System of equations and configurations in the Euclidean space

Abstract

In the 3-dimensional Euclidean space \({\textbf{E}}^3\), fix six pairwise distinct points

$$\begin{aligned} \begin{array}{ccc} A=(a_1,a_2,a_3), & B=(b_1,b_2,b_3), & C=(c_1,c_2,c_3), \\ D=(d_1,d_2,d_3), & E=(e_1,e_2,e_3), & F=(f_1,f_2,f_3) \end{array} \end{aligned}$$

together with two further points \(X^*=(x_1^*,x_2^*,x_3^*)\) and \(Y^*=(y_1^*,y_2^*,y_3^*)\) in \({\textbf{E}}^3\). We show that System \((*)\) consisting of the following six equations in the unknowns \(X=(x_1,x_2,x_3)\) and \(Y=(y_1,y_2,y_3)\)

$$\begin{aligned} \frac{1}{\Vert X-T\Vert ^2} +\frac{1}{\Vert Y-T\Vert ^2}=\frac{1}{\Vert X^*-T\Vert ^2} +\frac{1}{\Vert Y^*-T\Vert ^2}, \quad T\in \{A,B,C,D,E,F\} \end{aligned}$$

(1)

has only finitely many solutions provided that both of the following two conditions are satisfied:

1. (i)

   No four of the fixed points A, B, C, D, E, F are coplanar;

2. (ii)

   No four of the six spheres of center T and radius \(1/\sqrt{k_T}\) with

   $$\begin{aligned} k_T=\frac{1}{\Vert X^*-T\Vert ^2} +\frac{1}{\Vert Y^*-T\Vert ^2} \end{aligned}$$

   (2)

   share a common point in \({\textbf{E}}^3\).

Furthermore, we exhibit configurations \(ABCDEFX^*Y^*\), showing that (i) is also necessary. This result is an improvement on [2, Theorem 1] where the finiteness of solutions of System \((*)\) was only ensured for sufficiently generic choices of the points \(A,B,\ldots ,F,X^*,Y^*.\) The extended System \((**)\) associated to System \((*)\) consists of seven equations (1) where \(T\in \{A,B,C,D,E,E,F,G\}\) with a further point \(G=(g_1,g_2,g_3)\in {\textbf{E}}^3\). We show that if (i) and (ii) hold for \(T\in \{A,B,C,D,E,F\}\) and the associated extended System \((**)\) has some solutions other than \((X^*,Y^*)\) and \((Y^*,X^*)\), then G lies on a real affine surface only depending on \(\{A,B,\ldots ,F\}\). This result proves [2, Conjecture 1]. Motivation for studying the above problems comes from applications to genetics; see [2].