The problem consists in bounding the number of tetrahedra that share the same volume V, circumradius R, and face areas $A_1,A_2,A_3,A_4$. A symbolic computation approach is used to prove that the upper bound of the number of tetrahedra is eight. First, the problem is reduced to a counting root problem using the Cayley-Menger determinant formula for the volume of a simplex and some metric equations of the tetrahedron to construct a system of algebraic equations. Then, an investigation of its real roots is done by analyzing the discriminant sequence and extensive numerical computations.

中文翻译：

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通过符号和数值计算共享相同度量不变量的四面体的数量

问题在于限制共享相同体积V、外接半径R和面区域的四面体的数量${\mathrm{一种}}_1,{\mathrm{一种}}_2,{\mathrm{一种}}_3,{\mathrm{一种}}_4$. 用符号计算的方法证明四面体数目的上界为8。首先，使用单纯形体积的 Cayley-Menger 行列式公式和四面体的一些度量方程将问题简化为计数根问题，以构建代数方程组。然后，通过分析判别序列和广泛的数值计算来研究其真正的根源。