace of dimension $v\ge n-x$ meeting X is a non-empty set of dimension at least $v-n+x$.

Remark 14.

Take an integral variety $X\subset {\mathbb{P}}^n$ and set $x:=dimX$. Because ${\mathbb{P}}^n$ only has finitely many coordinate linear subspaces, a form of the theorem of Bertini provides the existence of a non-empty Zariski open subset T of $PGL(n+1)$ such that $e\left(g\right(X\left)\right)=x$ for every $g\in T$.

Proposition 4.

Take integral varieties $X\subset {\mathbb{P}}^n$, $Y\subset {\mathbb{P}}^n$ and set $x:=dimX$ and $y:=dimX$. Assume $x+y\le n$; then, there is a non-empty Zariski open subset T of $GL(n+1)$ such that ${\mathcal{J}}_n\cap (g\left(X\right)\times g\left(Y\right))\ne \varnothing$ for all $g\in T$. hence, the restriction of h to $g\left(X\right)\times g\left(Y\right)$ is a surjective morphism $h_{g\left(X\right),g\left(Y\right)}:g\left(X\right)\times g\left(Y\right)\to g\left(X\right)★g\left(Y\right)$. In particular, for all $g\in T$, the variety $g\left(X\right)★g\left(Y\right)$ is the image of $h(X\times Y)$ and we do not need to take the closure in its definition.

Proof. 

We use Remarks 6 and 14 and the fact that the intersection of two non-empty Zariski open subsets of $GL(n+1)$ is a non-empty Zariski open subset of $GL(n+1)$. □

For all integers $0<e<n$, let $(n,e)$ denote the set of all integral e-dimensional varieties $X\subset {\mathbb{P}}^n$ such that X is dimensionally transversal to every coordinate linear subspace, i.e., for all $0\le f<n$$dimV\cap X=max\{-1,e+f-n\}$ with the convention $dim\varnothing =-1$. For any integral variety $X\subset {\mathbb{P}}^n$, let $e\left(X\right)$ be the maximal number of zero-coordinates of some $p\in X$.

Remark 15.

The case $f=0$ of the definition of $(n,e)$ shows that no $X\in (n,e)$ contains a coordinate point.

Lemma 2.

Fix integers $n>e>0$ and let $X\subset {\mathbb{P}}^n$. We have $g\left(X\right)\in (n,e)$ for a general $g\in PGL(n+1)$.

Proof. 

As there are only finitely many coordinate linear subspaces, it is sufficient to use the theorem of Bertini. □

Proposition 5. 

Fix positive integers e and f such that $n\ge e+f$ and take integral varieties $X\subset {\mathbb{P}}^n$, $Y\subset {\mathbb{P}}^n$. Assume $e\left(X\right)+e\left(Y\right)\le n$. Then, the rational map $h:{\mathbb{P}}^n\times {\mathbb{P}}^n⤏{\mathbb{P}}^n$ induces a morphism $h_{X,Y}:X\times Y\to {\mathbb{P}}^n$ and $e(X★Y)\le e\left(X\right)+e\left(Y\right)$.

Proof. 

The results follow from the definition of h as the multiplication of all coordinates. □

Remark 16. 

We may iterate Proposition 5; in particular, if $k\ge 2$ and $ke\left(X\right)\le n$, then the variety $X^{★k}$ is the image of a morphism $X^k\to X^{★k}$.

Remark 17.

Take an integral variety $X\subset {\mathbb{P}}^n$ such that $X\times X⊈{\mathcal{J}}_n$. Because $P★Q=Q★P$ for every $(P,Q)\in X\times X$, the rational map $h_{X,X}$ factors through the degree 2 morphism, which is the quotient of $X\times X$ by the order 2 automorphism $(P,Q)\mapsto (Q,P)$. Set $x:=deg\left(X\right)$ and $a:=deg\left(X\right)$. If $dimX★X=2x$, then the upper bound for the integer $deg(X★X)$ coming from Remark 7 is $\left({\displaystyle \genfrac{}{}{0pt}{}{2x}{x}}\right)a^2/2$.

3. The Algebro-Geometric Part, II: Proofs of the Theorems

Proof of Theorem 1: 

Because $W⊈{\Delta }_{n-1}$, $X⊈{\Delta }_{n-1}$ and $Y⊈{\Delta }_{n-1}$. Fix $Q_0\in Y\cap U_n$. For each $Q\in Y\cap U_n$, $X★Q$ is $PD(n+1)$-equivalent to X. Hence, there is $f_{Q_0,Q}\in PD(n+1)$ such that $f_{Q_0,Q}:X★Q_0\to X★Q$ is an isomorphism. Because $dimW=dimX+dimY$ and $dimY=1$, there are only finitely many $Q_{\alpha }$, $\alpha \in S$, such that $X★Q_{\alpha }=X★Q$. Hence, the minimal algebraic subgroup G of $PD(n+1)$ containing all ${\left\{f_{Q_0,Q}\right\}}_{Q\in Y\cap U_n}$ has positive dimension. By assumption, there is $Q\in Y\cap U_n$ such that $f_{Q_0,Q}\left(W\right)\ne W$. Note that $f_{Q_0,Q}\left(W\right)\cap W\supseteq X★Q$. Because $f_{Q_0,Q}\left(W\right)\ne W$, we find that $f_{Q_0,Q}\left(W\right)\cap W$ has a codimension 1 component. By ([14], Th.2.2.5), this codimension 1 component has degree at most $d^2$. Hence, $a=deg\left(X\right)=deg(X★Q_0)\le d^2$. □

Fix an integer s such that $1\le s\le n-1$ and s integers $0\le i_1<\cdots <i_s\le n$. Set $H(i_1,\dots ,i_s):={\cap }_{j\in \{i_1,\dots ,i_s\}}{H}_j$, $e:=n+1-s$, and $\{j_1,\dots ,j_e\}:=\{0,\dots ,n\}\setminus \{i_1,\dots ,i_s\}$. Let $c_{i_1,\dots ,i_s}:{\mathbb{P}}^n\setminus H(j_1,\dots ,j_e)\to {\mathbb{P}}^n$ denote the morphism defined by the formula

$$e_{i_1,\dots ,i_s}\left([a_0:\cdots :a_n]\right)=[b_0:\cdots :b_n]$$

with $b_i=a_i$ if $i\notin \{j_1,\dots ,j_s\}$ and $b_i=0$ if $i\in \{j_1,\dots ,j_s\}$. Each fiber of $c_{i_1,\dots ,i_s}$ is an affine space of dimension s and its closure in ${\mathbb{P}}^n$ is an s-dimensional linear projective space.

Proof of Theorem 2: 

Because $W\cap {\Delta }_{n-3}=\varnothing$, the set $W\cap V_1$ is finite for each irreducible component $V_1$ of ${\Delta }_{n-2}$. Thus, we may assume that $W\cap {\Delta }_{n-2}$ is finite. Fix $i\in \{0,\dots ,n\}$. Because X and Y are projective curves, $X\cap H_i\ne \varnothing$ and $Y\cap H_i\ne \varnothing$ for each coordinate hyperplane $H_i$.

Claim 1:$X\cap {\Delta }_{n-3}=Y\cap {\Delta }_{n-3}=\varnothing$.

Proof of Claim 1: Assume $Y\cap V\ne \varnothing$ (for instance) for some codimension 3 coordinate linear subspace V and take $Q\in Y\cap V$. Fix any $P\in X\cap U_n$. The point $P★Q$ is well-defined; hence, $W\cap V\ne \varnothing$, which is a contradiction.

Claim 1 means that each $P=[a_0:\cdots :a_n]\in X\cup Y$ has $a_i=0$ for at most 2 indices i. Because each $O\in {\mathbb{P}}^n$ has $n+1>2+2$ homogeneous coordinates, $P★Q$ is defined for all $(P,Q)\in X\times Y$, i.e., $h_{X,Y}$ is a morphism.

Assume that part (b) fails and take $P=[a_0:\cdots :a_n]\in X\cup Y$ such that there are $0\le i<j\le n$ with $a_i=a_j=0$. Without loss of generality, we may assume $P\in X$. Fix $u\in \{0,\dots ,n\}\setminus \{i,j\}$. Because Y is a projective curve, $Y\cap H_u\ne \varnothing$. Fix $Q\in Y\cap H_u$. Then, from step (a) we have $P★Q\in W$. The point $P★Q$ is contained in the codimension 3 linear subspace $H_i\cap H_j\cap H_u$, which is a contradiction. □

Proof of Theorem 3: 

Set $L_i:=W\cap H_i$, $i=0,\dots ,n$. By Theorem 2 $h_{X,Y}$, is a morphism and each element of $X\cup Y$ has at most one zero as a coordinate. Moreover, W has exactly $\left({\displaystyle \genfrac{}{}{0pt}{}{n+1}{2}}\right)$ points $\left\{O_{ij}\right\}$, $i,j\in \{0,\dots ,n\}$ such that $i\ne j$ with two zero-coordinates, i.e., the points $O_{ij}:=W\cap H_i\cap H_j$. Fix $i\in \{0,\dots ,n\}$ and take $P\in X\cap H_i$ and $Q\in Y\cap H_i$. The set $L_i$ is a line containing the n points $O_{ij}$, $j\ne i$. Because $h_{X,Y}$ is a morphism and Y is irreducible, the closed set $P★Y$ is either a point or the line $L_i$. Write $P=[p_0:\cdots :p_n]$ with $p_i=0$ and $p_j\ne 0$ for all $j\ne 0$, and let ${\gamma }_P\in PD(n+1)$ be the automorphism induced by the multiplication by 1 on the i-th coordinate and the multiplication by $p_j^{-1}$ on the coordinate of $x_j$.

Assume $P★Y=\left\{O\right\}$ for some $O\in W\cap H_i$. Writing $O=[a_0,\dots ,a_n]$ with $a_i=0$, in this case Y is the line $L:=\overline{c_i^{-1}\left({\gamma }_P\left(O\right)\right)}$, as P has a unique 0 among its coordinates. Because L is the line spanned by $\{{\gamma }_P\left(O\right),e_i\}$ and $e_i\notin Y$, we have a contradiction.

Now, assume $P★Y=L_i$. Because P has a unique zero among its coordinates, Y is contained in the plane ${\Pi }_i:=\overline{c_i^{-1}\left({\gamma }_P\left(L_i\right)\right)}$, i.e., the plane spanned by $e_i$ and ${\gamma }_P\left(L_i\right)$. Because ${\Pi }_i$ is a plane, it contains at most three coordinate points. Fix j such that $e_j\notin {\Pi }_i$ and take $P_j\in X\cap H_j$. We find that Y is contained in the plane ${\Pi }_j$ spanned by $e_j$ and ${\gamma }_{P_j}\left(L_j\right)$. Because $e_j\notin {\Pi }_i$, Y is the line ${\Pi }_i\cap {\Pi }_j$.

In the same way, we can see that X is a line. Because X is a line and $X\cap {\Delta }_{n-2}=\varnothing$, the points $P_0,\dots ,P_n$ are distinct and uniquely determined by X. Because Y is uniquely determined by the lines ${\gamma }_{P_i}\left(L_j\right)$, Y is uniquely determined by W and X. □

Proof of Theorem 4: 

Set $W:=L★L$. The variety W is a plane ([3], Th. 2.1). Because $L\cap V=\varnothing$ for each codimension 2 coordinate linear subspace V, the coordinate of each element of W has at most two zeroes. We first apply Theorem 3 with $X=Y=L$ and then with $X=L$ and $Y=R$. □

Example 3.

Take a general line $L\subset {\mathbb{P}}^n$, $n\ge 3$ and set $W:=L★L$. Because $L\cap {\Delta }_{n-2}=\varnothing$, W is a plane ([3], Th. 2.1) and $W\cap {\Delta }_{n-3}=\varnothing$. Let $\alpha \in PD(n+1)$ be the element induced by multiplication by $-1$ for the variable $x_0$ and 1 for the other variables. Set $R:=\alpha \left(L\right)$. For general L, we have $R\ne L$. Obviously, $R★R=L★L$. If $n\ne 4$, we can see that in Theorem 3 W does not determine the set $\{X,Y\}$, although $(W,X)$ determines Y. The same is true if we take an order 2 element of $PD(n+1)$ instead of α.

Question 2: Take a general line $L\subset {\mathbb{P}}^n$, $n\ge 4$ and take any line $D\subset {\mathbb{P}}^n$ such that $L★L=D★D$. Is there an order 2 element $\gamma \in PD(n+1)$ such that $D=\gamma \left(L\right)$?

Remark 18.

Take $k\ge 2$ and set $W:=X_1★\cdots ★X_k$, with $X_i\subset {\mathbb{P}}^n$ as an integral variety; here, W does not change if we permute $X_1,\dots ,X_k$. Take ${\alpha }_i\in PD(n+1)$, $1\le i\le k$, such that ${\prod }_{i=1}^k{\alpha }_i={\mathrm{Id}}_{{\mathbb{P}}^n}$. We have $W={\alpha }_1\left(X_1\right)★\cdots ★{\alpha }_k\left(X_k\right)$. If we restrict this to powers of the same variety X, the same holds if ${\alpha }_i\left(X\right)={\alpha }_1\left(X\right)$ for all $i>1$. If X is not fixed by any non-trivial element of $PD(n+1)$, then we need to take ${\alpha }_i={\alpha }_1$ for all $i>1$, i.e., we need to take a k-root of the identity map. Over an algebraically closed field K with characteristic 0, the group $PD(n+1,K)$ has $k^n$ roots of unity.

Question 3: Fix 5 positive integers n, x, y, a, and b such that $x+y<n$. Let $\mathcal{W}$ be the family of all irreducible $W\subset {\mathbb{P}}^n$ such that $W⊈{\Delta }_{n-1}$, $dimW=x+y$, and $deg\left(W\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{x+y}{x}}\right)ab$ ([3], Cor. 5.1).

(a) Provide upper or lower bounds for the dimension of the set of all $W\in \mathcal{W}$ such that $W=X★Y$ for some integral varieties X and Y with $dim\left(X\right)=x$, $dim\left(Y\right)=y$, $deg\left(X\right)=a$ and $deg\left(Y\right)=b$.

(b) Fix $W\in \mathcal{W}$. What is the dimension $d_W$ of all pairs $(X,Y)$ such that $dimX=x$, $dimY=y$, $deg\left(X\right)=a$, $deg\left(Y\right)=b$ and $W=X★Y$? What is the maximum of all integers $d_W$? Is $d_W=0$ for some $W\in \mathcal{W}$?

(d) The same questions may be raised for the ★-products of $k>2$ varieties.

4. Over the Real Numbers

We take $K=\mathbb{R}$ and $\overline{K}=\mathbb{C}$. On ${\mathbb{P}}^n\left(\mathbb{R}\right)$ and ${\mathbb{P}}^n\left(\mathbb{C}\right)$, we have the Euclidean topology and the Zariski topology, both of which are useful. The Euclidean topology has a basis formed by the open balls, which are semialgebraic open sets. Recall that in the Euclidean topology the closure and open part of a semialgebraic set are semialgebraic ([11], Prop. 2.2.2) and that images of semialgebraic sets by a regular mapping are semialgebraic ([11], Prop. 2.2.7).

Because the multiplication in $\mathbb{R}$ and $\mathbb{C}$ is continuous for the Euclidean topology, if the channels have small errors or if the multiplications have small errors, then the observant reader can easily see that the picture in the screen is “near” to the true one. If we know the degrees and dimensions of the expected picture, it is easy to see the best approximation of the picture in the screen.

Recall that $\sigma$ is the complex conjugation and that if $X\subset {\mathbb{P}}^n$ is defined over $\mathbb{R}$, then $X\left(\mathbb{R}\right)=\{P\in X\left(\mathbb{C}\right)\mid \sigma \left(P\right)=P\}=X\left(\mathbb{C}\right)\cap {\mathbb{P}}^n\left(\mathbb{R}\right)$. Because $X\left(\mathbb{C}\right)$ and ${\mathbb{P}}^n\left(\mathbb{R}\right)$ are compact topological spaces for the Euclidean topology, $X\left(\mathbb{R}\right)$ is compact. Moreover, $X\left(\mathbb{R}\right)$ is a real algebraic variety of dimension at most $dimX={dim}_{\mathbb{C}}X\left(\mathbb{C}\right)$. However, in many cases $X\left(\mathbb{R}\right)=\varnothing$ (e.g., take the real conic $\{x_0^2+\cdots +x_n^2=0\}\subset {\mathbb{P}}^n$). If X is irreducible over $\mathbb{R}$ but not over $\mathbb{C}$, then the real algebraic set $X\left(\mathbb{R}\right)$ has dimension $<dimX$. Hence, in the next statements, e.g., Theorem 5, the assumptions imply that the real algebraic varieties are irreducible over $\mathbb{C}$.

Remark 19.

Take $X_1,\dots ,X_k$, $k\ge 2$, defined over $\mathbb{R}$, and set $W\left(\mathbb{C}\right):=X_1\left(\mathbb{C}\right)★\cdots ★X_k\left(\mathbb{C}\right)$. The variety W and its real structure are not changed if we permute the ★-factors. Moreover, W and its real structure are not changed if we take ${\alpha }_i\in PD(n+1,\mathbb{R})$, $1\le i\le k$ such that ${\prod }_{i=1}^k{\alpha }_i={\mathrm{Id}}_{{\mathbb{P}}^n}$ and use ${\alpha }_i\left(X_i\right)$ instead of $X_i$. Now, assume $X_i=X_1$ for all i. This was previously considered over $\mathbb{C}$ in Remark 18; the main difference over $\mathbb{R}$ is that if k is odd (resp. even), then the set of all $\alpha \in PD(n+1,\mathbb{R})$ (such that ${\alpha }^k={\mathrm{Id}}_{{\mathbb{P}}^n}$) has cardinality 1 (resp. $2^n$).

Assume that $X\subset {\mathbb{P}}^n$ is defined over $\mathbb{R}$ and that its embedding is defined over $\mathbb{R}$, i.e., assume that $X\left(\mathbb{C}\right)\subset {\mathbb{P}}^n\left(\mathbb{C}\right)$ is scheme-theoretically cut out by finitely homogeneous polynomials with real coefficients. Because the embedding of X is defined over $\mathbb{R}$, we have $X\left(\mathbb{R}\right)=X\left(\mathbb{C}\right)\cap {\mathbb{P}}^n\left(\mathbb{R}\right)$. The set $X\left(\mathbb{R}\right)$ is a compact real algebraic variety of dimension at most $dimX$. The set $X\left(\mathbb{R}\right)$ may be empty, e.g., the real smooth conic $\{x_0^2+x_1^2+x_2^2=0\}\subset {\mathbb{P}}^2\left(\mathbb{R}\right)$. For any X, let $X_{\mathrm{reg}}$ denote the set of its smooth points. If $X\left(\mathbb{R}\right)$ contains a smooth point p of $X\left(\mathbb{C}\right)$, i.e., if $p\in X_{\mathrm{reg}}\left(\mathbb{C}\right)\cap X\left(\mathbb{R}\right)$, then the real algebraic set $X\left(\mathbb{R}\right)$ in a neighborhood of p for the Euclidean topology is a smooth differentiable manifold of dimension ${dim}_{\mathbb{C}}X\left(\mathbb{C}\right)$. For any irreducible quasi-projective variety W defined over $\mathbb{R}$, we have $W_{\mathrm{reg}}\left(\mathbb{R}\right)\ne \varnothing$ if and only if $W\left(\mathbb{R}\right)$ is Zariski dense in $W\left(\mathbb{C}\right)$. For any set $S\subseteq {\mathbb{P}}^n\left(\mathbb{R}\right)$ (resp. $S\subseteq {\mathbb{P}}^n\left(\mathbb{C}\right)$), let ${\overline{S}}_{\mathrm{Zar}}$ and ${\overline{S}}_{\mathrm{euc}}$ denote the closure of S for the Zariski and Euclidean topologies, respectively. If $S\subset {\mathbb{P}}^n\left(\mathbb{C}\right)$ is constructable, i.e., a finite union of sets which are locally closed for the Zariski topology, then ${\overline{S}}_{\mathrm{Zar}}={\overline{S}}_{\mathrm{euc}}$.

Remark 20.

For any semialgebraic set $S\subset {\mathbb{R}}^n$, the Euclidean closure ${\overline{S}}_{\mathrm{euc}}$ and Zariski closure ${\overline{S}}_{\mathrm{Zar}}$ have the same dimension and S contains a non-empty open ball (for the Euclidean topology) of ${\overline{S}}_{\mathrm{Zar}}$ ([11], Prop. 2.8.4 or Cor. 2.8.9).

If $X_{i\mathrm{reg}}\left(\mathbb{R}\right)\ne \varnothing$ for all $i=1,\dots ,k$, then we may use the ★-product form of the Terracini Lemma ([3], Lemma 1.6), ([8], Lemma 2.12), ([9], Lemma 4.13). If $X_{i\mathrm{reg}}\left(\mathbb{R}\right)\ne \varnothing$ for all $i=1,\dots ,k$, then we have the following result.

Theorem 5.

Fix an integer $k\ge 2$ and integral varieties $X_i\left(\mathbb{C}\right)\subset {\mathbb{P}}^r\left(\mathbb{C}\right)$, $1\le i\le k$ such that $X_{i\mathrm{reg}}\left(\mathbb{R}\right)⊈{\Delta }_{n-1}\left(\mathbb{R}\right)$ for all i. Set $A:=X_1\left(\mathbb{R}\right){★}^{\circ }\cdots {★}^{\circ }{X}_k\left(\mathbb{R}\right)\subseteq {\mathbb{P}}^n\left(\mathbb{R}\right)$. Then, $X_1\left(\mathbb{C}\right)★\cdots ★X_k\left(\mathbb{C}\right)\subseteq {\mathbb{P}}^n\left(\mathbb{C}\right)$ is uniquely determined by one of the following semi-algebraic sets:

(a) 

$A\subseteq {\mathbb{P}}^n\left(\mathbb{R}\right)$

(b) 

${\overline{A}}_{\mathrm{euc}}\subseteq {\mathbb{P}}^n\left(\mathbb{R}\right)$

(c) 

${\overline{A}}_{\mathrm{Zar}}\subseteq {\mathbb{P}}^n\left(\mathbb{R}\right)$

(d) 

the Zariski closure of A in ${\mathbb{P}}^n\left(\mathbb{C}\right)$.

Moreover, any set of homogeneous polynomials with real coefficients describing ${\overline{A}}_{\mathrm{Zar}}\subseteq {\mathbb{P}}^n\left(\mathbb{R}\right)$ have $X_1\left(\mathbb{C}\right)★\cdots ★X_r\left(\mathbb{C}\right)$ as their common solutions in ${\mathbb{P}}^n\left(\mathbb{C}\right)$.

Obviously, $X_{i\mathrm{reg}}\left(\mathbb{R}\right)⊈{\Delta }_{n-1}\left(\mathbb{R}\right)$ if and only if $X_{i\mathrm{reg}}\left(\mathbb{C}\right)⊈{\Delta }_{n-1}\left(\mathbb{C}\right)$.

Example 4.

If $X_1\left(\mathbb{R}\right)={\mathbb{P}}^n$, then it is sufficient to have $X_i\left(\mathbb{R}\right)⊈{\Delta }_{n-1}\left(\mathbb{R}\right)$ for all $i>1$ to obtain $X_1\left(\mathbb{R}\right){★}^{\circ }\cdots {★}^{\circ }{X}_k\left(\mathbb{R}\right)={\mathbb{P}}^n\left(\mathbb{R}\right)$; hence, $X_1\left(\mathbb{C}\right)★\cdots ★X_k\left(\mathbb{C}\right)={\mathbb{P}}^n\left(\mathbb{C}\right)$. If $X_{1\mathrm{reg}}\left(\mathbb{R}\right)=\varnothing$, then we always have ${dim}_{\mathbb{R}}{X}_1\left(\mathbb{R}\right){★}^{\circ }\cdots {★}^{\circ }{X}_k\left(\mathbb{R}\right)<{\sum }_{i=1}^k{dim}_{\mathbb{C}}{X}_i\left(\mathbb{C}\right)$.

Example 5.

Take $n=2$. Let $L\subset {\mathbb{P}}^2\left(\mathbb{C}\right)$ be a line such that $L\ne \sigma \left(L\right)$. We have $L★\sigma \left(L\right)={\mathbb{P}}^2\left(\mathbb{C}\right)$, and the reducible conic $L\cup \sigma \left(L\right)$ is defined over $\mathbb{R}$. We obtain the same answer taking $R★R^{\prime }$ with R, $R^{\prime }$ as lines defined over $\mathbb{R}$ and intersecting $U_1$, except that $(L\cup \sigma (L\left)\right)\left(\mathbb{R}\right)$ is a single point while $R\left(\mathbb{R}\right){★}^{\circ }{R}^{\prime }\left(\mathbb{R}\right)$ contains a non-empty open subset of ${\mathbb{P}}^2\left(\mathbb{R}\right)$.

Example 6.

Take $n=3$ and L as a line such that $\sigma \left(L\right)\cap L=\varnothing$ and L meets no coordinate line. It is easy to check that $Q:=L★\sigma \left(L\right)$ is a smooth quadric. Obviously, $\sigma \left(Q\right)=Q$, i.e., Q is defined over $\mathbb{R}$. Because $\sigma \left(L\right)\cap L=\varnothing$, σ exchanges the two rulings of Q and $Q\left(\mathbb{R}\right)=\varnothing$.

Proposition 6.

Fix an integer $k\ge 2$ and integral projective varieties $X_i\subset {\mathbb{P}}^n$, $1\le i\le k$. Assume $X_{i\mathrm{reg}}\left(\mathbb{R}\right)\ne \varnothing$ for all i. Then, ${dim}_{\mathbb{C}}{X}_1\left(\mathbb{C}\right)★\cdots ★X_k\left(\mathbb{C}\right)={dim}_{\mathbb{R}}{X}_1\left(\mathbb{R}\right){★}^{\circ }\cdots {★}^{\circ }{X}_k\left(\mathbb{R}\right)$.

Proof. 

For any smooth point p of some variety $W\subset {\mathbb{P}}^n\left(\mathbb{C}\right)$, let $T_{p}W$ denote its Zariski tangent space. The set $T_{p}W$ is a complex projective space containing p and ${dim}_{\mathbb{C}}{T}_{p}W=dimW$. Now, assume that W is defined over $\mathbb{R}$ and that $p\in W\left(\mathbb{R}\right)$. The set $T_{p}W$ is the complex vector space associated with the ${dim}_{\mathbb{C}}W$ real vector space $T_{p}W\cap {\mathbb{P}}^n\left(\mathbb{R}\right)=T_{p}W\cap \sigma \left(T_{p}W\right)$. By the Terracini Lemma ([3], Lemma 1.6), the dimension over $\mathbb{C}$ (the irreducible projective variety $X_1\left(\mathbb{C}\right)★\cdots ★X_k\left(\mathbb{C}\right)$) may be computed in the following way. There is a non-empty Zariski open subset U of $X_{1\mathrm{reg}}\left(C\right)\times \cdots \times X_{k\mathrm{reg}}\left(\mathbb{C}\right)$ with the property that for all $(p_1,\dots ,p_k)\in U$ the integer $dim⟨{\cup }_{i=1}^k{L}_i⟩$ is the same, where $L_i$ is the ★-product of $T_{p_i}{X}_i\left(\mathbb{C}\right)$ and all $p_j$, $j\in \{1,\dots ,k\}\setminus \left\{i\right\}$. By the Terracini Lemma ([3], Lemma 1.6), this integer is ${dim}_{\mathbb{C}}{X}_1\left(\mathbb{C}\right)★\cdots ★X_k\left(\mathbb{C}\right)$. Because $X_{i\mathrm{reg}}\left(\mathbb{R}\right)\ne \varnothing$, $X_{i\mathrm{reg}}\left(\mathbb{R}\right)$ is Zariski dense in $X_{i\mathrm{reg}}\left(\mathbb{C}\right)$, in the Zariski open set U we may find $(p_1,\dots ,p_k)\in U$ with $p_i\in X_{i\mathrm{reg}}\left(\mathbb{R}\right)$ for all i. Note that $⟨{\cup }_{i=1}^k{L}_i⟩$ is the complexification of the real projective space spanned over $\mathbb{R}$ by all real projective spaces $L_i\left(\mathbb{R}\right)$, where $L_i\left(\mathbb{R}\right)$ is the ★-product of $T_{p_i}{X}_i\left(\mathbb{C}\right)\cap {\mathbb{P}}^n\left(\mathbb{R}\right)$ and all $p_j$, $j\in \{1,\dots ,k\}\setminus \left\{i\right\}$. □

Lemma 3.

Take $k\ge 2$ and geometrically integral projective varieties $X_i\left(\mathbb{C}\right)\subset {\mathbb{P}}^n\left(\mathbb{C}\right)$, $1\le i\le k$, defined over $\mathbb{C}$ such that $X_{i\mathrm{reg}}\left(\mathbb{R}\right)\notin {\Delta }_{n-1}\left(\mathbb{R}\right)$ for all i. Then, the semialgebraic set $X_1\left(\mathbb{R}\right){★}^{\circ }\cdots {★}^{\circ }{X}_k\left(\mathbb{R}\right)\subseteq {\mathbb{P}}^n\left(\mathbb{R}\right)$ has real dimension ${dim}_{\mathbb{C}}{X}_1\left(\mathbb{C}\right)★\cdots ★X_k\left(\mathbb{C}\right)$, while $X_1\left(\mathbb{C}\right)★\cdots ★X_k\left(\mathbb{C}\right)$ is the Zariski closure in ${\mathbb{P}}^n\left(\mathbb{C}\right)$ of $X_1\left(\mathbb{R}\right){★}^{\circ }\cdots {★}^{\circ }{X}_k\left(\mathbb{R}\right)\subseteq {\mathbb{P}}^n\left(\mathbb{R}\right)$.

Proof. 

The assertion on the dimensions is true by Proposition 13. For any variety $W\left(\mathbb{R}\right)$ defined over $\mathbb{R}$, the semialgebraic set $W\left(\mathbb{R}\right)$ has real dimension at most ${dim}_{\mathbb{C}}W\left(\mathbb{C}\right)$. Because $X_1\left(\mathbb{C}\right)★\cdots ★X_k\left(\mathbb{C}\right)$ is irreducible and contains $X_1\left(\mathbb{R}\right){★}_{\mathbb{R}}^{\circ }\cdots {★}_{\mathbb{R}}^{\circ }{X}_k\left(\mathbb{R}\right)$, we have $X_1\left(\mathbb{C}\right)★\cdots ★X_k\left(\mathbb{C}\right)$ as the Zariski closure in ${\mathbb{P}}^n\left(\mathbb{C}\right)$ of $X_1\left(\mathbb{R}\right){★}_{\mathbb{R}}^{\circ }\cdots {★}_{\mathbb{R}}^{\circ }{X}_k\left(\mathbb{R}\right)\subset {\mathbb{P}}^n\left(\mathbb{R}\right)$. □

Proof of Theorem 5: 

The Zariski closure inside ${\mathbb{P}}^n\left(\mathbb{C}\right)$ is provided by the polynomials with real coefficients, providing the Zariski closure in (c).

The semialgebraic set in (b) is the Euclidean closure in ${\mathbb{P}}^n\left(\mathbb{R}\right)$ of the semialgebraic set in (a). The semialgebraic set in (c) is the Zariski closure inside ${\mathbb{P}}^n\left(\mathbb{R}\right)$ of both the set in (b) and the set in (d). Thus, it is sufficient to prove that $X_1\left(\mathbb{C}\right)★\cdots ★X_k\left(\mathbb{C}\right)\subseteq {\mathbb{P}}^n\left(\mathbb{C}\right)$ is the Zariski closure in ${\mathbb{P}}^n\left(\mathbb{C}\right)$ of the semialgebraic set $A\subseteq {\mathbb{P}}^n\left(\mathbb{R}\right)$. This is true by Lemma 3. □

Thus, we can see that with mild assumptions on $X_i\left(\mathbb{R}\right)$, $1\le i\le k$, the ★-product of the complexifications is uniquely determined by the real ★-product of their real parts. The following examples shows that the converse does not hold, even when we avoid taking the closure.

Example 7.

Take $n=2$. Let $X_1$ be the real conic $x_0^2+x_1^2+x_2^2=0$ with $X_1\left(\mathbb{R}\right)=\varnothing$ and let $X_2$ be a general real line. Obviously, $X_1\left(\mathbb{C}\right){★}^{\circ }{X}_2\left(\mathbb{C}\right)$ contains the complement of finitely many points of ${\mathbb{P}}^2\left(\mathbb{C}\right)$, while $X_1\left(\mathbb{R}\right){★}^{\circ }{X}_2\left(\mathbb{R}\right)=\varnothing$.

Example 8.

Take irreducible $X\subset {\mathbb{P}}^n$ defined over $\mathbb{R}$ and take any $P\in X\left(\mathbb{C}\right)\setminus X\left(\mathbb{R}\right)$. Set $O:=P★\sigma \left(P\right)$. Because ★ is commutative, $\sigma \left(O\right)=O$; hence, $O\in (X\left(\mathbb{C}\right)★X\left(\mathbb{C}\right))\cap {\mathbb{P}}^n\left(\mathbb{R}\right)$. In many cases (though not all cases, e.g., this not true if X is a line), there are no $P_1,P_2\in X\left(\mathbb{R}\right)$ such that $O=P_1★P_2$.

When we have an even number of ★-powers of the same variety, the following construction provides many ghosts.

Example 9.

Fix an integral variety $X\subset {\mathbb{P}}^n$ defined over $\mathbb{R}$ (such that $X\left(\mathbb{C}\right)⊈{\Delta }_{n-1}\left(\mathbb{C}\right)$ and the rational map $h_{X,X}$ is birational) onto its image up to the exchange map (Remark 17); i.e., setting $x:=dimX$ and $a:=deg\left(X\right)$, we have $dimX★X=2x$ and $deg(X★X)=\left({\displaystyle \genfrac{}{}{0pt}{}{2x}{x}}\right)a^2/2$. Fix any $P\in X\left(\mathbb{C}\right)\setminus X\left(\mathbb{R}\right)$ such that $(P,\sigma \left(P\right))\notin {\mathcal{J}}_n$. The set of all such P contains a non-empty Euclidean open subset of $X\left(\mathbb{C}\right)\setminus X\left(\mathbb{R}\right)$. Note that $P★\sigma \left(P\right)\in {\mathbb{P}}^n\left(\mathbb{R}\right)$ for any $P\in X\left(\mathbb{C}\right)$ such that $(P,\sigma \left(P\right))\notin {\mathcal{J}}_n$. Hence, the set $X\left(\mathbb{C}\right){★}^{\circ }X\left(\mathbb{C}\right)\cap {\mathbb{P}}^n\left(\mathbb{R}\right)$ contains a Euclidean $2x$-dimensional manifold U containing no point of $A:=X\left(\mathbb{R}\right){★}^{\circ }X\left(\mathbb{R}\right)$. We may also assume $U\cap {\overline{A}}_{\mathrm{euc}}=\varnothing$, restricting the manifold U if necessary. It is reasonable to say that U is a part of a real ghost of the complex ★-product. It only arises here because we processed the complex data before restriction to the real screen ${\mathbb{P}}^n\left(\mathbb{R}\right)$.

4.1. Full Screens

Now, we consider a different problem. The task is checking whether $X_1\left(\mathbb{C}\right){★}^{\circ }\cdots {★}^{\circ }{X}_k\left(\mathbb{C}\right)$ is Zariski dense in ${\mathbb{P}}^n\left(\mathbb{C}\right)$ using the real screen ${\mathbb{P}}^n\left(\mathbb{R}\right)$.

For instance, we do not need the shape of $X_1\left(\mathbb{C}\right)★\cdots ★X_k\left(\mathbb{C}\right)$, as we can run a test to be sure that our k screens cover all the screen. Having $X_1\left(\mathbb{C}\right)★\cdots ★X_k\left(\mathbb{C}\right)={\mathbb{P}}^n\left(\mathbb{C}\right)$ means that k channels are sufficient to provide all of the data we will need. Now, we prove that we can perform the test using the real screen ${\mathbb{P}}^n\left(\mathbb{R}\right)$.

Lemma 4.

Let $U\subseteq {\mathbb{C}}^m$, $m\ge 1$ be a non-empty Zariski open subset such that $\sigma \left(U\right)=U$. Then, $U\cap {\mathbb{R}}^m$ is a Zariski dense open subset of ${\mathbb{R}}^m$.

Proof. 

Let $y_1,\dots ,y_m$ denote the variables in ${\mathbb{R}}^m$ (hence, ${\mathbb{C}}^m$). The set ${\mathbb{C}}^m\setminus U$ is Zariski closed in ${\mathbb{C}}^m$; hence, it is the zero-set of finitely many polynomials $f_1(y_1,\dots ,y_m),\dots ,$$f_s(y_1,\dots ,y_m)$. For any $f\in \mathbb{C}[y_1,\dots ,y_m]$, let $f^{\sigma }$ denote the polynomial obtained from f, taking the complex conjugation of all coefficients of f. Set $f\left[1\right](y_1,\dots ,y_m):=(f(y_1,\dots ,y_m)+f^{\sigma }(y_1,\dots ,y_m))/2$ and $f\left[2\right](y_1,\dots ,y_m):=(-f(y_1,\dots ,y_m)+f^{\sigma }(y_1,\dots ,y_m))/2i$. Note that $f(y_1,\dots ,y_m)$ and $f^{\sigma }(y_1,\dots ,y_m)$ are uniquely determined by the polynomials $f\left[1\right](y_1,\dots ,y_m)$ and $f\left[2\right](y_1,\dots ,y_m)$ and that $f\left[1\right](y_1,\dots ,y_m)$ and $f\left[2\right](y_1,\dots ,y_m)$ have real coefficients. Because $\sigma ({\mathbb{C}}^m\setminus U)={\mathbb{C}}^m\setminus U$, ${\mathbb{C}}^m\setminus U$ is the zero locus of the polynomials $f_i(y_1,\dots ,y_m)$ and $f_i^{\sigma }(y_1,\dots ,y_m)$, $1\le i\le s$, and consequently of the polynomials $f_i\left[1\right](y_1,\dots ,y_m)$ and $f_i\left[2\right](y_1,\dots ,y_m)$, we have $1\le i\le s$ with real coefficients. The set $U\cap {\mathbb{R}}^m$ is Zariski open in ${\mathbb{R}}^m$ because its complement is the zero-locus of $2s$ real polynomials.

Hence, the lemma is true for the integer m if and only if $U\cap {\mathbb{R}}^m\ne \varnothing$. In the inductive proof on m, it can be directly seen that $U\cap {\mathbb{R}}^m$ is Zariski dense in ${\mathbb{R}}^m$.

Both parts of the lemma are true for $m=1$, as $\mathbb{C}\setminus U$ is a finite set. Thus, we may assume $m>1$ and use induction on the integer m. Let $G(1,m)\left(\mathbb{C}\right)$ be the Grassmannian of all one-dimensional linear subspaces of ${\mathbb{C}}^m$. The variety $G(1,m)\left(\mathbb{C}\right)$ is defined over $\mathbb{R}$, $G(1,m)\left(\mathbb{C}\right)={\mathbb{P}}^{m-1}\left(\mathbb{C}\right)$, and $G(1,m)\left(\mathbb{R}\right)={\mathbb{P}}^{m-1}\left(\mathbb{R}\right)$. Let $V\subseteq G(1,m)\left(\mathbb{C}\right)$ be the set of all $L\in G(1,m)\left(\mathbb{C}\right)$ such that $L\cap U\ne \varnothing$. Because U is a non-empty Zariski open subset of ${\mathbb{C}}^m$, V is a non-empty Zariski open subset of $G(1,m)\left(\mathbb{C}\right)$. Since $\sigma \left(U\right)=U$, $\sigma \left(V\right)=V$. Per the inductive assumption, $G(1,m)\left(\mathbb{R}\right)\cap V$ contains a Zariski open subset of $G(1,m)\left(\mathbb{R}\right)$. For each $R\in G(1,m)\cap V$, the set $R\left(\mathbb{R}\right)\setminus R\left(\mathbb{R}\right)\cap U$ is finite; hence, ${\cup }_{R\in S}(R\left(\mathbb{R}\right)\setminus R\left(\mathbb{R}\right)\cap U)$ is contained in a Zariski subset of ${\mathbb{R}}^m$ of dimension at most $m-1$. The set ${\cup }_{R\in G(1,m)\setminus W}R$ is a proper Zariski closed subset of ${\mathbb{R}}^m$. Thus, the lemma is true for the integer m. □

Proposition 7.

Take $X_i\left(\mathbb{C}\right)\subset {\mathbb{P}}^n\left(\mathbb{C}\right)$, $1\le i\le k$ defined over $\mathbb{R}$ and such that $X_1\left(\mathbb{C}\right)★\cdots ★X_k\left(\mathbb{C}\right)={\mathbb{P}}^n\left(\mathbb{C}\right)$. Then, $(X_1\left(\mathbb{C}\right){★}^{\circ }\cdots {★}^{\circ }{X}_k\left(\mathbb{C}\right))\cap {\mathbb{P}}^n\left(\mathbb{R}\right)$ is Zariski dense in ${\mathbb{P}}^n\left(\mathbb{R}\right)$.

Proof. 

Apply Lemma 4. Note that if f in its proof is homogeneous, then $f^{\sigma }$, $f\left[1\right]$, and $f\left[2\right]$ are homogeneous. □

4.2. Non-Existence of Real Ghosts

In ([3], §5.1), there is a section called “Large dimensions” in which (under suitable generality assumptions) the variety $X_1\left(\mathbb{C}\right)★\cdots ★X_k\left(\mathbb{C}\right)$ and the Segre embedding of $X_1\left(\mathbb{C}\right)\times \cdots \times X_k\left(\mathbb{C}\right)$ are projectively equivalent. The assumptions are very strong; each $X_i$ is embedded in a projective subspace $M_i\subset {\mathbb{P}}^n\left(\mathbb{C}\right)$, and these k linear spaces $M_1,\dots ,M_k$ are linearly independent, i.e., they span a linear space of dimension $k-1+{\sum }_{i=1}^{k}dim{M}_i$. Because $dim{X}_i\le dim{M}_i$ with strict inequality if $X_i$ is not a linear space, the ambient space ${\mathbb{P}}^n\left(\mathbb{C}\right)$ is huge. If we take each $M_i$ defined over $\mathbb{R}$ with the embedding $X_i\subset M_i$ defined over $\mathbb{R}$, then we will have no real ghost, as in this case $X_1\left(\mathbb{C}\right){★}^{\circ }\cdots {★}^{\circ }=X_1\left(\mathbb{C}\right)★\cdots ★X_k\left(\mathbb{C}\right)$ and $X_1\left(\mathbb{R}\right){★}^{\circ }\cdots {★}^{\circ }{X}_k\left(\mathbb{R}\right)=X_1\left(\mathbb{C}\right)★\cdots ★X_k\left(\mathbb{C}\right)\cap {\mathbb{P}}^n\left(\mathbb{R}\right)$.

In ([3], §5.2) for $k=2$ there is a proof in which, by using a specific linear projection (hence, a smaller n, though not by much), we still find that $X_1\left(\mathbb{C}\right)★X_2\left(\mathbb{C}\right)\cong X_1\left(\mathbb{C}\right)\times X_2\left(\mathbb{C}\right)$.

The first question below (Question 4) is over an arbitrary field.

Question 4: Find a better lower bound for n. Extend the case $k=2$ to the case $k>2$ using linear projections.

Question 5: Address Question 4 over $\mathbb{C}$ using general linear projections and prove that real ghosts do not appear after a general real projection.

Question 4 (or Question 5) over $\mathbb{C}$ is raised again as Question 7 in the Conclusions section. For any linear space $V\subset {\mathbb{P}}^N\left(\mathbb{C}\right)$, let ${\ell }_V:{\mathbb{P}}^N\left(\mathbb{C}\right)\setminus V\to {\mathbb{P}}^{N-dimV-1}\left(\mathbb{C}\right)$ denote the linear projection. Suppose that for a fixed $k\ge 2$ and integers $x_i:=dim{M}_i$, $1\le i\le s$ we have a positive solution for a certain integer n, i.e., we find a complex linear space $V\subset {\mathbb{P}}^N\left(\mathbb{C}\right)$ such that $dimV=N-n-1$, $V\cap M_i=\varnothing$, and $E_1\times \cdots \times E_k\cap {\mathcal{J}}_{N,k}=\varnothing$ (taking $E_i:={\ell }_V\left(M_i\right)$) and where the induced morphism $E_1\times \cdots \times E_k\to E_1★\cdots ★E_k$ is an isomorphism. Let $G(N-n,N+1)\left(\mathbb{C}\right)$ be the Grassmannian of all $(N-n-1)$-dimensional complex linear subspaces of ${\mathbb{P}}^N\left(\mathbb{C}\right)$. The set of all linear spaces $M\in G(N-n,N+1)\left(\mathbb{C}\right)$ satisfying all of the conditions satisfied by V is a Zariski open subset U of $G(N-n,N+1)\left(\mathbb{C}\right)$. Because $V\in U$, $U\ne \varnothing$, and $G(N-n,N+1)\setminus U$ has lower dimension, a random $V_1\in G(N-n,N+1)\left(\mathbb{C}\right)$ should be in U. We can certify that we have $V_2$ defined over $\mathbb{R}$; indeed, the set $G(N-n,N+1)\left(\mathbb{R}\right)$ is Zariski dense in $G(N-n,N+1)\left(\mathbb{C}\right)$. Hence, we have $V_2\in U$ defined over $\mathbb{R}$. Moreover, $U\cap G(N-n,N+1)\left(\mathbb{R}\right)$ is Zariski open and Zariski dense in $G(N-n,N+1)\left(\mathbb{R}\right)$, meaning that a random element of $G(N-n,N+1)\left(\mathbb{R}\right)$ should work. We have seen that if $V\in U\cap G(N-n,N+1)\left(\mathbb{R}\right)$ and $Y_i\left(\mathbb{C}\right)\subset M_i\left(\mathbb{C}\right)$ are arbitrary integral varieties defined over $\mathbb{R}$, then the varieties ${\ell }_V\left(Y_1\left(\mathbb{C}\right)\right),\dots ,{\ell }_V(Y_k\left(\mathbb{C}\right)$ have no real ghosts.

Question 6: Fix $X_1\left(\mathbb{C}\right),\dots ,X_k\left(\mathbb{C}\right)$ defined over $\mathbb{R}$. Assume $dim{X}_1\left(\mathbb{C}\right)+\cdots +dim{X}_k\left(\mathbb{C}\right)<n$ and take general $(g_1,\dots ,g_k)\in PGL{(n+1,\mathbb{R})}^k$. Then, prove that the variety $g_1\left(X_1\left(\mathbb{C}\right)\right)★\cdots ★g_k\left(X\left(\mathbb{C}\right)\right)$ has no real ghost.

Example 9 shows that if $k=2$ and $X_2=X_1$, then in Question 6 it is essential to take a general $(g_1,\dots ,g_k)$. Indeed, $g\left(X_1\left(\mathbb{C}\right)\right)★g\left(X_1\left(\mathbb{C}\right)\right)$ has real ghosts for a general $g\in PGL(n+1,\mathbb{R})$.

4.3. Suggestion for Mitigations and Killing the Ghosts

Below, we present four suggestions.

(a) If the complex data are mixed, as in Example 8, then our best suggestion is to use a small channel to send (and save) a small part of $X_1\left(\mathbb{R}\right),\dots ,X_k\left(\mathbb{R}\right)$. This should be sufficient to guess the ghost part.

(b) Taking the setup in Example 9, to identify some real ghost, send random points $P★\sigma \left(P\right)$; because these points P are random, they are in $X\left(\mathbb{C}\right)\setminus X\left(\mathbb{R}\right)$ and hopefully $(P,\sigma \left(P\right))\notin {\mathcal{J}}_n$. In this case, at minimum $P★\sigma \left(P\right)$ is outside the interior of $\overline{X\left(\mathbb{R}\right){★}^{\circ }X\left(\mathbb{R}\right)}$, and is probably in the interior of a ghost.

(c) Supposing that we know the exact the variety $X_1,\dots ,X_k$ and preprocessing is available, it is possible to describe the set $X_1\left(\mathbb{R}\right){★}^{\circ }{★}^{\circ }\cdots X_k\left(\mathbb{R}\right)$ (or at least its Euclidean closure) using k real channels. In this way, it is possible to determine whether or not new data coming from k complex channels belong to ghosts.

(d) Supposing that we know the dimensions and degrees (very low) of the varieties $X_1\left(\mathbb{C}\right),\dots ,X_k\left(\mathbb{C}\right)$, only very few data from k distinct (real or complex) channels (without mixing the data) are needed to reconstruct $X_1\left(\mathbb{C}\right),\dots X_k\left(\mathbb{C}\right)$, in which case we are back to (c). For instance, in order to reconstruct a line (assuming that we are looking for a line), it is sufficient to know two of its points; moreover, if the points have only a small error from the “true” point, the reconstructed line will be near the true line.

5. Conclusions

The Hadamard product of $k\ge 2$ elements of an n-dimensional (real or complex) projective space with a fixed coordinate system is the coordinate-wise product of these elements. There are k channels, each of them providing packets of $(n+1)$-ples of real (or complex) numbers. Then, some device makes the entry-wise multiplication of the k packets received by the k channels and shows it on another device, called the screen. The screen sees a sequence of $(n+1)$-ples of numbers. Now, assume that the data on the i-th channel are restricted to a set $X_i$. From $k\ge 2$ embedded varieties $X_i\subset {\mathbb{P}}^n$, we obtain an irreducible variety $X_1★\cdots ★X_k$; its definition requires it to take a closure in the Zariski topology. In Section 2 and Section 3, we prove several theorems for complex projective varieties. In the fourth section, we study the case of real algebraic varieties. Under mild assumptions, we prove that by taking real data from k channels and then multiplying them, we can reconstruct the data over both $\mathbb{R}$ and over $\mathbb{C}$. We show that this “reconstruction” works even if small errors are introduced by the channels or multiplications. If we instead start from k real varieties $X_1,\dots ,X_k$, take k channels obtaining data for the complex points of $X_1,\dots ,X_k$, and multiply to obtain elements of ${\mathbb{P}}^n\left(\mathbb{C}\right)$, then the part appearing on ${\mathbb{P}}^n\left(\mathbb{R}\right)$ contains the products of the real data. However, there are sometimes large parts, which we call real ghosts, that do not come from the real points of the varieties. We provide examples in which real ghosts are certified to occur and prove cases in which we certify that real ghosts do not occur. The latter case is related to Question 7 below.

Fix integers $k\ge 2$ and $x_i>0$, $1\le i\le k$, take $N:=k-1+x_1+\cdots +x_k$, and let $M_1,\dots ,M_k$ be a linear subspace of ${\mathbb{P}}^N\left(\mathbb{C}\right)$ such that $M_1+\cdots +M_k={\mathbb{P}}^N\left(\mathbb{C}\right)$. For any linear space $V\subset {\mathbb{P}}^N\left(\mathbb{C}\right)$, let ${\ell }_V:{\mathbb{P}}^N\left(\mathbb{C}\right)\setminus V\to {\mathbb{P}}^{N-dimV-1}$ denote the linear projection. Assume $V\cap M_i=\varnothing$ for all i (this is always satisfied if V is general and $dimV+x_i<N$ for all i); then, we obtain k linear spaces $E_i:={\ell }_V\left(M_i\right)\subset {\mathbb{P}}^{N-dimV-1}$, $1\le i\le k$.

Question 7: Find a “large” integer s such that, for a general s-dimensional linear space V, the rational map $E_1\times \cdots \times E_k⤏E_1★\cdots ★E_k$ is an isomorphism.

In the fourth section (Over the Real Numbers, subsection Full Screens), we consider a different problem. We test whether the complex images cover the full complex screen ${\mathbb{P}}^n\left(\mathbb{R}\right)$ using only the real screen ${\mathbb{P}}^n\left(\mathbb{R}\right)$. For this query, it does not matter whether there are real ghosts.