Volume of a simplex as a multivalued algebraic function of the areas of its two-faces

Abstract

For n greater than or equal to 4, the square of the volume of an n-simplex satisfies a polynomial relation with coefficients depending on the squares of the areas of 2-faces of this simplex. First, we compute the minimal degree of such polynomial relation. Second, we prove that the volume an n-simplex satisfies a monic polynomial relation with coefficients depending on the areas of 2-faces of this simplex if and only if n is even and at least 6, and we study the leading coefficients of polynomial relations satisfied by the volume for other n.