We prove, in the case of hyperbolic 3-space, a couple of conjectures raised by J. J. Seidel in On the volume of a hyperbolic simplex, Stud. Sci. Math. Hung. 21 (1986), 243–249. Seidel’s first conjecture states that the volume of an ideal tetrahedron in hyperbolic 3-space is determined by (the permanent and the determinant of) a certain Gram matrix G of its vertices; Seidel’s fourth conjecture claims that the mentioned volume is a monotonic function of both the permanent and the determinant of G. A stronger form of the first conjecture is obtained: the permanent and the determinant of G are actually coordinates on the space of all ideal tetrahedra.

中文翻译：

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双曲面三空间中的SEIDEL猜想

在双曲3空间的情况下，我们证明了JJ Seidel在关于双曲单纯形的体积Stud中提出的几个猜想。科学 数学。挂了 21（1986），243–249。Seidel的第一个猜想指出，双曲3空间中理想四面体的体积取决于其顶点的某个Gram矩阵G（的永久值和行列式）。Seidel的第四个猜想声称，提到的体积是G的永久性和行列式的单调函数。获得第一个猜想的更强形式：G的永久性和行列式实际上是所有理想四面体的空间上的坐标。