Configurations in the Euclidean plane associated to a system of equations

TL;DR

This paper studies a system of equations in the Euclidean plane involving four fixed points and associated parameters, establishing conditions for finitely many solutions and providing a counterexample with many solutions.

Contribution

It characterizes when the system has finitely many solutions based on geometric conditions and presents a configuration with many solutions, extending a space-based problem to the plane.

Findings

System has finitely many solutions under certain geometric conditions.

Counterexample shows many solutions can occur despite conditions.

Results connect planar configurations to applications in genetics.

Abstract

In the Euclidean plane ${\bf{E}}^2$, fix four pairwise distinct points \begin{equation*} \label{eqA} \begin{array}{ccc} A=(a_1,a_2),\ B=(b_1,b_2),\ C=(c_1,c_2),\ D=(d_1,d_2), \end{array} \end{equation*} together with four non-zero real numbers $k_A,k_B,k_C,k_D$. We show that System (*) consisting of the following four equations in the unknowns $X=(x_1,x_2)$ and $Y=(y_1,y_2)$ \begin{equation*} \label{egy} \frac{1}{\|X-T\|^2} +\frac{1}{\|Y-T\|^2}=k_T, \quad T\in\{A,B,C,D\} \end{equation*} has finitely many solutions $(X,Y)$ (counting also those with complex coordinates) provided that both of the following two conditions are satisfied: ($i$) no three of the fixed points $A,B,C,D$ are coplanar; ($ii$) no three of the four circles of center $T$ and radius $1/\sqrt{k_T}$ with share a common point in ${\bf{E}}^2$. Furthermore, we exhibit a configuration $ABCD$ showing that System (*) satisfying $(i)$ and $(ii)$ may have many real solutions $(X,Y)$. This result is the planar version of an analog problem in the Euclidean space arising from applications to genetics, investigated in the recent papers \cite{cif} and \cite{ak2024}.