System of equations and configurations in the Euclidean space

Abstract

In the 3-dimensional Euclidean space E3, fix six pairwise distinct points (Formula presented.) together with two further points X∗=(x1∗,x2∗,x3∗) and Y∗=(y1∗,y2∗,y3∗) in E3. We show that System (∗) consisting of the following six equations in the unknowns X=(x1,x2,x3) and Y=(y1,y2,y3) (Formula presented.) has only finitely many solutions provided that both of the following two conditions are satisfied: No four of the fixed points A, B, C, D, E, F are coplanar; No four of the six spheres of center T and radius 1/kT with (Formula presented.) share a common point in E3. 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,…,F,X∗,Y∗. The extended System (∗∗) associated to System (∗) consists of seven equations (1) where T∈{A,B,C,D,E,E,F,G} with a further point G=(g1,g2,g3)∈E3. We show that if (i) and (ii) hold for T∈{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,…,F}. This result proves [2, Conjecture 1]. Motivation for studying the above problems comes from applications to genetics; see [2].