Given a tetrahedron, we can compute its volume, its circumradius (radius of the sphere going through the four vertices), and the surface areas of the four faces. Mazur [1] asked, effectively, if this is injective; that is, given these six quantities, is there only one corresponding tetrahedron? Lisoněk [2] rapidly produced a counter-example, a set of values corresponding to two tetrahedra, and asked whether the mapping from tetrahedra to these quantities was at worst finite-to-one. Yang and Zeng [3] again disproved this, but their example had three faces with equal areas. There have been other results too technical to mention here; see the paper for details.

In this paper, for all possible combinations of equal/unequal face areas, the authors first calculate the possible number of complex solutions by an ingenious application of the methods of parametric Groebner bases. These bounds are precise, in that they have examples. They then bound the number of real solutions by Hermite’s root-counting methods. An open question is whether the real solutions’ bounds are precise.

The computations are in Mathematica, and the author has published the worksheets used.