System of equations and configurations in the Euclidean space

In the 3-dimensional Euclidean space \({\textbf{E}}^3\) ${\mathbf{E}}^3$ , fix six pairwise distinct points \(\begin{aligned} \begin{array}{ccc} A=(a_1,a_2,a_3), & B=(b_1,b_2,b_3), & C=(c_1,c_2,c_3), \\ D=(d_1,d_2,d_3), & E=(e_1,e_2,e_3), & F=(f_1,f_2,f_3) \end{array} \end{aligned}\) $$\begin{array}{c}\hfill \begin{array}{ccc}A=(a_1,a_2,a_3),& B=(b_1,b_2,b_3),& C=(c_1,c_2,c_3),\\ D=(d_1,d_2,d_3),& E=(e_1,e_2,e_3),& F=(f_1,f_2,f_3)\end{array}\end{array}$$ together with two further points \(X^*=(x_1^*,x_2^*,x_3^*)\) $X^{\ast }=(x_1^{\ast },x_2^{\ast },x_3^{\ast })$ and \(Y^*=(y_1^*,y_2^*,y_3^*)\) $Y^{\ast }=(y_1^{\ast },y_2^{\ast },y_3^{\ast })$ in \({\textbf{E}}^3\) ${\mathbf{E}}^3$ . We show that System \((*)\) $(\ast )$ consisting of the following six equations in the unknowns \(X=(x_1,x_2,x_3)\) $X=(x_1,x_2,x_3)$ and \(Y=(y_1,y_2,y_3)\) $Y=(y_1,y_2,y_3)$ 1 \(\begin{aligned} \frac{1}{\Vert X-T\Vert ^2} +\frac{1}{\Vert Y-T\Vert ^2}=\frac{1}{\Vert X^*-T\Vert ^2} +\frac{1}{\Vert Y^*-T\Vert ^2}, \quad T\in \{A,B,C,D,E,F\} \end{aligned}\) $$\begin{array}{c}\hfill \frac{1}{{\Vert X-T\Vert }^2}+\frac{1}{{\Vert Y-T\Vert }^2}=\frac{1}{\Vert X^{\ast }{-T\Vert }^2}+\frac{1}{\Vert Y^{\ast }{-T\Vert }^2},T\in \{A,B,C,D,E,F\}\end{array}$$ has only finitely many solutions provided that both of the following two conditions are satisfied: (i)

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

(ii)

No four of the six spheres of center T and radius \(1/\sqrt{k_T}\) $1/\sqrt{k_T}$ with 2 \(\begin{aligned} k_T=\frac{1}{\Vert X^*-T\Vert ^2} +\frac{1}{\Vert Y^*-T\Vert ^2} \end{aligned}\) $$\begin{array}{c}\hfill k_T=\frac{1}{\Vert X^{\ast }{-T\Vert }^2}+\frac{1}{\Vert Y^{\ast }{-T\Vert }^2}\end{array}$$ share a common point in \({\textbf{E}}^3\) ${\mathbf{E}}^3$ .

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