Estimating the number of tetrahedra determined by volume, circumradius and four face areas using Groebner basis

Given any set of six positive parameters, the number of tetrahedra, all having these values as their volume, circumradius and four face areas, is studied. We identify all parameters that determine infinitely many tetrahedra. On the other hand, we classify parameters that determine finitely many tetrahedra and find only four different upper bounds, zero, six, eight, and nine, on the numbers of tetrahedra. In each case, the upper bound is sharp in the complex domain.In this paper, the upper bounds are obtained through checking the dimensions of various quotient algebras of ideals by counting monomials. This is done by computing Groebner bases with block orders. Partitioning the parameter space into several cases, we find either the dimension or an upper bound of it for the quotient algebra in each case. From that, various upper bounds on the number of tetrahedra are obtained. To show the upper bounds are sharp, we pick rational parameters and study the number of tetrahedra through Hermite's root counting method.