# A limit of nonplanar 5-body central configurations is nonplanar

Alain Albouy1, Antonio Carlos Fernandes2

1 IMCCE, CNRS-UMR8028, Observatoire de Paris

77, avenue Denfert-Rochereau, 75014 Paris, France

<EMAIL_ADDRESS>

2 Instituto de Matemática e Computação,

Universidade Federal de Itajubá, Minas Gerais, Brazil

<EMAIL_ADDRESS>

Abstract. Moeckel, (1990), Moeckel and Simó, (1995) proved that, while
continuously changing the masses, a 946-body planar central configuration
bifurcates into a spatial central configuration. We show that this kind of
bifurcation does not occur with 5 bodies. Question 17 in the list Albouy & al,
(2012) is thus answered negatively.

1\. Introduction

The aim of this article is to prove the following Theorem, where $n$-body
configurations are thought of as configurations of positions
$(q_{1},\dots,q_{n})\in(\mathbb{R}^{p})^{n}$ together with positive masses
$(m_{1},\dots,m_{n})\in\mathbb{R}^{n}$. The dimension of a configuration is
the dimension of the space generated by the vectors
$(q_{2}-q_{1},q_{3}-q_{1},\dots,q_{n}-q_{1})$. By “limit” we mean the limit of
a converging sequence in the space $(\mathbb{R}^{p})^{n}\times\mathbb{R}^{n}$.
Each position tends to the limiting position and each mass tends to a limiting
mass, which is assumed to be positive.

1.1. Theorem. A 5-body central configuration which is the limit of a sequence
of 5-body central configurations of dimension $d$ is of dimension $d$.

The corresponding result is true for $n\leq 4$, unknown for $6\leq n\leq 945$,
false for $n=946$ (see §4.3). Recall that a configuration of $n$ bodies with
given positive masses is central if and only if it is, for some $\lambda>0$, a
critical point of the amended Newtonian force function $U+\lambda I/2$, where

$U=\sum_{1\leq i<j\leq n}\frac{m_{i}m_{j}}{r_{ij}},\quad
I=\frac{1}{M}\sum_{1\leq i<j\leq n}m_{i}m_{j}r_{ij}^{2},$ $None$
$M=m_{1}+\cdots+m_{n},\quad r_{ij}=\|q_{i}-q_{j}\|.$

The configuration considered in the next theorem is a planar central
configuration with $n=5$ and $\lambda=M$. The choice $\lambda=M$ does not
restrict the generality. It simply scales the central configurations, in such
a way that for example the equilateral Lagrange configuration ($n=3$, $d=2$)
has its three sides equal to 1, whatever the masses.

1.2. Theorem. Consider in $\mathbb{R}^{3}$ a 5-body planar configuration with
positions $q_{i}\in\mathbb{R}^{3}$ and masses $m_{i}>0$, $i=1,\dots,5$, which
is a critical point of the function
$\sum_{i<j}m_{i}m_{j}(r_{ij}^{-1}+r_{ij}^{2}/2)$, where
$r_{ij}=\|q_{i}-q_{j}\|$. Let an orthonormal frame ${\rm O}xyz$ be such that
$q_{i}=(x_{i},y_{i},0)$. The Hessian quadratic form of this function is
strictly negative on any $(v_{1},\dots,v_{5})\in(\mathbb{R}^{3})^{5}$ with
$v_{i}=(0,0,\zeta_{i})$, where $(\zeta_{1},\dots,\zeta_{5})$ is not a linear
combination of $(x_{1},\dots,x_{5})$, $(y_{1},\dots,y_{5})$ and $(1,\dots,1)$.

Note that these excluded linear combinations are in the kernel of the Hessian
quadratic form. They generate infinitesimal translations and rotations of the
configuration. Let us compare our result with the following one.

1.3. Theorem (Moeckel, 1990). Every planar central configuration of $n\geq 4$
masses has at least one negative eigenvalue with eigenvector normal to the
manifold of planar configurations.

Let us call a $(v_{1},\dots,v_{n})$ with all the vectors $v_{i}$ orthogonal to
the configuration a vertical vector. Moeckel proved that for a planar $n$-body
central configuration there is a vertical eigenvector of the Hessian quadratic
form with a negative eigenvalue. We state in 1.2 that in the case $n=5$ there
are two independent vertical eigenvectors with negative eigenvalue. There
cannot be more than two in this case, since the maximal number of independent
vertical eigenvectors with nonzero eigenvalue is $n-3$. Theorem 1.2 together
with the following proposition gives Theorem 1.1.

1.4. Proposition. If an $n$-body central configuration of dimension $d$ is the
limit of a sequence of $n$-body central configurations of dimension $>d$, then
the Hessian quadratic form of the amended force function has a nontrivial
vertical degeneracy.

By a vertical degeneracy we mean a vertical vector in the kernel of the
Hessian quadratic form. By nontrivial we mean that this vector does not
generate a translation or a rotation. The next results are elementary
consequences of a block diagonal structure of the Hessian matrix.

1.5. Proposition. If $(u_{1},\dots,u_{n})\in(\mathbb{R}^{p})^{n}$ belongs to
the kernel of the Hessian quadratic form of the amended force function, and if
we project each $u_{i}$ on a vertical direction, we obtain a vertical vector
in this kernel.

1.6. Proposition. A vertical vector
$(u_{1},\dots,u_{n})\in(\mathbb{R}^{p})^{n}$ belongs to the kernel of the
Hessian quadratic form of the amended force function if and only if the list
of its components $(z_{1},\dots,z_{n})\in\mathbb{R}^{n}$ on any vertical
direction belongs to the kernel of the shifted Wintner-Conley matrix.

We will prove these results in reverse order in the next sections. We will
first recall the construction of the Wintner-Conley matrix and of the shifted
Wintner-Conley matrix. We will show in a future work that a nontrivial
degeneracy of the latter has surprising consequences if $d=n-3$.

2\. The Wintner-Conley matrix.

Here we prove Propositions 1.5 and 1.6 after recalling some known facts. The
Newton equations of the $n$-body problem are

$\ddot{q}_{i}=-\gamma_{i},\qquad i=1,\dots,n,$

where

$\gamma_{i}=\sum_{j\neq i}m_{j}S_{ij}(q_{i}-q_{j}),\qquad
S_{ij}=r_{ij}^{-3},\qquad r_{ij}=\|q_{i}-q_{j}\|.$

2.1. Definition. A configuration of point bodies with positive masses is
called a central configuration if there exist $\lambda\in\mathbb{R}$ (the
multiplier) and $q_{0}\in\mathbb{R}^{p}$ (the center of attraction) such that
$\lambda(q_{i}-q_{0})=\gamma_{i}$.

2.2. Proposition. In a central configuration with positive masses, the center
of attraction $q_{0}$ is the center of mass of the configuration, and the
multiplier $\lambda$ is positive.

Proof. We have $\sum m_{i}\gamma_{i}=0$ and consequently
$\lambda(q_{G}-q_{0})=0$ where $q_{G}=(\sum m_{i}q_{i})/M$ is the center of
mass. Consider a body $q_{i}$ on the boundary of the convex hull of the
configuration. The attraction vector $-\gamma_{i}$ is the sum of vectors all
pointing toward the same half-space, which contains the center of mass. So
$\lambda\neq 0$, $q_{0}=q_{G}$ and finally $\lambda>0$. QED

2.3. Proposition. A configuration with positive masses is central with
multiplier $\lambda$ if and only if it is a critical point of $U+\lambda I/2$.

Proof. From the formulas $(1.1)$ we compute the derivative of $U+\lambda I/2$
in the direction $u=(u_{1},\dots,u_{n})$:

$\langle{\rm d}(U+\lambda
I/2),(u_{1},\dots,u_{n})\rangle=-\sum_{i,j,i<j}m_{i}m_{j}(S_{ij}-{\lambda\over
M})\langle q_{i}-q_{j},u_{i}-u_{j}\rangle$ $None$ $=-\sum_{i,j,i\neq
j}m_{i}m_{j}(S_{ij}-{\lambda\over M})\langle
q_{i}-q_{j},u_{i}\rangle=-\sum_{i=1}^{n}m_{i}\langle\gamma_{i}-\lambda(q_{i}-q_{G}),u_{i}\rangle.$

This quantity should be zero for all $u$. QED

Hessian quadratic form. Let us differentiate once more and express the Hessian
bilinear form

$\langle{\rm d}^{2}(U+\lambda
I/2),(u_{1},\dots,u_{n})\otimes(v_{1},\dots,v_{n})\rangle=-\sum_{i,j,i<j}m_{i}m_{j}(S_{ij}-{\lambda\over
M})\langle u_{i}-u_{j},v_{i}-v_{j}\rangle$
$+3\sum_{i,j,i<j}m_{i}m_{j}r_{ij}^{-5}\langle
q_{i}-q_{j},u_{i}-u_{j}\rangle\langle q_{i}-q_{j},v_{i}-v_{j}\rangle$
$=\sum_{i,j,i\neq j}m_{i}m_{j}\bigl{(}-(S_{ij}-{\lambda\over M})\langle
u_{i}-u_{j},v_{i}\rangle+3r_{ij}^{-5}\langle
q_{i}-q_{j},u_{i}-u_{j}\rangle\langle q_{i}-q_{j},v_{i}\rangle\bigr{)}$
$=\sum_{i}m_{i}\langle g_{i},v_{i}\rangle\quad\hbox{with}$
$g_{i}=\sum_{j,j\neq i}m_{j}\bigl{(}-(S_{ij}-{\lambda\over
M})(u_{i}-u_{j})+3r_{ij}^{-5}\langle
q_{i}-q_{j},u_{i}-u_{j}\rangle(q_{i}-q_{j})\bigr{)}.$ $None$

The configuration is conventionally horizontal. If we take horizontal
$u_{i}$’s and vertical $v_{i}$’s the bilinear form is zero. If both the
$u_{i}$’s and the $v_{i}$’s are vertical, there is only the first term, where
there are only first derivatives of $U$. This term may be expressed with the
Wintner-Conley matrix.

The Wintner-Conley matrix $Z$ gives a matrix form to the Newton equations
$\ddot{q}_{i}=-\gamma_{i}$. It gives the $\gamma_{i}$’s as follows

$(\gamma_{1},\dots,\gamma_{n})=(q_{1},\dots,q_{n})Z$

where each $q_{i}$ and each $\gamma_{i}$ is a column vector and $Z$ is the
matrix

$Z=\pmatrix{\Sigma_{1}&-m_{1}S_{12}&-m_{1}S_{13}&\dots&-m_{1}S_{1n}\cr-
m_{2}S_{12}&\Sigma_{2}&-m_{2}S_{23}&\dots&-m_{2}S_{2n}\cr-
m_{3}S_{13}&-m_{3}S_{23}&\Sigma_{3}&\dots&-m_{3}S_{3n}\cr\dots&\dots&\dots&\dots&\dots\cr-
m_{n}S_{1n}&-m_{n}S_{2n}&-m_{n}S_{3n}&\dots&\Sigma_{n}},\qquad\Sigma_{i}=\sum_{j\neq
i}m_{j}S_{ij}.$ $None$

The equation of motion is now $\ddot{q}=-qZ$ while the equation of central
configurations is $\lambda(q-q_{G})=qZ$, with, for a configuration in
$\mathbb{R}^{3}={\rm O}xyz$,

$q=(q_{1},q_{2},\dots,q_{n})=\pmatrix{x_{1}&x_{2}&\cdots&x_{n}\cr
y_{1}&y_{2}&\cdots&y_{n}\cr z_{1}&z_{2}&\cdots&z_{n}}.$ $None$

We may remove the $q_{G}$ term by simply restricting to configurations with
the center of mass at zero. Then the equation is $\lambda q=qZ$. The rows of
the matrix $q$ are eigenvectors with eigenvalue $\lambda$ of the matrix $Z$
(conventionally acting at the right). If we replace $Z$ by $Z-\lambda{\rm
Id}$, where ${\rm Id}$ is the identity matrix, the rows of $q$ are now in the
left kernel. But $(1,1,\dots,1)$ is not in this kernel. We will prefer to
define a shifted Wintner-Conley matrix with $(1,1,\dots,1)$ in its left
kernel. The shifted matrix will then give the terms in $\lambda$ of $(2.1)$
and $(2.2)$.

2.4. Quotient space. A row of the matrix $q$ representing a configuration with
center of mass at the origin is called a disposition. It is an element of the
$n-1$ dimensional vector space of the $(x_{1},\dots,x_{n})\in\mathbb{R}^{n}$
such that $\sum m_{i}x_{i}=0$. Another definition was emphasized in Albouy and
Chenciner, (1998) and Albouy, (2003) as giving more general and simpler
equations. A disposition is a configuration $(x_{1},\dots,x_{n})$ on the real
line, considered up to translations. The space of dispositions is now a
quotient space instead of a subspace. A linear form on this space is an
invariant linear form on $\mathbb{R}^{n}$, i.e., a
$(\xi_{1},\dots,\xi_{n})\in\mathbb{R}^{n}$ such that
$\sum_{i=1}^{n}\xi_{i}=0$. We see that the columns of $Z$ satisfy this
condition. If we take the equation $\lambda q=qZ$ as the definition of a
central configuration, and if we adopt the second definition of a disposition,
then the new definition is correct for real $m_{i}$’s such
$M=m_{1}+\cdots+m_{n}=0$. In the case, there is no center of mass. For
example, an equilateral triangle is a central configuration for any
$(m_{1},m_{2},m_{3})\in\mathbb{R}^{3}$, even if $M=0$.

Instead of subtracting $\lambda{\rm Id}$ to $Z$, as Wintner indeed proposed,
we introduce another matrix which induces the identity on the disposition
space, namely, the matrix ${\cal I}$ such that $0=(1,\dots,1)\,{\cal I}$ and
such that $x=x\,{\cal I}$ for any $x=(x_{1},\dots,x_{n})$ with $\sum
m_{i}x_{i}=0$:

${\cal I}={1\over M}\pmatrix{M-m_{1}&-m_{1}&-m_{1}&\dots&-m_{1}\cr-
m_{2}&M-m_{2}&-m_{2}&\dots&-m_{2}\cr-
m_{3}&-m_{3}&M-m_{3}&\dots&-m_{3}\cr\dots&\dots&\dots&\dots&\dots\cr-
m_{n}&-m_{n}&-m_{n}&\dots&M-m_{n}},\qquad M=\sum_{i=1}^{n}m_{i}.$

2.5. Definition. The shifted Wintner-Conley matrix of a central configuration
with positive masses is the matrix ${\check{Z}}=Z-\lambda{\cal I}$.

2.6. Properties. The basic properties of ${\check{Z}}$ are the following. We
have $0=(1,\dots,1)\,{\check{Z}}$. The configuration $q$ is a central
configuration with multiplier $\lambda$ if and only if $q{\check{Z}}=0$. The
$i$-th column

$(u{\check{Z}})_{i}=\sum_{j,j\neq i}m_{j}(S_{ij}-{\lambda\over
M})(u_{i}-u_{j}),$

where $u=(u_{1},\dots,u_{n})\in(\mathbb{R}^{p})^{n}$. Let $\mu$ be the mass
matrix

$\mu=\pmatrix{m_{1}&0&\dots&0\cr 0&m_{2}&\dots&0\cr\dots&\dots&\dots&\dots\cr
0&0&\dots&m_{n}}.$

Then $Z\mu$, ${\cal I}\mu$ and ${\check{Z}}\mu$ are symmetric. Let
$v=(v_{1},\dots,v_{n})\in(\mathbb{R}^{p})^{n}$. We get

$\sum_{i,j,i<j}m_{i}m_{j}(S_{ij}-{\lambda\over M})\langle
u_{i}-u_{j},v_{i}-v_{j}\rangle={\rm\,trace\,}({}^{t}\\!vu{\check{Z}}\mu).$

Proof of Proposition 1.6. According to $(2.2)$, a vector
$u=(u_{1},\dots,u_{n})$ belongs to the kernel of the Hessian quadratic form if
and only if $g=(g_{1},\dots,g_{n})=0$. If $u$ is vertical, the second term of
$g$ vanishes. The first term $u{\check{Z}}$ should vanish. Each row of this
matrix should vanish. Assume for example a configuration $(2.4)$ in ${\rm
O}xyz$ which is 2-dimensional, conventionally in the ${\rm O}xy$ plane,
satisfying consequently $z_{1}=\cdots=z_{n}=0$. Let

$u=(u_{1},u_{2},\dots,u_{n})=\pmatrix{\xi_{1}&\xi_{2}&\cdots&\xi_{n}\cr\eta_{1}&\eta_{2}&\cdots&\eta_{n}\cr\zeta_{1}&\zeta_{2}&\cdots&\zeta_{n}}.$

The first two rows are zero, and $(\zeta_{1},\dots,\zeta_{n})$ is a
disposition in the left kernel of ${\check{Z}}$. If the planar configuration
was in $\mathbb{R}^{4}$ instead of $\mathbb{R}^{3}$, the fourth coordinate
would produce another disposition in the same kernel. QED

Proof of Proposition 1.5. Assume that a vector $u=(u_{1},\dots,u_{n})$ is in
the kernel of the Hessian quadratic form. It satisfies
$g=(g_{1},\dots,g_{n})=0$. Let us take the same 3-dimensional example as in
the previous proof. The projection of $u$ on the vertical direction ${\rm O}z$
is

$\hat{u}=\pmatrix{0&0&\cdots&0\cr
0&0&\cdots&0\cr\zeta_{1}&\zeta_{2}&\cdots&\zeta_{n}}.$

If we replace $u$ by $\hat{u}$ in $(2.2)$, we find again $g_{i}=0$, since the
$x$ and $y$ coordinates of $g_{i}$ are sum of two zero terms, and the $z$
coordinates are unchanged. QED

A vector $u$ in the kernel of the Hessian quadratic form gives $\hat{u}$ in
the same kernel, and $\zeta=(\zeta_{1},\dots,\zeta_{n})$ in the left kernel of
${\check{Z}}$. If $\hat{u}$ is nontrivial in the sense of §1, then a $\zeta$
is nontrivial in the sense that it is not a linear combination of the known
vectors in the left kernel of ${\check{Z}}$. We may restate this fact as
follows.

2.7. Proposition. A central configuration $q$ satisfies ${\rm
rank\,}{\check{Z}}+{\rm rank\,}q\leq n-1$. It has a nontrivial vertical
degeneracy if and only if ${\rm rank\,}{\check{Z}}+{\rm rank\,}q\leq n-2$.

Proof. In the left kernel of ${\check{Z}}$ we find the vector $(1,\dots,1)$
and, since $q{\check{Z}}=0$, each row of the matrix $q$, which gives the first
inequality. According to $(2.2)$, $-{\check{Z}}\mu$ is the vertical Hessian
quadratic form. The vector $(1,\dots,1)$ generates an infinitesimal
translation of the configuration. The rows of $q$ generate infinitesimal
rotations. If there are other vectors in the left kernel of ${\check{Z}}$,
there are other degeneracies of the vertical Hessian quadratic form: these are
nontrivial vertical degeneracies. QED

3\. Changes of dimension. Two proofs of Proposition 1.4.

Short proof of Proposition 1.4. We consider a sequence $q^{\kappa}$,
$\kappa\in\mathbb{N}$, of configurations of dimension $>d$, which converges to
a $d$-dimensional configuration $q^{\infty}$, and a sequence of mass vectors
$m^{\kappa}$, which converges to a mass vector $m^{\infty}$. These
configurations are central. In particular, they do not have collisions and the
mass vectors have positive entries. According to 2.6,
$q^{\kappa}{\check{Z}}^{\kappa}=0$ and $q^{\infty}{\check{Z}}^{\infty}=0$. The
rank is lower semi-continuous. The rank of ${\check{Z}}$ cannot increase in
the limit. According to Proposition 2.7, ${\check{Z}}^{\infty}$ has a
nontrivial kernel, i.e., the central configuration has a vertical degeneracy.
QED

Constructive argument. The previous argument does not tell what is the
nontrivial kernel. Clearly, it is associated to a deformation of the
configuration which changes the dimension.

Here is an argument which gives a vector in the kernel. Again, we consider a
sequence $q^{\kappa}$, $\kappa\in\mathbb{N}$, of configurations in a space of
dimension $>d$, which converges to a $d$-dimensional configuration
$q^{\infty}$. We renumber the bodies such that the $d+1$ first bodies of
$q^{\infty}$ form a nonflat simplex (of dimension $d$). Then by a small
rotation and a small translation of each configuration $q^{\kappa}$ we obtain
another sequence $\hat{q}^{\kappa}$ which also converges to $q^{\infty}$ and
has the following property: (i) for any $\kappa\in\mathbb{N}$, for any
$i=1,\dots,d+1$, the first $i$ bodies of $\hat{q}^{\kappa}$ generate the same
affine subspace as the first $i$ bodies of $q^{\infty}$. So, whatever
$\kappa\in\mathbb{N}$, $\hat{q}_{1}^{\kappa}=q^{\infty}_{1}$,
$\hat{q}_{2}^{\kappa}$ is on the line $q^{\infty}_{1}q^{\infty}_{2}$,
$\hat{q}^{\kappa}_{3}$ is on the plane
$q^{\infty}_{1}q^{\infty}_{2}q^{\infty}_{3}$, etc. The bodies
$\hat{q}_{1}^{\kappa},\dots,\hat{q}_{d+1}^{\kappa}$ are in the “horizontal”
space. All their “vertical” coordinates are zero. The vertical coordinates of
the other bodies converge to zero.

Assume now that the configurations $q^{\kappa}$ and $q^{\infty}$ are all
central. There is now a sequence of mass vectors $m^{\kappa}$. Our hypothesis
is that for any $i$, $i=1,\dots,n$, $m_{i}^{\kappa}\to m_{i}^{\infty}>0$.
Consequently also the multipliers $\lambda^{\kappa}\to\lambda^{\infty}>0$, and
the total mass $M^{\kappa}\to M^{\infty}>0$. We rescale to whole sequence in
order to get $\lambda^{\infty}=M^{\infty}$. By slightly rescaling again each
$q^{\kappa}$ we get this second property: (ii) for any $\kappa\in\mathbb{N}$,
$\lambda^{\kappa}=M^{\kappa}$.

Applying both the small rotation and the change of scale, we obtain from the
sequence $q^{\kappa}$ another sequence with the properties (i) and (ii). In
what follows we will assume, without loss of generality, that the sequence
$q^{\kappa}$ has the above properties (i) and (ii).

The hypothesis of Proposition 1.4 is: for any $\kappa$, $\dim q^{\kappa}>d$.
This hypothesis implies that after renumbering the $n-d-1$ last bodies, at
least one of the vertical coordinates of $q_{n}^{\kappa}$ takes infinitely
many values when $\kappa\to+\infty$. Let us call $t$ such a coordinate. It
will be the coordinate $x_{l}$ of the following proposition, in the case
$x_{l}(t)=t$.

3.1. Proposition (Hampton and Moeckel, 2006). Let
$\mathbb{T}=(\mathbb{C}\setminus\\{0\\})^{r}$. Suppose that a system of $s$
polynomial equations $f_{i}(x)=0$ defines an infinite variety
$V\subset\mathbb{T}$. Then there is a nonzero rational vector
$\alpha=(\alpha_{1},\dots,\alpha_{r})$, a point
$a=(a_{1},\dots,a_{r})\in\mathbb{T}$, and Puiseux series
$x_{j}(t)=a_{j}t^{\alpha_{j}}+\cdots$, $j=1,\dots,r$, convergent in some
punctured neighborhood $U$ of $t=0$, such that
$f_{i}(x_{1}(t),\dots,x_{r}(t))=0$ in $U$, $i=1,\dots,s$. Moreover, if the
projection from $V$ onto the $x_{l}$-axis is dominant, there exists such a
series solution with $x_{l}(t)=t$ and another with $x_{l}(t)=t^{-1}$.

The usual existence results of Puiseux series are too weak for our purpose,
while this general proposition is exactly what we need. We apply it to the
system $q{\check{Z}}=0$ which characterizes the central configurations and
which is also: $\sum_{j\neq i}m_{j}(S_{ij}-1)(q_{i}-q_{j})=0$, for any
$i=1,\dots,n$. Recall that $S_{ij}=\|q_{i}-q_{j}\|^{-3}$. The system will be
polynomial if the variables are the vectors $q_{1},\dots,q_{n}$, the numbers
$m_{1},\dots,m_{n}$ and the numbers $S_{ij}$, $1\leq i<j\leq n$. We add to the
system defining the central configuration the polynomial constraints
$S_{ij}^{2}\|q_{i}-q_{j}\|^{6}=1$.

We consider a neighborhood of the limiting $d$-dimensional central
configuration $q^{\infty}$. We consider as the new variables, suitable with
Proposition 3.1, the increments $q_{i}-q^{\infty}_{i}$,
$m_{i}-m^{\infty}_{i}$, $S_{ij}-S_{ij}^{\infty}$ of the old variables. We
restrict the increments of the $q_{i}$’s, $i=1,\dots,d+1$, in agreement with
property (i). The system is again polynomial in these new variables. In order
to agree with the hypothesis $V\subset\mathbb{T}$, we remove from the list of
variables any scalar increment which is identically zero. Then, we can apply
3.1 to the new list of variables.

3.2. Proposition. If an $n$-body central configuration $q^{\infty}$ of
dimension $d$ is the limit of a sequence of $n$-body central configurations of
dimension $>d$, then, after renumbering the bodies, there is a Taylor series
in $\varepsilon\in\mathbb{C}$ converging near $\varepsilon=0$ and
parametrizing central configurations by expressing the configuration matrix as
$q=q^{\infty}+\varepsilon q^{[1]}+\varepsilon^{2}q^{[2]}+\cdots$ and the mass
vector as $m=m^{\infty}+\varepsilon m^{[1]}+\varepsilon^{2}m^{[2]}+\cdots$.
Moreover, the $q_{i}$’s, $i=1,\dots,d+1$, generate a $d$-dimensional affine
space which is independent of the value of $\varepsilon$, which we call the
horizontal space. One of the vertical coordinates of $q_{n}$ is
$\varepsilon^{k}$, for some positive integer number $k$.

Note that Proposition 3.1 states the existence of a Puiseux series while
Proposition 3.2 states the existence of a Taylor series. Indeed the variable
of 3.2 is $\varepsilon=t^{1/k}$ where $t$ is given in 3.1. Moreover, there are
no negative powers in the series since we are dealing with increments which
are small as $\varepsilon\to 0$.

Constructive proof of Proposition 1.4. We expand the equation of central
configurations until the first order $i$ in $\varepsilon$ such that $q^{[i]}$
is not horizontal (the case $i=1$ will reprove 1.5). We write

$0=q{\check{Z}}=(q^{\infty}+\varepsilon
q^{[1]}+\varepsilon^{2}q^{[2]}+\cdots)({\check{Z}}^{\infty}+\varepsilon{\check{Z}}^{[1]}+\varepsilon^{2}{\check{Z}}^{[2]}+\cdots).$

Call $z=(z_{1},\dots,z_{n})\in\mathbb{R}^{n}$ the list of vertical coordinates
of $q=(q_{1},\dots,q_{n})$ that is nonzero at order $i$:
$z^{\infty}=z^{[1]}=\cdots=z^{[i-1]}=0$, $z^{[i]}\neq 0$. The above equation
is true after replacing the matrix $q$ by its row $z$. Consequently
$z^{[i]}{\check{Z}}^{\infty}=0$. Note that Proposition 3.1 does not discuss
the existence of a real branch, so $z^{[i]}$ may not be real. But then its
real part and its imaginary part are both in the left kernel of the real
matrix ${\check{Z}}^{\infty}$. So, we have a vertical degeneracy. Property (i)
as restated in Proposition 3.2 implies that $z_{1}=\cdots=z_{d+1}$ at any
order in $\varepsilon$. If $z^{[i]}$ was a linear combination of the lists of
horizontal coordinates of $q^{\infty}$, the vectors
$q_{j}^{\infty}-q_{1}^{\infty}$, $j=2,\dots,d+1$, would be linearly dependent.
But we assumed the contrary. The vertical degeneracy is nontrivial. QED

4\. Context of Theorem 1.2

We first recall two inequalities which are used to determine a sign in
Dziobek’s formulas. We consider the matrix $Z=(Z_{ij})_{1\leq i,j\leq n}$ of
$(2.3)$ and the matrix $Z-\lambda{\cal
I}={\check{Z}}=({\check{Z}}_{ij})_{1\leq i,j\leq n}$.

4.1. Proposition (Albouy, 2003). Consider a central configuration with
positive masses. Then for any $i$, $j$, $i\neq j$,
${\check{Z}}_{ij}+{\check{Z}}_{ji}\leq{\check{Z}}_{ii}+{\check{Z}}_{jj}$, with
equality if and only if all the other bodies are equidistant from bodies
$q_{i}$ and $q_{j}$.

We will not reproduce the proof. Here are alternative expressions. The
nonnegative quantity
${\check{Z}}_{ii}+{\check{Z}}_{jj}-{\check{Z}}_{ij}-{\check{Z}}_{ji}=Z_{ii}+Z_{jj}-Z_{ij}-Z_{ji}-2\lambda$.
It is also, with the notation $\Sigma_{i}$ of $(2.3)$,

$\Sigma_{i}+\Sigma_{j}+(m_{i}+m_{j})S_{ij}-2\lambda=2(m_{i}+m_{j})S_{ij}+\sum_{k\neq
i,j}m_{k}(S_{ik}+S_{jk})-2\lambda,$

or finally, with the notation ${\check{S}}_{ij}=S_{ij}-\lambda/M$,
$2(m_{i}+m_{j}){\check{S}}_{ij}+\sum_{k\neq
i,j}m_{k}({\check{S}}_{ik}+{\check{S}}_{jk})$. Moeckel deduced his Theorem 1.3
from the following proposition.

4.2. Proposition (Moeckel, 1990). For any central configuration with positive
masses, ${\rm\,trace\,}{\check{Z}}\geq 0$. The equality only occurs for the
equilateral simplex, for which ${\check{Z}}=0$.

Proof. This is a corollary of Proposition 4.1. We know that
$\sum_{i}{\check{Z}}_{ij}=0$. Consequently $-{\rm\,trace\,}{\check{Z}}$ is the
sum of all the nondiagonal entries of ${\check{Z}}$. Summing up the inequality
of 4.1 for all $i$, $j$, $i<j$, we find
$-{\rm\,trace\,}{\check{Z}}\leq(n-1){\rm\,trace\,}{\check{Z}}$. QED

4.3. Previous works. Wintner, (1941) introduces the shifted matrix
$\Gamma=\lambda{\rm Id}-Z$ at $\S 356$. He expresses the equation of central
configurations as we did. He writes “Actually, $r$ is always precisely the
multiplicity of the root $0$ of the characteristic equation of $\Gamma$.”
There is a misprint since Wintner defined $r$ as the rank of $\Gamma$. Albouy,
(2003) claims that Wintner was stating the impossibility of a “vertical
degeneracy”. This was indeed assuming that the misprinted $r$ was the rank of
the other matrix, which represents the configuration. But more probably the
misprinted $r$ is simply $n-r$ and Wintner’s statement is trivial. This
paragraph is somewhat neglected by Wintner, which is consistent with his
conclusion: “Nevertheless, the characterization of central configurations in
terms of the matrix $\Gamma$ is often unmanageable, and will not be used in
what follows.” However, in Pacella, (1987), Albouy and Chenciner, (1998) and
in what follows, several results are deduced from this characterization.

This short paragraph by Wintner may be inspired by the long article Meyer,
(1933), where the eigenvectors of the same matrix are discussed. At §24, on
page 152, Meyer claims to prove that for a collinear central configuration of
$n$ bodies, the matrix $Z$, which he writes on page 133, has $n-2$ eigenvalues
strictly greater than $\lambda$. The vertical degeneracy of a collinear
central configuration is impossible. These two results are true, as later
proved by Conley and published in Pacella, (1987). Meyer’s argument is the
last paragraph of his §24. He translates the lower rank possibility into an
equivalent statement about the kernel, which he expresses through the
Krediet–Laura–Andoyer equations111Krediet, (1892) seems to have been the first
author to publish the so-called Andoyer equations for an arbitrary number of
bodies in the plane. See his equation (8).. But he does not explain why this
should be “impossible”. In Palmore, (1982), the same claim is made, also with
unclear arguments “by comparing the terms”.

Previously Saari, (1980) had sketched an argument that would discard the
vertical degeneracy in any dimension. Moeckel, (1990) showed a counter
example, a configuration made of two regular polygons of 473 bodies each.
Moeckel and Simó, (1995) further describe this configuration. Note that
bifurcations of 5 bodies configurations are common in the plane (see Meyer and
Schmidt, 1988 or Xia, 1991) and in 3 dimensions (see Alvarez-Ramírez & al,
2016, Santos & al, 2017).

Flat Dziobek configurations. A Dziobek configuration is, according to the most
usual definition, an $n$-body central configuration of dimension exactly
$n-2$. The shifted Wintner-Conley matrix ${\check{Z}}$ is of rank one. Our
Theorem 1.2 claims that for $n=5$, the reciprocal statement is true: if
${\check{Z}}$ is of rank one, the dimension of the central configuration is 3.
For $n\geq 6$, we do not know if there are flat Dziobek configurations, i.e.,
central configurations of dimension $n-3$ such that ${\rm
rank\,}{\check{Z}}=1$. They would be central configurations with a vertical
degeneracy. Note that Albouy, (2003) extends the usual definition of a Dziobek
configuration: They are defined by the condition ${\rm rank\,}{\check{Z}}=1$.
With this definition, flat Dziobek configurations are Dziobek configurations,
while with the most usual definition, they are not. See also the remark of §7
in this reference, and note an inconsistency concerning this definition: the
proof of Proposition 12 indeed assumes the usual definition.

4.4. Proposition. A configuration with positive masses is a central
configuration satisfying ${\rm rank\,}{\check{Z}}=1$ if and only if there is a
$\lambda\in\mathbb{R}$, a nonzero
$(\Delta_{1},\dots,\Delta_{n})\in\mathbb{R}^{n}$ such that $\sum\Delta_{i}=0$,
$\sum\Delta_{i}q_{i}=0$, and, for all $i$ and $j$, $i<j$,
$m_{i}m_{j}(S_{ij}-\lambda/M)=-\Delta_{i}\Delta_{j}$.

Proof. As the matrix ${\check{Z}}\mu$ is symmetric of rank one, it is of the
form $\pm(\Delta_{i}\Delta_{j})_{i,j}$. As $(1,\dots,1){\check{Z}}=0$,
$\sum_{i}\Delta_{i}=0$. The condition for a central configuration is
$q{\check{Z}}=0$, which is equivalent to $\sum_{i}\Delta_{i}q_{i}=0$.
According to Proposition 4.2, the $\pm$ sign is $+$. QED

4.5. Proposition (Dziobek, 1900; Schmidt, 1988). An $n$-body configuration of
dimension exactly $n-2$, with positive masses, is central if and only if there
is a $\lambda\in\mathbb{R}$, a nonzero $(\Delta_{1},\dots,\Delta_{n})$ such
that $\sum\Delta_{i}=0$, $\sum\Delta_{i}q_{i}=0$, and, for all $i$ and $j$,
$m_{i}m_{j}(S_{ij}-\lambda/M)=-\Delta_{i}\Delta_{j}$.

Proof. We have $q$ and $(1,\dots,1)$ in the left kernel, so ${\check{Z}}$ is
of rank one. We conclude by Proposition 4.4. QED

Remark. If the dimension of a configuration is exactly $n-2$, the first two
conditions $\sum\Delta_{i}=0$, $\sum\Delta_{i}q_{i}=0$ determine uniquely, up
to a factor, a nonzero $(\Delta_{1},\dots,\Delta_{n})$. We call the
$\Delta_{i}$’s the homogeneous barycentric coordinates of the configuration.
Here the choice of factor is removed by the third equation: the $\Delta_{i}$’s
are uniquely determined.

Remark. This proof of Dziobek’s main formula is really short. The idea is from
Albouy, (1997). Its simplicity contradicts Wintner’s opinion (see 4.3). The
original proof by Dziobek relies on an identity involving Cayley’s
determinant, which is often considered as difficult (Saari, 2005). Dziobek
indeed made a sign mistake (Cors and Roberts, 2012). Brehm, (1908), Meyer,
(1933), Wintner, (1941) and Meyer and Schmidt, (1988) praised Dziobek’s work,
but they did not write his identity nor his main formula. Schmidt, (1988)
writes them and extends them to the case of a 3-dimensional 5-body
configuration. Moeckel, (2001) gives a simple proof of the extended Dziobek
identity together with a very elementary way to prove Dziobek’s main formula
without Cayley determinant (the same as advertized later in Albouy, 2003).

Remark. If ${\rm rank\,}{\check{Z}}=1$ and if $S_{ij}=\lambda/M$, then
$\Delta_{i}=0$ or $\Delta_{j}=0$ and consequently at least $n-1$ of the mutual
distances are $(\lambda/M)^{-1/3}$. The same relation $S_{ij}=\lambda/M$ is
still remarkable if ${\rm rank\,}{\check{Z}}>1$, but the conclusions are not
as strong. Gidea and Llibre, (2010), Chen and Hsiao, (2012) independently
discovered a one-parameter family of 5-body planar central configurations with
$q_{2}-q_{1}=q_{3}-q_{2}$ and $S_{24}=S_{25}=\lambda/M$.

4.6. Proposition (Albouy, 2003). In a central configuration
$q_{1},\dots,q_{n}$ with ${\rm rank\,}{\check{Z}}=1$ satisfying
$\Delta_{n}=0$, the bodies $q_{1},\dots,q_{n-1}$ are equidistant from $q_{n}$
and they form a central configuration with ${\rm rank\,}{\check{Z}}=1$.

Proof. We simply make $\Delta_{n}=0$ in the equations of Proposition 4.4. We
get the same equations with body $n$ ignored and, for all $i\leq n-1$,
$S_{in}=\lambda/M$. QED

The normalized multiplier $\lambda/M$ of the $n-1$-body central configuration
should be the normalized multiplier of the $n$-body central configuration
since in both cases, if $1\leq i<j\leq n-1$,
$m_{i}m_{j}(S_{ij}-\lambda/M)=-\Delta_{i}\Delta_{i}$. Note that $M$ is
different in both cases. Such configurations do exist for 5 bodies in
dimension 3. They were introduced by Erich Ludwig Brehm. This reciprocal
statement may be extracted from Brehm’s formulas:

4.7. Proposition (Brehm, 1908, p. 28). Given any 4-body cocircular central
configuration $(q_{1},\dots,q_{4})$ with positive masses $(m_{1},\dots,m_{4})$
and a mass $m_{5}>0$, there exists, up to reflection, a unique nonplanar
5-body central configuration $(q_{1},\dots,q_{5})$ with same
$q_{1},\dots,q_{4}$ and with masses $(m_{1},\dots,m_{5})$. In this
configuration, $r_{15}=r_{25}=r_{35}=r_{45}$. This common distance does not
depend on $m_{5}$. The height $h$ of the resulting pyramid satisfies
$r_{15}\sqrt{2}/2<h<r_{15}\sqrt{3}/2$.

Proof. Let us first see the uniqueness. A 3-dimensional 5-body central
configuration satisfies the conditions of Proposition 4.5. The first four
bodies generate an affine plane, so they have homogeneous barycentric
coordinates, i.e., there is a $(\Delta_{1},\dots,\Delta_{4})\neq 0$ such that
$\sum_{i=1}^{4}\Delta_{i}=0$ and $\sum_{i=1}^{4}\Delta_{i}q_{i}=0$. Joining
$\Delta_{5}=0$, we get the homogeneous barycentric coordinates
$(\Delta_{1},\dots,\Delta_{5})$ of the 5-body configuration. As
$\Delta_{5}=0$, for $i<5$, $S_{i5}=r_{i5}^{-3}=\lambda/M$. So the pyramid is
unique. About the existence, note that if the distance
$r_{15}=(\lambda/M)^{-1/3}$ was too small, the pyramid would be impossible. So
the existence is a consequence of a lower bound on $r_{15}$ or on the height
$h$. A cocircular configuration is convex. Assume the cyclic order $1234$. As
explained by Dziobek, $\Delta_{1},\Delta_{3}<0<\Delta_{2},\Delta_{4}$. Then
$m_{1}m_{3}(r_{13}^{-3}-r_{15}^{-3})=-\Delta_{1}\Delta_{3}<0$ and
$m_{1}m_{2}(r_{12}^{-3}-r_{15}^{-3})=-\Delta_{1}\Delta_{2}>0$, so
$r_{12}<r_{15}<r_{13}$. The length of a side of the convex quadrilateral is
smaller than $r_{15}$, which is smaller than the length of a diagonal. If the
radius of the circle is $r$, the greatest side should be greater than
$\sqrt{2}r$, and the diagonals smaller than $2r$. So, $\sqrt{2}r<r_{15}<2r$,
or $2(r_{15}^{2}-h^{2})<r_{15}^{2}<4(r_{15}^{2}-h^{2})$, which gives Brehm’s
inequalities. In another way, $r<h<\sqrt{3}r$. QED

4.8. Proposition (Alvarez-Ramírez & al, 2013). A flat Dziobek 5-body central
configuration with positive masses and with $\Delta_{5}=0$ is impossible.

Proof. Proposition 4.4 gives us the $\Delta_{i}$’s. Then we can repeat the
argument in the proof of 4.7. The planar configuration would have $h=0$, which
is excluded by the inequalities. This is a proof. Let us give a second proof.
Proposition 4.6 shows that the configuration has four bodies on a circle and
the fifth body at the center. Theorem 7 of Alvarez-Ramírez & al, (2013) states
that for such configuration ${\rm rank\,}{\check{Z}}=2$. QED

5\. Proof of Theorem 1.2.

Idea of the proof. We will show a contradiction between the equations
$\sum\Delta_{i}=0$, $\sum\Delta_{i}q_{i}=0$, which constrain
$(\Delta_{1},\dots,\Delta_{5})$, and Dziobek’s equations
$S_{ij}=z-\delta_{i}\delta_{j}$, $1\leq i<j\leq 5$, where
$\delta_{i}=\Delta_{i}/m_{i}$. In other words, we will find contradictions
between the “barycentric point of view” (we call the $\Delta_{i}$’s the
homogeneous barycentric coordinates of the $q_{i}$’s) and the “Distance-
Dziobek point of view”.

A well-known example of such a contradiction proves the statement: A
2-dimensional 4-body central configuration cannot have three bodies on a line.
By Proposition 4.5 there would be a $(\Delta_{1},\dots,\Delta_{4})$ with
$\sum\Delta_{i}=0$, $\sum\Delta_{i}q_{i}=0$, $S_{ij}=z-\delta_{i}\delta_{j}$.
If the three points $q_{1}$, $q_{2}$, $q_{3}$ are on a line, $\Delta_{4}=0$
according to the first two equations. But $\delta_{4}=0$ and
$S_{ij}=z-\delta_{i}\delta_{j}$ implies that $q_{1}$, $q_{2}$, $q_{3}$ are on
a circle (this is indeed a case of Proposition 4.6). This is a contradiction:
since $q_{1}$, $q_{2}$, $q_{3}$ are distinct points, they cannot be on a
circle and on a line. The same kind of contradiction allows to prove the
symmetry of the equal mass 4-body central configurations (see Albouy, 1995).

Barycentric point of view. We shall describe the possible
$(\Delta_{1},\dots,\Delta_{5})$ for a planar 5-body configuration. They form a
plane $\Pi\subset\mathbb{R}^{5}$ defined by the three independent linear
equations $\sum\Delta_{i}=0$, $\sum\Delta_{i}x_{i}=0$,
$\sum\Delta_{i}y_{i}=0$. We shall describe all the possible planes $\Pi$ and
all the possible combinations of signs of the $\Delta_{i}$’s on each plane.

If we had $\Delta_{1}=0$ on all a plane $\Pi$, there would be a plane of
homogeneous barycentric coordinates $(\Delta_{2},\dots,\Delta_{5})$ of bodies
2 to 5. These bodies would consequently be collinear. This is impossible in a
2-dimensional 5-body central configuration, according to the perpendicular
bisector theorem. So $\Delta_{1}=0$ defines a vectorial line in $\Pi$. The
plane $\Pi$ is divided into sectors by the conditions $\Delta_{i}=0$,
$i=1,\dots,5$. The question is to determine in which cyclic order these
conditions happen. Two lines $\Delta_{i}=0$ and $\Delta_{j}=0$ may coincide,
as we will see in Table B. A coincidence of three lines corresponds to a
collision $q_{i}=q_{j}$. But collisions are excluded in our problem.

Figure 1. Barycentric signs

When for example $\Delta_{1}=0$, we observe the configuration formed by bodies
$2345$. As $(\Delta_{2},\dots,\Delta_{5})$ are homogeneous barycentric
coordinates of this configuration, the $\Delta_{i}$’s should have consistent
signs, illustrated in Figure 1. If the quadrilateral is not convex, the
exterior signs are the same and they differ from the interior sign: the
interior body has positive barycentric coordinates in the barycentric frame of
the exterior bodies. If the quadrilateral is convex, the signs should
alternate cyclically (compare Saari, 2005, §3.3).

$\matrix{1&2&3&4&5\cr
0&+&-&+&-\cr+&+&-&+&-\cr+&0&-&+&-\cr+&-&-&+&-\cr+&-&0&+&-\cr+&-&+&+&-\cr+&-&+&0&-\cr+&-&+&-&-\cr+&-&+&-&0\cr+&-&+&-&+\cr
0&-&+&-&+\cr}\quad\left|\quad\matrix{1&2&3&4&5\cr
0&+&-&+&-\cr+&+&-&+&-\cr+&0&-&+&-\cr+&-&-&+&-\cr+&-&0&+&-\cr+&-&+&+&-\cr+&-&+&+&0\cr+&-&+&+&+\cr+&-&+&0&+\cr+&-&+&-&+\cr
0&-&+&-&+\cr}\quad\right|\quad\matrix{1&2&3&4&5\cr
0&+&-&+&-\cr+&+&-&+&-\cr+&+&0&+&-\cr+&+&+&+&-\cr+&0&+&+&-\cr+&-&+&+&-\cr+&-&+&+&0\cr+&-&+&+&+\cr+&-&+&0&+\cr+&-&+&-&+\cr
0&-&+&-&+\cr}$

Table A

$\matrix{1&2&3&4&5\cr
0&+&-&+&-\cr+&+&-&+&-\cr+&0&-&+&-\cr+&-&-&+&-\cr+&-&0&+&-\cr+&-&+&+&-\cr+&-&+&0&0\cr+&-&+&-&+\cr
0&-&+&-&+}\quad\left|\quad\matrix{1&2&3&4&5\cr
0&+&-&+&-\cr+&+&-&+&-\cr+&0&0&+&-\cr+&-&+&+&-\cr+&-&+&+&0\cr+&-&+&+&+\cr+&-&+&0&+\cr+&-&+&-&+\cr
0&-&+&-&+}\right.$

Table B

Suppose first that the 5-body configuration is convex, in cyclic order 12345.
If $\Delta_{1}=0$ then $(\Delta_{2},\dots,\Delta_{5})$ are homogeneous
barycentric coordinates of bodies 2 to 5, which form a convex configuration in
the cyclic order 2345. Consequently their signs are $+-+-$ or $-+-+$. These
two choices are equivalent. We write these signs as the first row of Table A1.
The following reasoning gives the other rows of Table A1. The second row is a
small perturbation of the condition $\Delta_{i}=0$, which gives $\Delta_{i}>0$
and does not change the other signs. The case $\Delta_{i}<0$ would give the
same table in reverse order. In the third row the only choice is a 0 at body
2, since only this choice gives correct signs $+-+-$ for the remaining bodies,
which also form a convex quadrilateral. We check at each step that this
reasoning gives a unique choice for the next vanishing $\Delta_{i}$. The last
row is the first row with all the signs changed: by continuing we would follow
the same sequence again with all the signs changed.

In Table B1 we assume that bodies 123 are on a line in this order, 1345 being
convex in this cyclic order. In Table A2, body 2 passes slightly inside. As
$\Delta_{5}=0$, we find the signs $+-++$ of a concave configuration. In Table
B2, we push body 5 on the side 14. We see that the cases with three collinear
bodies are obtained from a generic case by suppression of an intermediate
step. The allowed nongeneric cases do not produce new combinations of five
signs compared to the generic cases. The last type of generic case is
described as the configuration in Table A3, a triangle with 134 outside, and
25 inside, such that 2345 is convex in this cyclic order. The other two
quadrilaterals which include 2 and 5 are concave.

Remark. Esther Klein remarked in 1933 that by extracting a chosen point of a
5-point planar configuration one can leave a convex quadrilateral. She assumed
that 3 points are not on a same line. The question was first extended in Erdös
and Szekeres, (1935).

Distance-Dziobek point of view. We have obtained all the combinations of signs
of the $\Delta_{i}$’s. We will now see their incompatibilities with Dziobek’s
equations $S_{ij}=z-\delta_{i}\delta_{j}$. The method is inspired from
Fernandes & al, (2017). We recall our notation: $r_{ij}=\|q_{i}-q_{j}\|$ and
$S_{ij}=r_{ij}^{-3}$, $\delta_{i}=\Delta_{i}/m_{i}$.

5.1. Proposition. A convex planar configuration of 4 points
$(q_{1},q_{2},q_{3},q_{4})$ in this cyclic order, such that there exists $z>0$
and $(\delta_{1},\delta_{2},\delta_{3},\delta_{4})$ with signs $++--$, with
$S_{ij}=z-\delta_{i}\delta_{j}$, is impossible.

Proof. The $r_{ij}$ are ordered as the $\delta_{i}\delta_{j}$. There would be
two large distances $r_{12}$ and $r_{34}$, corresponding to opposite sides.
Let the smallest of both be $r_{12}$. According to the four small distances
$r_{13}$, $r_{23}$, $r_{14}$, $r_{24}$, and to the cyclic order, bodies 3 and
4 should be both in an ogive (see Figure 2) delimited by the segment 12 and by
two circles, centered respectively in bodies 1 and 2, passing respectively
through 2 and 1. But the greatest distance in the ogive is $r_{12}$. The
largest side 34 does not fit in the ogive. Contradiction. QED

Figure 2. Proofs of 5.1 and 5.2

5.2. Proposition. A concave planar configuration of 4 points
$(q_{1},q_{2},q_{3},q_{4})$, the fourth being inside the triangle of the first
three, such that there exists $z>0$ and
$(\delta_{1},\delta_{2},\delta_{3},\delta_{4})$ with signs $-+++$, with
$S_{ij}=z-\delta_{i}\delta_{j}$, is impossible.

Proof. Due to the concavity hypothesis, $q_{1}$ is “behind” (see Figure 2) the
vertex $q_{4}$ of the triangle $q_{2}q_{3}q_{4}$. But $r_{12}$ and $r_{13}$
are “small” distances, since $\delta_{1}\delta_{2}$ and $\delta_{1}\delta_{3}$
are negative, while $r_{24}$ and $r_{34}$ are “great” distances, since
$\delta_{2}\delta_{4}$ and $\delta_{3}\delta_{4}$ are positive. This is a
contradiction. More precisely, we may observe that
$r_{24}+r_{34}<r_{12}+r_{13}$ since the ellipse with foci $q_{2}$ and $q_{3}$,
passing through $q_{4}$, is bisecting the angle 243. QED

5.3. Proposition. If a convex planar configuration of 4 points
$(q_{1},q_{2},q_{3},q_{4})$ is such that there exists $z>0$ and
$(\delta_{1},\delta_{2},\delta_{3},\delta_{4})$ such that
$0<\delta_{4}\leq\delta_{3}\leq\delta_{2}\leq\delta_{1}$ and that
$S_{ij}=z-\delta_{i}\delta_{j}$, then the cyclic orders may be 1432 or 1324.
In the first case, the angle $132$ is acute and we have the estimate
$2r_{34}<r_{12}$. In the second case, the angle $132$ is obtuse and
$\sqrt{3}r_{34}\leq r_{12}$.

Proof. We choose coordinates ${\rm O}xy$ with $q_{2}=(-1,0)$ and
$q_{1}=(1,0)$. The $r_{ij}$ are ordered as the $\delta_{i}\delta_{j}$:

$r_{34}\leq r_{24}\leq(r_{14}\hbox{ or }r_{23})\leq r_{13}\leq r_{12}.$

The point $q_{3}=(x_{3},y_{3})$ is in the half-ogive of Figure 3, with
$x_{3}\leq 0$ and $y_{3}\geq 0$. We draw the circle of center $q_{2}$ passing
through $q_{3}$ and the level curve of

$f(x,y)=\bigl{(}(x+1)^{2}+y^{2}\bigr{)}^{-3/2}-\bigl{(}(x-1)^{2}+y^{2}\bigr{)}^{-3/2}$

passing through $q_{3}$. We have $f(x_{3},y_{3})\geq 0$. For any positive
value of $f$ the level curve is a closed curve on the left half-plane. The
only intersection of the level curve and the circle in the upper semi-plane is
$q_{3}$. Here, the value of $f$ on a point moving on the circle
counterclockwise from the ${\rm O}x$ axis to the ${\rm O}x$ axis is
increasing.

Figure 3. Proof of 5.3, first case

First case. The point $q_{4}$ is in the upper half plane. If
$\delta_{1}=\delta_{2}$, then $x_{3}=x_{4}=0$ and the configuration is not
convex. So $\delta_{2}<\delta_{1}$. The point $q_{4}$ should be in the same
half-ogive as $q_{3}$, inside the circle (since $r_{24}<r_{23}$) and outside
the level curve since

${S_{24}-S_{14}\over S_{23}-S_{13}}={\delta_{4}\over\delta_{3}}\leq 1,$

which means that $f(x_{4},y_{4})\leq f(x_{3},y_{3})$. It belongs to the region
A (see Figure 3, left). But the convexity condition requires in particular
$q_{4}$ above the triangle $q_{1}q_{3}q_{2}$. The region A is inside this
triangle, except if $q_{3}$ is above the unit circle of the plane. In this
case, the angle $q_{2}q_{3}q_{1}$ is acute, and the segment $q_{3}q_{1}$
crosses the disk of center $q_{2}$ passing through $q_{3}$ (see Figure 3,
right). To get a simple bound on $r_{34}$, we observe that $q_{4}$ is located
between the circle of center $q_{2}$ passing through $q_{3}$, the segment from
$q_{3}$ to $q_{1}$ and the ${\rm O}y$ axis. Within this domain, due to the
negative slope, the farthest point from $q_{3}=(x_{3},y_{3})$ is the
intersection of the segment with ${\rm O}y$, if the domain is cut by ${\rm
O}y$. If the domain is not cut by ${\rm O}y$, this distance will still be an
upper bound of $r_{34}$. This bound is $r_{13}x_{3}/(x_{3}-1)$. It is maximal
at $q_{3}=q_{2}$ where it is $r_{12}/2$.

Second case. The point $q_{4}$ is in the lower half plane. The domain for the
point $q_{4}$ is restricted by the perpendicular bisector of $(q_{2},q_{3})$,
by the ${\rm O}x$ and ${\rm O}y$ axes and by the circle of center $q_{2}$
passing through $q_{3}$. Due to the perpendicular bisector, $q_{3}$ should be
within the unit disk (see Figure 4). The angle $132$ is obtuse. The
perpendicular bisector crosses the ${\rm O}y$ axis at
$y=(x_{3}^{2}+y_{3}^{2}-1)/(2y_{3})<0$. As the distance $r_{34}$ is increased
when going farther along the perpendicular bisector, the point $q_{4}=(0,y)$
maximizes the distance to $q_{3}$. The distance $r_{34}$ is again maximized if
$x_{3}=0$ and $y_{3}=\sqrt{3}/3$, in which case $x_{4}=0$, $y_{4}=y=-y_{3}$
gives the upper bound $2\sqrt{3}/3$ for the distance $r_{34}$. If
$y_{3}<\sqrt{3}/3$, the constraint by the circle give $y_{4}>-y_{3}$ and a
shorter distance $r_{34}$. If $y_{3}>\sqrt{3}/3$, the constraint by the
perpendicular bisector gives $y_{4}>(y_{3}^{2}-1)/(2y_{3})$ and a shorter
distance $r_{34}$. QED

Figure 4. Proof of 5.3, second case

5.4. Proposition. A planar configuration of 5 points
$(q_{1},q_{2},q_{3},q_{4},q_{5})$ such that the first four points form a
convex quadrilateral with the fifth point inside, and such that there exists
$z>0$ and $(\delta_{1},\delta_{2},\delta_{3},\delta_{4},\delta_{5})$ with
signs $++++-$, with $S_{ij}=z-\delta_{i}\delta_{j}$, is impossible.

Figure 5. Proof of 5.4

Proof. We may renumber the points in such a way that
$\delta_{5}<0<\delta_{4}\leq\delta_{3}\leq\delta_{2}\leq\delta_{1}$. The order
of the distances satisfies

$(r_{15}\hbox{ or }{r_{12}\over 2})\leq r_{25}\leq r_{35}\leq
r_{45}<r_{34}\leq r_{24}\leq(r_{14}\hbox{ or }r_{23})\leq r_{13}\leq r_{12}.$

The inequality $r_{12}/2\leq r_{25}$ comes from the triangular inequality
$r_{12}\leq r_{15}+r_{25}$. Proposition 5.3 applies to the first four points.
Since $r_{12}/2<r_{34}$, the first case is excluded. In the second case, we
should have $x_{5}>0$ and $q_{5}$ should be below the perpendicular bisector
of $q_{2}q_{3}$ (see Figure 5). In particular, $y_{5}<y_{4}$. So, $q_{5}$ is
not in the interior of the convex quadrilateral $q_{1}q_{3}q_{2}q_{4}$. QED

Impossibility of the 5-body flat Dziobek configurations. We assume that there
are 5 noncollinear, non coinciding, positions $q_{1},\dots,q_{5}$ in the
plane, a positive number $z$ and 5 nonzero numbers
$\delta_{1},\dots,\delta_{5}$, such that $S_{ij}=z-\delta_{i}\delta_{j}$,
where $S_{ij}=\|q_{i}-q_{j}\|^{-3}$.

Furthermore, we assume that 5 homogeneous barycentric coordinates
$\Delta_{i}$, $i=1,\dots,5$ of the 5 positions $q_{i}$ are such that
$\Delta_{i}/\delta_{i}>0$.

These two conditions are satisfied for a flat Dziobek configuration with
positive masses $m_{1},\dots,m_{5}$. In this case we have
$\Delta_{i}/\delta_{i}=m_{i}$ and $z=\lambda/M$ where $\lambda$ is the
multiplier of the central configuration and $M=m_{1}+\cdots+m_{5}$. A zero
$\delta_{i}$ (or $\Delta_{i}$) is impossible according to Proposition 4.8. We
will examine the rows without zeros in Table A. We do not need to examine
Table B or other non-generic cases since the rows without zero are the same,
and the impossibilities of Propositions 5.1, 5.2 and 5.4 are easily extended
to the boundary defined by the collinearity of 3 bodies.

Table A1, second row, is impossible, since if we ignore body 4, the remaining
quadrilateral satisfies the hypothesis of Proposition 5.1. The other rows of
this table are similar. In the fourth row, we ignore body 5. In the sixth row,
we ignore body 1 (note that this ignored $+$ sign is equidistant from the
zeros of the same column.) Etc.

Table A2, second row, is impossible, since if we ignore body 5, the
quadrilateral satisfies the hypothesis of Proposition 5.2. The fourth row is
similar, ignoring body 4. The sixth row is impossible according to Proposition
5.1, ignoring body 1. The eighth row is excluded by Proposition 5.4. The tenth
row is excluded by 5.1, ignoring body 3.

Table A3, second row, is impossible according to 5.2, ignoring body 5. The
fourth row is excluded by 5.2 ignoring body 4. The sixth row, by 5.1 ignoring
body 1. The eighth row, by 5.2 ignoring body 3. The tenth row, by 5.2 ignoring
body 2. Theorem 1.2 is proved.

6\. Proof of Theorem 1.1.

Let us begin with the changes of dimension which are impossible whatever the
number $n$ of bodies.

The maximal dimension for an $n$-body configuration is $n-1$. In this
dimension the equation for central configurations $q{\check{Z}}=0$ implies
${\check{Z}}=0$ and $S_{ij}=\lambda/M$ for any $i,j$, $1\leq i<j\leq n$
(compare Saari, 1980). The configuration is a regular simplex of sides
$r_{ij}=(\lambda/M)^{-1/3}$. Clearly, a central configuration of dimension
$\leq n-2$ cannot be a limit of regular simplexes of dimension $n-1$.

The minimal dimension of a configuration without collision is one. As
mentioned in §4.3, Conley proved that a nontrivial vertical degeneracy of a
collinear central configuration is impossible. According to Proposition 1.4, a
limit of central configurations of dimension $d>1$ cannot be of dimension one.
A stronger result is known, which delimitates, independently of the masses, a
neighborhood of the collinear central configurations where no central
configuration of higher dimension can be found: According to Moeckel, (1990),
if a noncollinear central configuration is given, one cannot find a direction
$L$ such that for all $i,j,1\leq i<j\leq n$, the angle
$(q_{i}q_{j},L)\leq\pi/4$.

These results are sufficient to exclude the changes of dimension if $n=4$. If
$n=5$, there only remains to prove that a planar central configuration cannot
be a limit of spatial central configurations. This is proved by Theorem 1.2
and Proposition 1.4.

Acknowledgements. This work has benefited from various discussions in January
2023, at the seminar organized at UFPE, Recife, by Eduardo S.G. Leandro. We
thank Alan A. Santos and Marcelo P. Santos for their useful comments.

Data Availibility Statement. Data sharing was not applicable to this article
as no datasets were generated or analyzed during the current study.

Statements and Declarations. We wish to thank the Brazilian-French Network in
Mathematics, the projects FAPEMIG APQ 03149-18 and CNPq 306568/2021-7 for
giving us opportunities to meet. The authors declare that they have no
conflict of interest concerning the research presented in this work.

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